Physics Book - User contributions [en]

Solution for Simple Harmonic Oscillator

2022-04-25T03:46:25Z

Lim, Xuen Zhen: Added hyperlink to Morse potential

<b>Claimed by Lim, Xuen Zhen (Spring 2022)</b>

==Introduction==
One of the many situations which can be analyzed with the Schrodinger’s Equation is the one-dimensional simple harmonic oscillator. This situation draws an analogy between a quantum-mechanical particle’s oscillating movement and the movement of a common [//www.physicsbook.gatech.edu/Spring_Potential_Energy classical oscillator: a spring]. Like its classical spring counterpart described under Hooke's Law, a quantum harmonic oscillator has the force function <math> F = -k x </math> and the associated potential function <math> V = \frac{1}{2} k x^2 </math>, with <math> k </math> being the force constant (spring constant in classical case). Despite the simplicity of a harmonic oscillator's smooth parabolic potential, it acts as an important foundation to solving more complicated quantum systems due to it being one of the few quantum-mechanical systems with an exact, known analytical solution.

==Mathematical Setup==
We may use the time-independent Schrodinger's equation to represent the state of a quantum particle in the harmonic potential by substituting the potential <math> V </math> with <math> \frac{1}{2} k x^2 </math>.
<br>
<math>\frac{-\hbar^2}{2m} \frac{d^2 \Psi}{d x^2} + \frac{1}{2} k x^2 \Psi = E \Psi</math>
<br>
The solution to this equation are the wave function <math> \Psi </math> and the energy function <math> E </math> that satisfies the above conditions.

==Deriving the Solution==
===Finding Ground State Wave Function===
For quantum harmonic oscillator, there are no boundaries between different regions, so one condition for the wave function is it must approach 0 as <math> x→+∞ </math> and <math> x→-∞ </math>. A simple general wave function that satisfies this requirement is <math> \Psi (x) = A e^{-ax^2} </math>. We begin the derivation with finding the second order differential of the general wave equation.
<br><br>
<math> \frac{d \Psi}{d x} = -2 a x (A e^{-ax^2}) = -2 a x \Psi </math>
<br><br>
<math> \frac{d^2 \Psi}{d x^2} = -2 a (A e^{-ax^2})-2 a x (-2 a x)(A e^{-ax^2}) = (-2 a+4a^2x^2)A e^{-ax^2} = (-2 a+4a^2x^2)\Psi </math>
<br>

Substituting the differential equation into the time-independent Schrodinger equation produces

<math> \frac{-\hbar^2}{2m} (-2 a+4a^2x^2)\Psi + \frac{1}{2} k x^2 \Psi = E \Psi </math>
<br>
<math> \frac{\hbar^2 a}{m}\Psi-\frac{2 a^2 \hbar^2 x^2}{m}\Psi + \frac{1}{2} k x^2 \Psi = E \Psi </math>
<br>
<math> \frac{\hbar^2 a}{m}-\frac{2 a^2 \hbar^2}{m}x^2 + \frac{1}{2} k x^2 = E </math>
<br>

One common misconception to be aware of is that this is not an equation to be solved for <math> x </math>. <math> x </math> is posing as a free variable here, and the solution is one that makes the Schrodinger equation true for any value of <math> x </math>. To allow this equation to be consistent for any <math> x </math>, the coefficients to <math> x^2 </math> must cancel out, leaving the remaining constants to be equal to each other.

<span style="color:blue"><math> \frac{\hbar^2 a}{m}</math></span><math> - </math><span style="color:red"><math>\frac{2 a^2 \hbar^2}{m}x^2 + \frac{1}{2} k x^2</math></span><math> = </math><span style="color:blue"><math> E </math></span>

<span style="color:red"><math>-\frac{2 a^2 \hbar^2}{m} + \frac{1}{2} k = 0 </math></span> and <span style="color:blue"><math> \frac{\hbar^2 a}{m} = E </math></span>

Remember, <math>k</math>, <math>\hbar</math>, and <math>m</math> are variables which we already know the values, <math>a</math> and <math>E</math> are the only unknowns we are trying to find. We will first solve for <math>a</math> in the equation on the left to help find the wave function.

<math>-\frac{2 a^2 \hbar^2}{m} + \frac{1}{2} k = 0 </math>
<br>
<math>\frac{2 a^2 \hbar^2}{m} = \frac{1}{2} k </math>
<br>
<math> a^2 = \frac{k m}{4 \hbar^2} </math>
<br>
<math> a = \frac{\sqrt{k m}}{2 \hbar} </math>

Substituting <math>a</math>, the wave function now looks like

<math> \Psi (x) = A e^{-(\sqrt{k m}/2\hbar) x^2} </math>

We know the total probability of finding the particle from the entire space (<math>-∞</math> to <math>+∞</math>) is 1. Using this condition, the normalization constant <math>A</math> can be calculated. (Note that since there are no complex component in this equation, the complex conjugate is simply the equation itself, and |\Psi|^2 is the square of the wave function.)

<math> 1 = \int_{-∞}^{+∞} |\Psi|^2 dx = \int_{-∞}^{+∞} A^2 e^{-(\sqrt{k m}/\hbar) x^2} dx </math>

Using the [//en.wikipedia.org/wiki/Gaussian_integral Gaussian Integral], <math> 1 = \int_{-∞}^{+∞} e^{-ax^2} dx = \sqrt{\frac{\pi}{a}} </math>, the integral is turned into

<math>1 = A^2 \sqrt{\frac{\hbar \pi}{\sqrt{k m}}} </math>
<br>
<math >A^2 = \sqrt{\frac{\sqrt{k m}}{\hbar \pi}} = \sqrt{\frac{\sqrt{(m w_{0}^2)m}}{\hbar \pi}} = \sqrt{\frac{m w_{0}}{\hbar \pi}}</math>
<br>
<math >A = (\frac{m w_{0}}{\hbar \pi})^{1/4} </math>

The complete ground state wave function is

<math >\Psi (x) = (\frac{m w_{0}}{\hbar \pi})^{1/4} e^{-(\sqrt{k m}/2\hbar) x^2}</math>

Note that the <math>A</math> and <math >\Psi (x)</math> found here is only valid for ground state of harmonic oscillator. The general solution to the wave function is of the form <math> \Psi_{n} (x) = A f_{n} (x) e^{-ax^2} </math>, with <math>f_{n} (x)</math> being a polynomial in which the highest power of <math>x</math> is <math>x^n</math>.

One interesting trait of this wave function is, just like the finite potential well, the probability density can pentrate into the forbidden region beyond the classical turning points (the boundary of the region at which the potential energy becomes higher than the energy).

===Finding energy function===

With <math>a</math> derived above, we can easily substitute it into the second equation and find the energy <math>E</math>.

<math> E = (\frac{h^2}{m}) (\frac{\sqrt{k m}}{2 \hbar}) </math>
<br>
<math> E = \frac{1}{2} \hbar \sqrt{\frac{k}{m}} </math>

We can write the energy in terms of the classical frequency <math>ω</math><sub>0</sub>=<math>\sqrt{\frac{k}{m}}</math>.

<math> E = \frac{1}{2} \hbar ω</math><sub>0</sub>

By quantizing the energy, we then create the energy function for a simple harmonic oscillator.

<math> E </math><sub>n</sub> <math>= (\frac{1}{2}+n) \hbar ω</math><sub>0</sub>   <math>n=0,1,2,...</math>

Note that, contrary to one-dimensional energy function of a potential well, the energy function here are uniformly spaced with the interval <math>∆E=\hbar w_{0}</math>.

==Applications==
While a one-dimensional quantum harmonic oscillator with smooth potential is virtually inexistant in the nature, there exists some systems that behave akin to one, such as a vibrating diatomic molecule like HCl. A complex potential with arbitrary smooth curve can also usually be approximated as a harmonic potential near its stable equilibrium point, also known as the minimum.

===Morse Potential===
[//en.wikipedia.org/wiki/Morse_potential Morse potential], named after physicist Philip M. Morse, is a interatomic interaction model for the potential energy of a diatomic molecule like the quantum harmonic oscillator. However, it is a better approximation for the vibrational structure of a molecule than quantum harmonic oscillator as it includes the effects of bond breaking as <math>x→+∞</math>, rather than letting the potential increases exponentially to infinity as seen in a harmonic oscillator. The potential function for Morse potential is:

<math>V(x) = A (1-e^{-a(x-x_{0})})^2</math>
<ul>
<li><math>A</math> is the depth of the well (defined relative to the dissociated atoms)</li>
<li><math>x</math> is the distance between atoms</li>
<li><math>x_{0}</math> is the equilibrium bond distance (the distance where the potential is at minimum)</li>
<li><math>a</math> controls the width of the potential (smaller a leads to wider well)</li>
</ul>

The Morse potential function, by itself, is a complicated function that would make the Schrodinger equation difficult to solve. However, when overlaying the Morse potential curve on top of the harmonic oscillator potential curve, we can observe that they are similar near the point of equilibrium <math>x_{0}</math>. Due to this, we may relate the Morse potential to harmonic oscillator when working at low energy levels to get a rather accurate approximation of the wave function and energy function conveniently. To achieve this, we need a simplified, approximated function of Morse potential around the point of equilibrium, which can be done by performing [//en.wikipedia.org/wiki/Taylor_series Taylor Series] expansion about <math>x_{0}</math>.

<math>V(x_{0})+\frac{d}{d x} V(x_{0}) x + \frac{1}{2}\frac{d^2}{d x^2} V(x_{0}) x^2 + ... </math>
<br>
<math>= 0 + 0•x + \frac{1}{2} 2 a^2 A x^2 + ...</math>
<br>
<math>≈ \frac{1}{2} 2 a^2 A x^2</math>
<br>

We may, of course, keep expanding the Taylor series. But the reason we stopped here is due to its resemblance to the potential function of harmonic oscillator, which would be very useful for simplifying the derivation of solution to Schrodinger equation as we can re-use the solutions previously derived for harmonic oscillator and quickly arrive at an approximate solution. We can relate the two potential functions as seen below.

<math>V_{morse} (x)≈ \frac{1}{2}</math> <span style="color:red"><math>2 a^2 A</math></span> <math>x^2</math>        
<math>V_{harmonic} (x)= \frac{1}{2}</math> <span style="color:red"><math>k</math></span> <math>x^2</math>
<br>

which shows that we may approximate the Schrodinger equation by treating <math>k</math> as <math>2 a^2 A</math> since these are both constants. We can find the solution for ground state wave function and energy level function by simply replacing the <math>k</math> in the solutions for harmonic oscillator with <math>2 a^2 A</math>. Note that the <math>m</math> used for Morse potential related derivations must be the reduced mass constant between the two particles in the system <math>m=\frac{m_{1} m_{2}}{m_{1}+m_{2}}</math> as we are working with a two-particle system rather than a singular one.

Finding the approximate energy level function of Morse potential at low energy levels:

<math> E_{n} = (n+\frac{1}{2}) \hbar \sqrt{\frac{k}{m}} </math>
<br>
<math> E_{n} = (n+\frac{1}{2}) \hbar a \sqrt{\frac{2 A}{m}} </math>

Note that the <math>m</math> used here must be the reduced mass constant between the two particles in the system <math>m=\frac{m_{1} m_{2}}{m_{1}+m_{2}}</math>.

Applying similar logic to approximate the ground state wave function of Morse potential:

<math >\Psi (x) = (\frac{m w_{0}}{\hbar \pi})^{1/4} e^{-(\sqrt{k m}/2\hbar) x^2}</math>
<br>
<math >\Psi (x) = (\frac{m \sqrt{k/m}}{\hbar \pi})^{1/4} e^{-(\sqrt{k m}/2\hbar) x^2}</math>
<br>
<math >\Psi (x) = (\frac{m a\sqrt{2 A/m}}{\hbar \pi})^{1/4} e^{-(a\sqrt{2 A m}/2\hbar) x^2}</math>
<br>

Solution for Simple Harmonic Oscillator

2022-04-25T03:41:54Z

Lim, Xuen Zhen: Up the hierarchy of every title and subtitle by one.

<b>Claimed by Lim, Xuen Zhen (Spring 2022)</b>

==Introduction==
One of the many situations which can be analyzed with the Schrodinger’s Equation is the one-dimensional simple harmonic oscillator. This situation draws an analogy between a quantum-mechanical particle’s oscillating movement and the movement of a common [//www.physicsbook.gatech.edu/Spring_Potential_Energy classical oscillator: a spring]. Like its classical spring counterpart described under Hooke's Law, a quantum harmonic oscillator has the force function <math> F = -k x </math> and the associated potential function <math> V = \frac{1}{2} k x^2 </math>, with <math> k </math> being the force constant (spring constant in classical case). Despite the simplicity of a harmonic oscillator's smooth parabolic potential, it acts as an important foundation to solving more complicated quantum systems due to it being one of the few quantum-mechanical systems with an exact, known analytical solution.

==Mathematical Setup==
We may use the time-independent Schrodinger's equation to represent the state of a quantum particle in the harmonic potential by substituting the potential <math> V </math> with <math> \frac{1}{2} k x^2 </math>.
<br>
<math>\frac{-\hbar^2}{2m} \frac{d^2 \Psi}{d x^2} + \frac{1}{2} k x^2 \Psi = E \Psi</math>
<br>
The solution to this equation are the wave function <math> \Psi </math> and the energy function <math> E </math> that satisfies the above conditions.

==Deriving the Solution==
===Finding Ground State Wave Function===
For quantum harmonic oscillator, there are no boundaries between different regions, so one condition for the wave function is it must approach 0 as <math> x→+∞ </math> and <math> x→-∞ </math>. A simple general wave function that satisfies this requirement is <math> \Psi (x) = A e^{-ax^2} </math>. We begin the derivation with finding the second order differential of the general wave equation.
<br><br>
<math> \frac{d \Psi}{d x} = -2 a x (A e^{-ax^2}) = -2 a x \Psi </math>
<br><br>
<math> \frac{d^2 \Psi}{d x^2} = -2 a (A e^{-ax^2})-2 a x (-2 a x)(A e^{-ax^2}) = (-2 a+4a^2x^2)A e^{-ax^2} = (-2 a+4a^2x^2)\Psi </math>
<br>

Substituting the differential equation into the time-independent Schrodinger equation produces

<math> \frac{-\hbar^2}{2m} (-2 a+4a^2x^2)\Psi + \frac{1}{2} k x^2 \Psi = E \Psi </math>
<br>
<math> \frac{\hbar^2 a}{m}\Psi-\frac{2 a^2 \hbar^2 x^2}{m}\Psi + \frac{1}{2} k x^2 \Psi = E \Psi </math>
<br>
<math> \frac{\hbar^2 a}{m}-\frac{2 a^2 \hbar^2}{m}x^2 + \frac{1}{2} k x^2 = E </math>
<br>

One common misconception to be aware of is that this is not an equation to be solved for <math> x </math>. <math> x </math> is posing as a free variable here, and the solution is one that makes the Schrodinger equation true for any value of <math> x </math>. To allow this equation to be consistent for any <math> x </math>, the coefficients to <math> x^2 </math> must cancel out, leaving the remaining constants to be equal to each other.

<span style="color:blue"><math> \frac{\hbar^2 a}{m}</math></span><math> - </math><span style="color:red"><math>\frac{2 a^2 \hbar^2}{m}x^2 + \frac{1}{2} k x^2</math></span><math> = </math><span style="color:blue"><math> E </math></span>

<span style="color:red"><math>-\frac{2 a^2 \hbar^2}{m} + \frac{1}{2} k = 0 </math></span> and <span style="color:blue"><math> \frac{\hbar^2 a}{m} = E </math></span>

Remember, <math>k</math>, <math>\hbar</math>, and <math>m</math> are variables which we already know the values, <math>a</math> and <math>E</math> are the only unknowns we are trying to find. We will first solve for <math>a</math> in the equation on the left to help find the wave function.

<math>-\frac{2 a^2 \hbar^2}{m} + \frac{1}{2} k = 0 </math>
<br>
<math>\frac{2 a^2 \hbar^2}{m} = \frac{1}{2} k </math>
<br>
<math> a^2 = \frac{k m}{4 \hbar^2} </math>
<br>
<math> a = \frac{\sqrt{k m}}{2 \hbar} </math>

Substituting <math>a</math>, the wave function now looks like

<math> \Psi (x) = A e^{-(\sqrt{k m}/2\hbar) x^2} </math>

We know the total probability of finding the particle from the entire space (<math>-∞</math> to <math>+∞</math>) is 1. Using this condition, the normalization constant <math>A</math> can be calculated. (Note that since there are no complex component in this equation, the complex conjugate is simply the equation itself, and |\Psi|^2 is the square of the wave function.)

<math> 1 = \int_{-∞}^{+∞} |\Psi|^2 dx = \int_{-∞}^{+∞} A^2 e^{-(\sqrt{k m}/\hbar) x^2} dx </math>

Using the [//en.wikipedia.org/wiki/Gaussian_integral Gaussian Integral], <math> 1 = \int_{-∞}^{+∞} e^{-ax^2} dx = \sqrt{\frac{\pi}{a}} </math>, the integral is turned into

<math>1 = A^2 \sqrt{\frac{\hbar \pi}{\sqrt{k m}}} </math>
<br>
<math >A^2 = \sqrt{\frac{\sqrt{k m}}{\hbar \pi}} = \sqrt{\frac{\sqrt{(m w_{0}^2)m}}{\hbar \pi}} = \sqrt{\frac{m w_{0}}{\hbar \pi}}</math>
<br>
<math >A = (\frac{m w_{0}}{\hbar \pi})^{1/4} </math>

The complete ground state wave function is

<math >\Psi (x) = (\frac{m w_{0}}{\hbar \pi})^{1/4} e^{-(\sqrt{k m}/2\hbar) x^2}</math>

Note that the <math>A</math> and <math >\Psi (x)</math> found here is only valid for ground state of harmonic oscillator. The general solution to the wave function is of the form <math> \Psi_{n} (x) = A f_{n} (x) e^{-ax^2} </math>, with <math>f_{n} (x)</math> being a polynomial in which the highest power of <math>x</math> is <math>x^n</math>.

One interesting trait of this wave function is, just like the finite potential well, the probability density can pentrate into the forbidden region beyond the classical turning points (the boundary of the region at which the potential energy becomes higher than the energy).

===Finding energy function===

With <math>a</math> derived above, we can easily substitute it into the second equation and find the energy <math>E</math>.

<math> E = (\frac{h^2}{m}) (\frac{\sqrt{k m}}{2 \hbar}) </math>
<br>
<math> E = \frac{1}{2} \hbar \sqrt{\frac{k}{m}} </math>

We can write the energy in terms of the classical frequency <math>ω</math><sub>0</sub>=<math>\sqrt{\frac{k}{m}}</math>.

<math> E = \frac{1}{2} \hbar ω</math><sub>0</sub>

By quantizing the energy, we then create the energy function for a simple harmonic oscillator.

<math> E </math><sub>n</sub> <math>= (\frac{1}{2}+n) \hbar ω</math><sub>0</sub>   <math>n=0,1,2,...</math>

Note that, contrary to one-dimensional energy function of a potential well, the energy function here are uniformly spaced with the interval <math>∆E=\hbar w_{0}</math>.

==Applications==
While a one-dimensional quantum harmonic oscillator with smooth potential is virtually inexistant in the nature, there exists some systems that behave akin to one, such as a vibrating diatomic molecule like HCl. A complex potential with arbitrary smooth curve can also usually be approximated as a harmonic potential near its stable equilibrium point, also known as the minimum.

===Morse Potential===
Morse potential, named after physicist Philip M. Morse, is a interatomic interaction model for the potential energy of a diatomic molecule like the quantum harmonic oscillator. However, it is a better approximation for the vibrational structure of a molecule than quantum harmonic oscillator as it includes the effects of bond breaking as <math>x→+∞</math>, rather than letting the potential increases exponentially to infinity as seen in a harmonic oscillator. The potential function for Morse potential is:

<math>V(x) = A (1-e^{-a(x-x_{0})})^2</math>
<ul>
<li><math>A</math> is the depth of the well (defined relative to the dissociated atoms)</li>
<li><math>x</math> is the distance between atoms</li>
<li><math>x_{0}</math> is the equilibrium bond distance (the distance where the potential is at minimum)</li>
<li><math>a</math> controls the width of the potential (smaller a leads to wider well)</li>
</ul>

The Morse potential function, by itself, is a complicated function that would make the Schrodinger equation difficult to solve. However, when overlaying the Morse potential curve on top of the harmonic oscillator potential curve, we can observe that they are similar near the point of equilibrium <math>x_{0}</math>. Due to this, we may relate the Morse potential to harmonic oscillator when working at low energy levels to get a rather accurate approximation of the wave function and energy function conveniently. To achieve this, we need a simplified, approximated function of Morse potential around the point of equilibrium, which can be done by performing [//en.wikipedia.org/wiki/Taylor_series Taylor Series] expansion about <math>x_{0}</math>.

<math>V(x_{0})+\frac{d}{d x} V(x_{0}) x + \frac{1}{2}\frac{d^2}{d x^2} V(x_{0}) x^2 + ... </math>
<br>
<math>= 0 + 0•x + \frac{1}{2} 2 a^2 A x^2 + ...</math>
<br>
<math>≈ \frac{1}{2} 2 a^2 A x^2</math>
<br>

We may, of course, keep expanding the Taylor series. But the reason we stopped here is due to its resemblance to the potential function of harmonic oscillator, which would be very useful for simplifying the derivation of solution to Schrodinger equation as we can re-use the solutions previously derived for harmonic oscillator and quickly arrive at an approximate solution. We can relate the two potential functions as seen below.

<math>V_{morse} (x)≈ \frac{1}{2}</math> <span style="color:red"><math>2 a^2 A</math></span> <math>x^2</math>        
<math>V_{harmonic} (x)= \frac{1}{2}</math> <span style="color:red"><math>k</math></span> <math>x^2</math>
<br>

which shows that we may approximate the Schrodinger equation by treating <math>k</math> as <math>2 a^2 A</math> since these are both constants. We can find the solution for ground state wave function and energy level function by simply replacing the <math>k</math> in the solutions for harmonic oscillator with <math>2 a^2 A</math>. Note that the <math>m</math> used for Morse potential related derivations must be the reduced mass constant between the two particles in the system <math>m=\frac{m_{1} m_{2}}{m_{1}+m_{2}}</math> as we are working with a two-particle system rather than a singular one.

Finding the approximate energy level function of Morse potential at low energy levels:

<math> E_{n} = (n+\frac{1}{2}) \hbar \sqrt{\frac{k}{m}} </math>
<br>
<math> E_{n} = (n+\frac{1}{2}) \hbar a \sqrt{\frac{2 A}{m}} </math>

Note that the <math>m</math> used here must be the reduced mass constant between the two particles in the system <math>m=\frac{m_{1} m_{2}}{m_{1}+m_{2}}</math>.

Applying similar logic to approximate the ground state wave function of Morse potential:

<math >\Psi (x) = (\frac{m w_{0}}{\hbar \pi})^{1/4} e^{-(\sqrt{k m}/2\hbar) x^2}</math>
<br>
<math >\Psi (x) = (\frac{m \sqrt{k/m}}{\hbar \pi})^{1/4} e^{-(\sqrt{k m}/2\hbar) x^2}</math>
<br>
<math >\Psi (x) = (\frac{m a\sqrt{2 A/m}}{\hbar \pi})^{1/4} e^{-(a\sqrt{2 A m}/2\hbar) x^2}</math>
<br>

Solution for Simple Harmonic Oscillator

2022-04-25T03:40:31Z

Lim, Xuen Zhen: Added even more content to morse potential.

<b>Claimed by Lim, Xuen Zhen (Spring 2022)</b>

===Introduction===
One of the many situations which can be analyzed with the Schrodinger’s Equation is the one-dimensional simple harmonic oscillator. This situation draws an analogy between a quantum-mechanical particle’s oscillating movement and the movement of a common [//www.physicsbook.gatech.edu/Spring_Potential_Energy classical oscillator: a spring]. Like its classical spring counterpart described under Hooke's Law, a quantum harmonic oscillator has the force function <math> F = -k x </math> and the associated potential function <math> V = \frac{1}{2} k x^2 </math>, with <math> k </math> being the force constant (spring constant in classical case). Despite the simplicity of a harmonic oscillator's smooth parabolic potential, it acts as an important foundation to solving more complicated quantum systems due to it being one of the few quantum-mechanical systems with an exact, known analytical solution.

===Mathematical Setup===
We may use the time-independent Schrodinger's equation to represent the state of a quantum particle in the harmonic potential by substituting the potential <math> V </math> with <math> \frac{1}{2} k x^2 </math>.
<br>
<math>\frac{-\hbar^2}{2m} \frac{d^2 \Psi}{d x^2} + \frac{1}{2} k x^2 \Psi = E \Psi</math>
<br>
The solution to this equation are the wave function <math> \Psi </math> and the energy function <math> E </math> that satisfies the above conditions.

===Deriving the Solution===
====Finding Ground State Wave Function====
For quantum harmonic oscillator, there are no boundaries between different regions, so one condition for the wave function is it must approach 0 as <math> x→+∞ </math> and <math> x→-∞ </math>. A simple general wave function that satisfies this requirement is <math> \Psi (x) = A e^{-ax^2} </math>. We begin the derivation with finding the second order differential of the general wave equation.
<br><br>
<math> \frac{d \Psi}{d x} = -2 a x (A e^{-ax^2}) = -2 a x \Psi </math>
<br><br>
<math> \frac{d^2 \Psi}{d x^2} = -2 a (A e^{-ax^2})-2 a x (-2 a x)(A e^{-ax^2}) = (-2 a+4a^2x^2)A e^{-ax^2} = (-2 a+4a^2x^2)\Psi </math>
<br>

Substituting the differential equation into the time-independent Schrodinger equation produces

<math> \frac{-\hbar^2}{2m} (-2 a+4a^2x^2)\Psi + \frac{1}{2} k x^2 \Psi = E \Psi </math>
<br>
<math> \frac{\hbar^2 a}{m}\Psi-\frac{2 a^2 \hbar^2 x^2}{m}\Psi + \frac{1}{2} k x^2 \Psi = E \Psi </math>
<br>
<math> \frac{\hbar^2 a}{m}-\frac{2 a^2 \hbar^2}{m}x^2 + \frac{1}{2} k x^2 = E </math>
<br>

One common misconception to be aware of is that this is not an equation to be solved for <math> x </math>. <math> x </math> is posing as a free variable here, and the solution is one that makes the Schrodinger equation true for any value of <math> x </math>. To allow this equation to be consistent for any <math> x </math>, the coefficients to <math> x^2 </math> must cancel out, leaving the remaining constants to be equal to each other.

<span style="color:blue"><math> \frac{\hbar^2 a}{m}</math></span><math> - </math><span style="color:red"><math>\frac{2 a^2 \hbar^2}{m}x^2 + \frac{1}{2} k x^2</math></span><math> = </math><span style="color:blue"><math> E </math></span>

<span style="color:red"><math>-\frac{2 a^2 \hbar^2}{m} + \frac{1}{2} k = 0 </math></span> and <span style="color:blue"><math> \frac{\hbar^2 a}{m} = E </math></span>

Remember, <math>k</math>, <math>\hbar</math>, and <math>m</math> are variables which we already know the values, <math>a</math> and <math>E</math> are the only unknowns we are trying to find. We will first solve for <math>a</math> in the equation on the left to help find the wave function.

<math>-\frac{2 a^2 \hbar^2}{m} + \frac{1}{2} k = 0 </math>
<br>
<math>\frac{2 a^2 \hbar^2}{m} = \frac{1}{2} k </math>
<br>
<math> a^2 = \frac{k m}{4 \hbar^2} </math>
<br>
<math> a = \frac{\sqrt{k m}}{2 \hbar} </math>

Substituting <math>a</math>, the wave function now looks like

<math> \Psi (x) = A e^{-(\sqrt{k m}/2\hbar) x^2} </math>

We know the total probability of finding the particle from the entire space (<math>-∞</math> to <math>+∞</math>) is 1. Using this condition, the normalization constant <math>A</math> can be calculated. (Note that since there are no complex component in this equation, the complex conjugate is simply the equation itself, and |\Psi|^2 is the square of the wave function.)

<math> 1 = \int_{-∞}^{+∞} |\Psi|^2 dx = \int_{-∞}^{+∞} A^2 e^{-(\sqrt{k m}/\hbar) x^2} dx </math>

Using the [//en.wikipedia.org/wiki/Gaussian_integral Gaussian Integral], <math> 1 = \int_{-∞}^{+∞} e^{-ax^2} dx = \sqrt{\frac{\pi}{a}} </math>, the integral is turned into

<math>1 = A^2 \sqrt{\frac{\hbar \pi}{\sqrt{k m}}} </math>
<br>
<math >A^2 = \sqrt{\frac{\sqrt{k m}}{\hbar \pi}} = \sqrt{\frac{\sqrt{(m w_{0}^2)m}}{\hbar \pi}} = \sqrt{\frac{m w_{0}}{\hbar \pi}}</math>
<br>
<math >A = (\frac{m w_{0}}{\hbar \pi})^{1/4} </math>

The complete ground state wave function is

<math >\Psi (x) = (\frac{m w_{0}}{\hbar \pi})^{1/4} e^{-(\sqrt{k m}/2\hbar) x^2}</math>

Note that the <math>A</math> and <math >\Psi (x)</math> found here is only valid for ground state of harmonic oscillator. The general solution to the wave function is of the form <math> \Psi_{n} (x) = A f_{n} (x) e^{-ax^2} </math>, with <math>f_{n} (x)</math> being a polynomial in which the highest power of <math>x</math> is <math>x^n</math>.

One interesting trait of this wave function is, just like the finite potential well, the probability density can pentrate into the forbidden region beyond the classical turning points (the boundary of the region at which the potential energy becomes higher than the energy).

====Finding energy function====

With <math>a</math> derived above, we can easily substitute it into the second equation and find the energy <math>E</math>.

<math> E = (\frac{h^2}{m}) (\frac{\sqrt{k m}}{2 \hbar}) </math>
<br>
<math> E = \frac{1}{2} \hbar \sqrt{\frac{k}{m}} </math>

We can write the energy in terms of the classical frequency <math>ω</math><sub>0</sub>=<math>\sqrt{\frac{k}{m}}</math>.

<math> E = \frac{1}{2} \hbar ω</math><sub>0</sub>

By quantizing the energy, we then create the energy function for a simple harmonic oscillator.

<math> E </math><sub>n</sub> <math>= (\frac{1}{2}+n) \hbar ω</math><sub>0</sub>   <math>n=0,1,2,...</math>

Note that, contrary to one-dimensional energy function of a potential well, the energy function here are uniformly spaced with the interval <math>∆E=\hbar w_{0}</math>.

===Applications===
While a one-dimensional quantum harmonic oscillator with smooth potential is virtually inexistant in the nature, there exists some systems that behave akin to one, such as a vibrating diatomic molecule like HCl. A complex potential with arbitrary smooth curve can also usually be approximated as a harmonic potential near its stable equilibrium point, also known as the minimum.

====Morse Potential====
Morse potential, named after physicist Philip M. Morse, is a interatomic interaction model for the potential energy of a diatomic molecule like the quantum harmonic oscillator. However, it is a better approximation for the vibrational structure of a molecule than quantum harmonic oscillator as it includes the effects of bond breaking as <math>x→+∞</math>, rather than letting the potential increases exponentially to infinity as seen in a harmonic oscillator. The potential function for Morse potential is:

<math>V(x) = A (1-e^{-a(x-x_{0})})^2</math>
<ul>
<li><math>A</math> is the depth of the well (defined relative to the dissociated atoms)</li>
<li><math>x</math> is the distance between atoms</li>
<li><math>x_{0}</math> is the equilibrium bond distance (the distance where the potential is at minimum)</li>
<li><math>a</math> controls the width of the potential (smaller a leads to wider well)</li>
</ul>

The Morse potential function, by itself, is a complicated function that would make the Schrodinger equation difficult to solve. However, when overlaying the Morse potential curve on top of the harmonic oscillator potential curve, we can observe that they are similar near the point of equilibrium <math>x_{0}</math>. Due to this, we may relate the Morse potential to harmonic oscillator when working at low energy levels to get a rather accurate approximation of the wave function and energy function conveniently. To achieve this, we need a simplified, approximated function of Morse potential around the point of equilibrium, which can be done by performing [//en.wikipedia.org/wiki/Taylor_series Taylor Series] expansion about <math>x_{0}</math>.

<math>V(x_{0})+\frac{d}{d x} V(x_{0}) x + \frac{1}{2}\frac{d^2}{d x^2} V(x_{0}) x^2 + ... </math>
<br>
<math>= 0 + 0•x + \frac{1}{2} 2 a^2 A x^2 + ...</math>
<br>
<math>≈ \frac{1}{2} 2 a^2 A x^2</math>
<br>

We may, of course, keep expanding the Taylor series. But the reason we stopped here is due to its resemblance to the potential function of harmonic oscillator, which would be very useful for simplifying the derivation of solution to Schrodinger equation as we can re-use the solutions previously derived for harmonic oscillator and quickly arrive at an approximate solution. We can relate the two potential functions as seen below.

<math>V_{morse} (x)≈ \frac{1}{2}</math> <span style="color:red"><math>2 a^2 A</math></span> <math>x^2</math>        
<math>V_{harmonic} (x)= \frac{1}{2}</math> <span style="color:red"><math>k</math></span> <math>x^2</math>
<br>

which shows that we may approximate the Schrodinger equation by treating <math>k</math> as <math>2 a^2 A</math> since these are both constants. We can find the solution for ground state wave function and energy level function by simply replacing the <math>k</math> in the solutions for harmonic oscillator with <math>2 a^2 A</math>. Note that the <math>m</math> used for Morse potential related derivations must be the reduced mass constant between the two particles in the system <math>m=\frac{m_{1} m_{2}}{m_{1}+m_{2}}</math> as we are working with a two-particle system rather than a singular one.

Finding the approximate energy level function of Morse potential at low energy levels:

<math> E_{n} = (n+\frac{1}{2}) \hbar \sqrt{\frac{k}{m}} </math>
<br>
<math> E_{n} = (n+\frac{1}{2}) \hbar a \sqrt{\frac{2 A}{m}} </math>

Note that the <math>m</math> used here must be the reduced mass constant between the two particles in the system <math>m=\frac{m_{1} m_{2}}{m_{1}+m_{2}}</math>.

Applying similar logic to approximate the ground state wave function of Morse potential:

<math >\Psi (x) = (\frac{m w_{0}}{\hbar \pi})^{1/4} e^{-(\sqrt{k m}/2\hbar) x^2}</math>
<br>
<math >\Psi (x) = (\frac{m \sqrt{k/m}}{\hbar \pi})^{1/4} e^{-(\sqrt{k m}/2\hbar) x^2}</math>
<br>
<math >\Psi (x) = (\frac{m a\sqrt{2 A/m}}{\hbar \pi})^{1/4} e^{-(a\sqrt{2 A m}/2\hbar) x^2}</math>
<br>

Solution for Simple Harmonic Oscillator

2022-04-25T03:21:58Z

Lim, Xuen Zhen: Added content to Morse potential application

<b>Claimed by Lim, Xuen Zhen (Spring 2022)</b>

===Introduction===
One of the many situations which can be analyzed with the Schrodinger’s Equation is the one-dimensional simple harmonic oscillator. This situation draws an analogy between a quantum-mechanical particle’s oscillating movement and the movement of a common [//www.physicsbook.gatech.edu/Spring_Potential_Energy classical oscillator: a spring]. Like its classical spring counterpart described under Hooke's Law, a quantum harmonic oscillator has the force function <math> F = -k x </math> and the associated potential function <math> V = \frac{1}{2} k x^2 </math>, with <math> k </math> being the force constant (spring constant in classical case). Despite the simplicity of a harmonic oscillator's smooth parabolic potential, it acts as an important foundation to solving more complicated quantum systems due to it being one of the few quantum-mechanical systems with an exact, known analytical solution.

===Mathematical Setup===
We may use the time-independent Schrodinger's equation to represent the state of a quantum particle in the harmonic potential by substituting the potential <math> V </math> with <math> \frac{1}{2} k x^2 </math>.
<br>
<math>\frac{-\hbar^2}{2m} \frac{d^2 \Psi}{d x^2} + \frac{1}{2} k x^2 \Psi = E \Psi</math>
<br>
The solution to this equation are the wave function <math> \Psi </math> and the energy function <math> E </math> that satisfies the above conditions.

===Deriving the Solution===
====Finding Ground State Wave Function====
For quantum harmonic oscillator, there are no boundaries between different regions, so one condition for the wave function is it must approach 0 as <math> x→+∞ </math> and <math> x→-∞ </math>. A simple general wave function that satisfies this requirement is <math> \Psi (x) = A e^{-ax^2} </math>. We begin the derivation with finding the second order differential of the general wave equation.
<br><br>
<math> \frac{d \Psi}{d x} = -2 a x (A e^{-ax^2}) = -2 a x \Psi </math>
<br><br>
<math> \frac{d^2 \Psi}{d x^2} = -2 a (A e^{-ax^2})-2 a x (-2 a x)(A e^{-ax^2}) = (-2 a+4a^2x^2)A e^{-ax^2} = (-2 a+4a^2x^2)\Psi </math>
<br>

Substituting the differential equation into the time-independent Schrodinger equation produces

<math> \frac{-\hbar^2}{2m} (-2 a+4a^2x^2)\Psi + \frac{1}{2} k x^2 \Psi = E \Psi </math>
<br>
<math> \frac{\hbar^2 a}{m}\Psi-\frac{2 a^2 \hbar^2 x^2}{m}\Psi + \frac{1}{2} k x^2 \Psi = E \Psi </math>
<br>
<math> \frac{\hbar^2 a}{m}-\frac{2 a^2 \hbar^2}{m}x^2 + \frac{1}{2} k x^2 = E </math>
<br>

One common misconception to be aware of is that this is not an equation to be solved for <math> x </math>. <math> x </math> is posing as a free variable here, and the solution is one that makes the Schrodinger equation true for any value of <math> x </math>. To allow this equation to be consistent for any <math> x </math>, the coefficients to <math> x^2 </math> must cancel out, leaving the remaining constants to be equal to each other.

<span style="color:blue"><math> \frac{\hbar^2 a}{m}</math></span><math> - </math><span style="color:red"><math>\frac{2 a^2 \hbar^2}{m}x^2 + \frac{1}{2} k x^2</math></span><math> = </math><span style="color:blue"><math> E </math></span>

<span style="color:red"><math>-\frac{2 a^2 \hbar^2}{m} + \frac{1}{2} k = 0 </math></span> and <span style="color:blue"><math> \frac{\hbar^2 a}{m} = E </math></span>

Remember, <math>k</math>, <math>\hbar</math>, and <math>m</math> are variables which we already know the values, <math>a</math> and <math>E</math> are the only unknowns we are trying to find. We will first solve for <math>a</math> in the equation on the left to help find the wave function.

<math>-\frac{2 a^2 \hbar^2}{m} + \frac{1}{2} k = 0 </math>
<br>
<math>\frac{2 a^2 \hbar^2}{m} = \frac{1}{2} k </math>
<br>
<math> a^2 = \frac{k m}{4 \hbar^2} </math>
<br>
<math> a = \frac{\sqrt{k m}}{2 \hbar} </math>

Substituting <math>a</math>, the wave function now looks like

<math> \Psi (x) = A e^{-(\sqrt{k m}/2\hbar) x^2} </math>

We know the total probability of finding the particle from the entire space (<math>-∞</math> to <math>+∞</math>) is 1. Using this condition, the normalization constant <math>A</math> can be calculated. (Note that since there are no complex component in this equation, the complex conjugate is simply the equation itself, and |\Psi|^2 is the square of the wave function.)

<math> 1 = \int_{-∞}^{+∞} |\Psi|^2 dx = \int_{-∞}^{+∞} A^2 e^{-(\sqrt{k m}/\hbar) x^2} dx </math>

Using the [//en.wikipedia.org/wiki/Gaussian_integral Gaussian Integral], <math> 1 = \int_{-∞}^{+∞} e^{-ax^2} dx = \sqrt{\frac{\pi}{a}} </math>, the integral is turned into

<math>1 = A^2 \sqrt{\frac{\hbar \pi}{\sqrt{k m}}} </math>
<br>
<math >A^2 = \sqrt{\frac{\sqrt{k m}}{\hbar \pi}} = \sqrt{\frac{\sqrt{(m w_{0}^2)m}}{\hbar \pi}} = \sqrt{\frac{m w_{0}}{\hbar \pi}}</math>
<br>
<math >A = (\frac{m w_{0}}{\hbar \pi})^{1/4} </math>

The complete ground state wave function is

<math >\Psi (x) = (\frac{m w_{0}}{\hbar \pi})^{1/4} e^{-(\sqrt{k m}/2\hbar) x^2}</math>

Note that the <math>A</math> and <math >\Psi (x)</math> found here is only valid for ground state of harmonic oscillator. The general solution to the wave function is of the form <math> \Psi_{n} (x) = A f_{n} (x) e^{-ax^2} </math>, with <math>f_{n} (x)</math> being a polynomial in which the highest power of <math>x</math> is <math>x^n</math>.

One interesting trait of this wave function is, just like the finite potential well, the probability density can pentrate into the forbidden region beyond the classical turning points (the boundary of the region at which the potential energy becomes higher than the energy).

====Finding energy function====

With <math>a</math> derived above, we can easily substitute it into the second equation and find the energy <math>E</math>.

<math> E = (\frac{h^2}{m}) (\frac{\sqrt{k m}}{2 \hbar}) </math>
<br>
<math> E = \frac{1}{2} \hbar \sqrt{\frac{k}{m}} </math>

We can write the energy in terms of the classical frequency <math>ω</math><sub>0</sub>=<math>\sqrt{\frac{k}{m}}</math>.

<math> E = \frac{1}{2} \hbar ω</math><sub>0</sub>

By quantizing the energy, we then create the energy function for a simple harmonic oscillator.

<math> E </math><sub>n</sub> <math>= (\frac{1}{2}+n) \hbar ω</math><sub>0</sub>   <math>n=0,1,2,...</math>

Note that, contrary to one-dimensional energy function of a potential well, the energy function here are uniformly spaced with the interval <math>∆E=\hbar w_{0}</math>.

===Applications===
While a one-dimensional quantum harmonic oscillator with smooth potential is virtually inexistant in the nature, there exists some systems that behave akin to one, such as a vibrating diatomic molecule like HCl. A complex potential with arbitrary smooth curve can also usually be approximated as a harmonic potential near its stable equilibrium point, also known as the minimum.

====Morse Potential====
Morse potential, named after physicist Philip M. Morse, is a interatomic interaction model for the potential energy of a diatomic molecule like the quantum harmonic oscillator. However, it is a better approximation for the vibrational structure of a molecule than quantum harmonic oscillator as it includes the effects of bond breaking as <math>x→+∞</math>, rather than letting the potential increases exponentially to infinity as seen in a harmonic oscillator. The potential function for Morse potential is:

<math>V(x) = D_{e} (1-e^{-a(x-x_{0})})^2</math>
<ul>
<li><math>D_{e}</math> is the depth of the well (defined relative to the dissociated atoms)</li>
<li><math>x</math> is the distance between atoms</li>
<li><math>x_{0}</math> is the equilibrium bond distance (the distance where the potential is at minimum)</li>
<li><math>a</math> controls the width of the potential (smaller a leads to wider well)</li>
</ul>

The Morse potential function, by itself, is a complicated function that would make the Schrodinger equation difficult to solve. However, when overlaying the Morse potential curve on top of the harmonic oscillator potential curve, we can observe that they are similar near the point of equilibrium <math>x_{0}</math>. Due to this, we may relate the Morse potential to harmonic oscillator when working at low energy levels to get a rather accurate approximation of the wave function and energy function conveniently. To achieve this, we need a simplified, approximated function of Morse potential around the point of equilibrium, which can be done by performing [//en.wikipedia.org/wiki/Taylor_series Taylor Series] expansion about <math>x_{0}</math>.

<math>V(x_{0})+\frac{d}{d x} V(x_{0}) x + \frac{1}{2}\frac{d^2}{d x^2} V(x_{0}) x^2 + ... </math>
<br>
<math>= 0 + 0•x + \frac{1}{2} 2 a^2 A x^2 + ...</math>
<br>
<math>≈ \frac{1}{2} 2 a^2 A x^2</math>
<br>

We may, of course, keep expanding the Taylor series. But the reason we stopped here is due to its resemblance to the potential function of harmonic oscillator, which would be very useful for simplifying the derivation of solution to Schrodinger equation as we can re-use the solutions previously derived for harmonic oscillator and quickly arrive at an approximate solution. We can relate the two potential functions by

<math>V_{morse} (x)≈ \frac{1}{2}</math> <math>2 a^2 A</math> <math>x^2</math>        
<math>V_{harmonic} (x)= \frac{1}{2}</math> <math>k</math> <math>x^2</math>
<br>

which shows that we may approximate the Schrodinger equation by treating <math>k</math> <math>2 a^2 A</math> since these are both constants.

Solution for Simple Harmonic Oscillator

2022-04-25T02:52:29Z

Lim, Xuen Zhen: /* Morse Potential */

Solution for Simple Harmonic Oscillator

2022-04-25T02:42:58Z

Lim, Xuen Zhen: Changed potential symbol from U to V

Solution for Simple Harmonic Oscillator

2022-04-25T02:38:50Z

Lim, Xuen Zhen: Added hyperlink to Gaussian Integral

Solution for Simple Harmonic Oscillator

2022-04-25T02:35:47Z

Lim, Xuen Zhen: Corrected hyperlink syntax

Solution for Simple Harmonic Oscillator

2022-04-25T02:29:05Z

Lim, Xuen Zhen: Corrected a href syntax error

Solution for Simple Harmonic Oscillator

2022-04-25T02:25:41Z

Lim, Xuen Zhen: Embedded a link to Physics 1 Spring Potential Energy page.

Solution for Simple Harmonic Oscillator

2022-04-25T02:21:48Z

Lim, Xuen Zhen: Added Finding Ground State Wave Function category

Solution for Simple Harmonic Oscillator

2022-04-25T02:20:30Z

Lim, Xuen Zhen: Added Morse Potential category to add content to in the future.

Solution for Simple Harmonic Oscillator

2022-04-25T02:06:15Z

Lim, Xuen Zhen: Added some clarification on the general solution to wave equation.

Solution for Simple Harmonic Oscillator

2022-04-25T01:51:54Z

Lim, Xuen Zhen: Saving changes just to be safe so I don't lose 3 hours worth of work.

Solution for Simple Harmonic Oscillator

2022-04-25T00:21:14Z

Lim, Xuen Zhen:

Solution for Simple Harmonic Oscillator

2022-04-24T23:41:33Z

Lim, Xuen Zhen: Added some initial content to derivation

Solution for Simple Harmonic Oscillator

2022-04-24T18:47:32Z

Lim, Xuen Zhen: Changed some phrasing

Solution for Simple Harmonic Oscillator

2022-04-24T18:45:39Z

Lim, Xuen Zhen: Minor grammatical correction

Solution for Simple Harmonic Oscillator

2022-04-24T18:45:05Z

Lim, Xuen Zhen: Writing some content into Applications to have an idea of how to kick things off.

Solution for Simple Harmonic Oscillator

2022-04-24T18:34:48Z

Lim, Xuen Zhen: Minor grammatical correction

Solution for Simple Harmonic Oscillator

2022-04-24T18:31:51Z

Lim, Xuen Zhen: Added content to Solution for Simple Harmonic Oscillator

Main Page

2022-04-24T18:31:20Z

Lim, Xuen Zhen: Removed author name from title of Solution for Simple Harmonic Oscillator.

__NOTOC__
= '''Georgia Tech Student Wiki for Introductory Physics.''' =

This resource was created so that students can contribute and curate content to help those with limited or no access to a textbook. When reading this website, please correct any errors you may come across. If you read something that isn't clear, please consider revising it for future students!

Looking to make a contribution?
#Pick one of the topics from intro physics listed below
#Add content to that topic or improve the quality of what is already there.
#Need to make a new topic? Edit this page and add it to the list under the appropriate category. Then copy and paste the default [[Template]] into your new page and start editing.

Please remember that this is not a textbook and you are not limited to expressing your ideas with only text and equations. Whenever possible embed: pictures, videos, diagrams, simulations, computational models (e.g. Glowscript), and whatever content you think makes learning physics easier for other students.

== Source Material ==
All of the content added to this resource must be in the public domain or similar free resource. If you are unsure about a source, contact the original author for permission. That said, there is a surprisingly large amount of introductory physics content scattered across the web. Here is an incomplete list of intro physics resources (please update as needed).
* A physics resource written by experts for an expert audience [https://en.wikipedia.org/wiki/Portal:Physics Physics Portal]
* A wiki written for students by a physics expert [http://p3server.pa.msu.edu/coursewiki/doku.php?id=183_notes MSU Physics Wiki]
* A wiki book on modern physics [https://en.wikibooks.org/wiki/Modern_Physics Modern Physics Wiki]
* The MIT open courseware for intro physics [http://ocw.mit.edu/resources/res-8-002-a-wikitextbook-for-introductory-mechanics-fall-2009/index.htm MITOCW Wiki]
* An online concept map of intro physics [http://hyperphysics.phy-astr.gsu.edu/hbase/hph.html HyperPhysics]
* Interactive physics simulations [https://phet.colorado.edu/en/simulations/category/physics PhET]
* OpenStax intro physics textbooks: [https://openstax.org/details/books/university-physics-volume-1 Vol1], [https://openstax.org/details/books/university-physics-volume-2 Vol2], [https://openstax.org/details/books/university-physics-volume-3 Vol3]
* The Open Source Physics project is a collection of online physics resources [http://www.opensourcephysics.org/ OSP]
* A resource guide compiled by the [http://www.aapt.org/ AAPT] for educators [http://www.compadre.org/ ComPADRE]
* The Feynman lectures on physics are free to read [http://www.feynmanlectures.caltech.edu/ Feynman]
* Final Study Guide for Modern Physics II created by a lab TA [https://docs.google.com/document/d/1_6GktDPq5tiNFFYs_ZjgjxBAWVQYaXp_2Imha4_nSyc/edit?usp=sharing Modern Physics II Final Study Guide]

== Resources ==
* Commonly used wiki commands [https://en.wikipedia.org/wiki/Help:Cheatsheet Wiki Cheatsheet]
* A guide to representing equations in math mode [https://en.wikipedia.org/wiki/Help:Displaying_a_formula Wiki Math Mode]
* A page to keep track of all the physics [[Constants]]
* A listing of [[Notable Scientist]] with links to their individual pages

<div style="float:left; width:30%; padding:1%;">

==Physics 1==
===Week 1===
<div class="toccolours mw-collapsible mw-collapsed">
====GlowScript 101====
<div class="mw-collapsible-content">
*[[Python Syntax]]
*[[GlowScript]]
</div>
</div>

<div class="toccolours mw-collapsible mw-collapsed">

====VPython====

<div class="mw-collapsible-content">
*[[VPython]]
*[[VPython basics]]
*[[VPython Common Errors and Troubleshooting]]
*[[VPython Functions]]
*[[VPython Lists]]
*[[VPython Loops]]
*[[VPython Multithreading]]
*[[VPython Animation]]
*[[VPython Objects]]
*[[VPython 3D Objects]]
*[[VPython Reference]]
*[[VPython MapReduceFilter]]
*[[VPython GUIs]]
</div>
</div>

<div class="toccolours mw-collapsible mw-collapsed">

====Vectors and Units====
<div class="mw-collapsible-content">
*[[Vectors]]
*[[SI Units]]
</div>
</div>

<div class="toccolours mw-collapsible mw-collapsed">

====Interactions====
<div class="mw-collapsible-content">
*[[Types of Interactions and How to Detect Them]]
</div>
</div>

<div class="toccolours mw-collapsible mw-collapsed">
====Velocity and Momentum====
<div class="mw-collapsible-content">
*[[Newton's First Law of Motion]]
*[[Mass]]
*[[Velocity]]
*[[Speed]]
*[[Speed vs Velocity]]
*[[Relative Velocity]]
*[[Derivation of Average Velocity]]
*[[2-Dimensional Motion]]
*[[3-Dimensional Position and Motion]]
</div>
</div>

===Week 2===
<div class="toccolours mw-collapsible mw-collapsed">
====Momentum and the Momentum Principle====
<div class="mw-collapsible-content">
*[[Linear Momentum]]
*[[Newton's Second Law: the Momentum Principle]]
*[[Impulse and Momentum]]
*[[Net Force]]
*[[Inertia]]
*[[Acceleration]]
*[[Relativistic Momentum]]

</div>
</div>

<div class="toccolours mw-collapsible mw-collapsed">

====Iterative Prediction with a Constant Force====
<div class="mw-collapsible-content">
*[[Iterative Prediction]]
</div>
</div>

===Week 3===
<div class="toccolours mw-collapsible mw-collapsed">

====Analytic Prediction with a Constant Force====
<div class="mw-collapsible-content">

*[[Kinematics]]
*[[Projectile Motion]]
</div>
</div>

<div class="toccolours mw-collapsible mw-collapsed">

====Iterative Prediction with a Varying Force====
<div class="mw-collapsible-content">
*[[Fundamentals of Iterative Prediction with Varying Force]]
*[[Spring_Force]]
*[[Simple Harmonic Motion]]


*[[Iterative Prediction of Spring-Mass System]]
*[[Terminal Speed]]
*[[Predicting Change in multiple dimensions]]
*[[Two Dimensional Harmonic Motion]]
*[[Determinism]]
</div>
</div>

===Week 4===
<div class="toccolours mw-collapsible mw-collapsed">
====Fundamental Interactions====
<div class="mw-collapsible-content">
*[[Gravitational Force]]
*[[Gravitational Force Near Earth]]
*[[Gravitational Force in Space and Other Applications]]
*[[3 or More Body Interactions]]

*[[Electric Force]]
*[[Introduction to Magnetic Force]]
*[[Strong and Weak Force]]
*[[Reciprocity]]
*[[Conservation of Momentum]]
</div>
</div>

===Week 5===
<div class="toccolours mw-collapsible mw-collapsed">
====Properties of Matter====
<div class="mw-collapsible-content">
*[[Kinds of Matter]]
*[[Ball and Spring Model of Matter]]
*[[Density]]
*[[Length and Stiffness of an Interatomic Bond]]
*[[Young's Modulus]]
*[[Speed of Sound in Solids]]
*[[Malleability]]
*[[Ductility]]
*[[Weight]]
*[[Hardness]]
*[[Boiling Point]]
*[[Melting Point]]
*[[Change of State]]
</div>
</div>

===Week 6===
<div class="toccolours mw-collapsible mw-collapsed">
====Identifying Forces====
<div class="mw-collapsible-content">
*[[Free Body Diagram]]
*[[Inclined Plane]]
*[[Compression or Normal Force]]
*[[Tension]]
</div>
</div>
<div class="toccolours mw-collapsible mw-collapsed">

====Curving Motion====
<div class="mw-collapsible-content">
*[[Curving Motion]]
*[[Centripetal Force and Curving Motion]]
*[[Perpetual Freefall (Orbit)]]
</div>
</div>

===Week 7===
<div class="toccolours mw-collapsible mw-collapsed">
====Energy Principle====
<div class="mw-collapsible-content">
*[[Energy of a Single Particle]]
*[[Kinetic Energy]]
*[[Work/Energy]]
*[[The Energy Principle]]
*[[Conservation of Energy]]
</div>
</div>

===Week 8===
<div class="toccolours mw-collapsible mw-collapsed">
====Work by Non-Constant Forces====
<div class="mw-collapsible-content">
*[[Work Done By A Nonconstant Force]]
</div>
</div>
<div class="toccolours mw-collapsible mw-collapsed">
====Potential Energy====
<div class="mw-collapsible-content">
*[[Potential Energy]]
*[[Potential Energy of Macroscopic Springs]]
*[[Spring Potential Energy]]
*[[Ball and Spring Model]]
*[[Gravitational Potential Energy]]
*[[Energy Graphs]]
*[[Escape Velocity]]
</div>
</div>

===Week 9===
<div class="toccolours mw-collapsible mw-collapsed">
====Multiparticle Systems====
<div class="mw-collapsible-content">
*[[Center of Mass]]
*[[Multi-particle analysis of Momentum]]
*[[Potential Energy of a Multiparticle System]]
*[[Work and Energy for an Extended System]]
*[[Internal Energy]]
**[[Potential Energy of a Pair of Neutral Atoms]]
</div>
</div>

===Week 10===
<div class="toccolours mw-collapsible mw-collapsed">
====Choice of System====
<div class="mw-collapsible-content">
*[[System & Surroundings]]
</div>
</div>
<div class="toccolours mw-collapsible mw-collapsed">
====Thermal Energy, Dissipation, and Transfer of Energy====
<div class="mw-collapsible-content">
*[[Thermal Energy]]
*[[Specific Heat]]
*[[Calorific Value(Heat of combustion)]]
*[[First Law of Thermodynamics]]
*[[Second Law of Thermodynamics and Entropy]]
*[[Temperature]]
*[[Transformation of Energy]]
*[[The Maxwell-Boltzmann Distribution]]
*[[Air Resistance]]
*[[The Third Law of Thermodynamics]]
</div>
</div>
<div class="toccolours mw-collapsible mw-collapsed">

====Rotational and Vibrational Energy====
<div class="mw-collapsible-content">
*[[Translational, Rotational and Vibrational Energy]]
*[[Rolling Motion]]
</div>
</div>

===Week 11===
<div class="toccolours mw-collapsible mw-collapsed">
====Different Models of a System====
<div class="mw-collapsible-content">
*[[Point Particle Systems]]
*[[Real Systems]]
</div>
</div>
<div class="toccolours mw-collapsible mw-collapsed">
====Friction====
<div class="mw-collapsible-content">
*[[Friction]]
*[[Static Friction]]
*[[Kinetic Friction]]
</div>
</div>

===Week 12===
<div class="toccolours mw-collapsible mw-collapsed">
====Conservation of Momentum====
<div class="mw-collapsible-content">
*[[Conservation of Momentum]]
</div>
</div>
<div class="toccolours mw-collapsible mw-collapsed">
====Collisions====
<div class="mw-collapsible-content">
*[[Newton's Third Law of Motion]]
*[[Collisions]]
*[[Elastic Collisions]]
*[[Inelastic Collisions]]
*[[Maximally Inelastic Collision]]
*[[Head-on Collision of Equal Masses]]
*[[Head-on Collision of Unequal Masses]]
*[[Scattering: Collisions in 2D and 3D]]
*[[Rutherford Experiment and Atomic Collisions]]
*[[Coefficient of Restitution]]
</div>
</div>

===Week 13===
<div class="toccolours mw-collapsible mw-collapsed">
====Rotations====
<div class="mw-collapsible-content">
*[[Rotational Kinematics]]
*[[Eulerian Angles]]
</div>
</div>
<div class="toccolours mw-collapsible mw-collapsed">

====Angular Momentum====
<div class="mw-collapsible-content">
*[[Total Angular Momentum]]
*[[Translational Angular Momentum]]
*[[Rotational Angular Momentum]]
*[[The Angular Momentum Principle]]
*[[Angular Impulse]]
*[[Predicting the Position of a Rotating System]]
*[[The Moments of Inertia]]
*[[Right Hand Rule]]
</div>
</div>

===Week 14===
<div class="toccolours mw-collapsible mw-collapsed">
====Analyzing Motion with and without Torque====
<div class="mw-collapsible-content">
*[[Torque]]
*[[Torque 2]]
*[[Systems with Zero Torque]]
*[[Systems with Nonzero Torque]]
*[[Torque vs Work]]
*[[Gyroscopes]]
</div>
</div>

===Week 15===
<div class="toccolours mw-collapsible mw-collapsed">
====Introduction to Quantum Concepts====
<div class="mw-collapsible-content">
*[[Bohr Model]]
*[[Energy graphs and the Bohr model]]
*[[Quantized energy levels]]
*[[Electron transitions]]
*[[Entropy]]
</div>
</div>
</div>

<div style="float:left; width:30%; padding:1%;">

==Physics 2==
===Week 1===
<div class="toccolours mw-collapsible mw-collapsed">
====3D Vectors====

<div class="mw-collapsible-content">
*[[Vectors]]
*[[Right-Hand Rule]]
*[[Right Hand Rule]]
</div>
</div>

<div class="toccolours mw-collapsible mw-collapsed">

====Electric field====
<div class="mw-collapsible-content">
*[[Electric Field]]
*[[Electric Field and Electric Potential]]
</div>
</div>

<div class="toccolours mw-collapsible mw-collapsed">

====Electric force====
<div class="mw-collapsible-content">
*[[Electric Force]]
*[[Lorentz Force]]
</div>
</div>

<div class="toccolours mw-collapsible mw-collapsed">

====Electric field of a point particle====
<div class="mw-collapsible-content">
*[[Point Charge]]
</div>
</div>

<div class="toccolours mw-collapsible mw-collapsed">

====Superposition====
<div class="mw-collapsible-content">
*[[Superposition Principle]]
*[[Superposition principle]]
</div>
</div>

<div class="toccolours mw-collapsible mw-collapsed">
====Dipoles====
<div class="mw-collapsible-content">
*[[Electric Dipole]]
*[[Magnetic Dipole]]
</div>
</div>

===Week 2===
<div class="toccolours mw-collapsible mw-collapsed">
====Interactions of charged objects====
<div class="mw-collapsible-content">
*[[Electric Field]]
*[[Electric Potential]]
*[[Electric Force]]
*[[Lorentz Force]]
</div>
</div>

<div class="toccolours mw-collapsible mw-collapsed">
====Tape experiments====
<div class="mw-collapsible-content">
*[[Polarization]]
*[[Electric Polarization]]
</div>
</div>

<div class="toccolours mw-collapsible mw-collapsed">
====Polarization====
<div class="mw-collapsible-content">
*[[Polarization]]
*[[Electric Polarization]]
*[[Polarization of an Atom]]
</div>
</div>

===Week 3===
<div class="toccolours mw-collapsible mw-collapsed">
====Conductors and Insulators====
<div class="mw-collapsible-content">
*[[Conductivity and Resistivity]]
*[[Insulators]]
*[[Potential Difference in an Insulator]]
*[[Conductors]]
*[[Polarization of a conductor]]
</div>
</div>

<div class="toccolours mw-collapsible mw-collapsed">

====Charging and Discharging====
<div class="mw-collapsible-content">
*[[Charge Transfer]]
*[[Electrostatic Discharge]]
*[[Charged Conductor and Charged Insulator]]
</div>
</div>

===Week 4===
<div class="toccolours mw-collapsible mw-collapsed">
====Field of a charged rod====
<div class="mw-collapsible-content">
*[[Field of a Charged Rod|Charged Rod]]
</div>
</div>

<div class="toccolours mw-collapsible mw-collapsed">

====Field of a charged ring/disk/capacitor====
<div class="mw-collapsible-content">
*[[Charged Ring]]
*[[Charged Disk]]
*[[Charged Capacitor]]
</div>
</div>

<div class="toccolours mw-collapsible mw-collapsed">
====Field of a charged sphere====
<div class="mw-collapsible-content">
*[[Charged Spherical Shell]]
*[[Field of a Charged Ball]]
</div>
</div>

===Week 5===
<div class="toccolours mw-collapsible mw-collapsed">
====Potential energy====
<div class="mw-collapsible-content">
*[[Potential Energy]]
</div>
</div>

<div class="toccolours mw-collapsible mw-collapsed">

====Electric potential====
<div class="mw-collapsible-content">
*[[Electric Potential]]
*[[Path Independence of Electric Potential]]
*[[Potential Difference Path Independence, claimed by Aditya Mohile]]
*[[Potential Difference in a Uniform Field]]
*[[Potential Difference of Point Charge in a Non-Uniform Field]]
</div>
</div>
<div class="toccolours mw-collapsible mw-collapsed">
====Sign of a potential difference====
<div class="mw-collapsible-content">
*[[Sign of a Potential Difference]]
</div>
</div>
<div class="toccolours mw-collapsible mw-collapsed">
====Potential at a single location====
<div class="mw-collapsible-content">
*[[Electric Potential]]
*[[Potential Difference at One Location]]
</div>
</div>

<div class="toccolours mw-collapsible mw-collapsed">

====Path independence and round trip potential====
<div class="mw-collapsible-content">
*[[Path Independence of Electric Potential]]
*[[Potential Difference Path Independence, claimed by Aditya Mohile]]
</div>
</div>

===Week 6===
<div class="toccolours mw-collapsible mw-collapsed">
====Electric field and potential in an insulator====
<div class="mw-collapsible-content">
*[[Potential Difference in an Insulator]]
*[[Electric Field in an Insulator]]
</div>
</div>
<div class="toccolours mw-collapsible mw-collapsed">
====Moving charges in a magnetic field====
<div class="mw-collapsible-content">
*[[Magnetic Field]]
*[[Magnetic Force]]
*[[Lorentz Force]]
</div>
</div>
<div class="toccolours mw-collapsible mw-collapsed">
====Biot-Savart Law====
<div class="mw-collapsible-content">
*[[Biot-Savart Law]]
*[[Biot-Savart Law for Currents]]
</div>
</div>
<div class="toccolours mw-collapsible mw-collapsed">

====Moving charges, electron current, and conventional current====
<div class="mw-collapsible-content">
*[[Moving Point Charge]]
*[[Current]]
</div>
</div>

===Week 7===
<div class="toccolours mw-collapsible mw-collapsed">
====Magnetic field of a wire====
<div class="mw-collapsible-content">
*[[Magnetic Field of a Long Straight Wire]]
</div>
</div>

<div class="toccolours mw-collapsible mw-collapsed">

====Magnetic field of a current-carrying loop====
<div class="mw-collapsible-content">
*[[Magnetic Field of a Loop]]
</div>
</div>

<div class="toccolours mw-collapsible mw-collapsed">
====Magnetic field of a Charged Disk====
<div class="mw-collapsible-content">
*[[Magnetic Field of a Disk]]
</div>
</div>

<div class="toccolours mw-collapsible mw-collapsed">
====Magnetic dipoles====
<div class="mw-collapsible-content">
*[[Magnetic Dipole Moment]]
*[[Bar Magnet]]
</div>
</div>

<div class="toccolours mw-collapsible mw-collapsed">
====Atomic structure of magnets====
<div class="mw-collapsible-content">
*[[Atomic Structure of Magnets]]
</div>
</div>

===Week 8===
<div class="toccolours mw-collapsible mw-collapsed">
====Steady state current====
<div class="mw-collapsible-content">
*[[Steady State]]
*[[Non Steady State]]
</div>
</div>

<div class="toccolours mw-collapsible mw-collapsed">
====Kirchoff's Laws====
<div class="mw-collapsible-content">
*[[Kirchoff's Laws]]
</div>
</div>

<div class="toccolours mw-collapsible mw-collapsed">
====Electric fields and energy in circuits====
<div class="mw-collapsible-content">
*[[Electric Potential Difference]]
</div>
</div>

<div class="toccolours mw-collapsible mw-collapsed">

====Macroscopic analysis of circuits====
<div class="mw-collapsible-content">
*[[Series Circuits]]
*[[Parallel Circuits]]
*[[Parallel Circuits vs. Series Circuits*]]
*[[Loop Rule]]
*[[Node Rule]]
*[[Fundamentals of Resistance]]
*[[Problem Solving]]
</div>
</div>

===Week 9===
<div class="toccolours mw-collapsible mw-collapsed">
====Electric field and potential in circuits with capacitors====
<div class="mw-collapsible-content">
*[[Charging and Discharging a Capacitor]]
*[[RC Circuit]]
*[[R Circuit]]
*[[AC and DC]]
</div>
</div>

<div class="toccolours mw-collapsible mw-collapsed">
====Magnetic forces on charges and currents====
<div class="mw-collapsible-content">
*[[Magnetic Force]]
*[[Lorentz Force]]
*[[Motors and Generators]]
*[[Applying Magnetic Force to Currents]]
*[[Magnetic Force in a Moving Reference Frame]]
*[[Right-Hand Rule]]
*[[Analysis of Railgun vs Coil gun technologies]]
</div>
</div>

<div class="toccolours mw-collapsible mw-collapsed">

====Electric and magnetic forces====
<div class="mw-collapsible-content">
*[[Electric Force]]
*[[Magnetic Force]]
*[[Lorentz Force]]
*[[VPython Modelling of Electric and Magnetic Forces]]
</div>
</div>

<div class="toccolours mw-collapsible mw-collapsed">

====Velocity selector====
<div class="mw-collapsible-content">
*[[Lorentz Force]]
*[[Combining Electric and Magnetic Forces]]
</div>
</div>

===Week 10===
<div class="toccolours mw-collapsible mw-collapsed">

====Hall Effect====
<div class="mw-collapsible-content">
*[[Hall Effect]]
<h1><strong>Alayna Baker Spring 2020</strong></h1>
[[File:Hall Effect 1.jpg]]
[[File:Hall Effect 2.jpg]]

*[[Right-Hand Rule]]
*[[Motional Emf]]
*[[Magnetic Force]]
*[[Magnetic Torque]]
</div>
</div>

<div class="toccolours mw-collapsible mw-collapsed">

]]]====Motional EMF====

<div class="mw-collapsible-content">
*[[Motional Emf]]
<h1><strong>Adeline Boswell Fall 2019</strong></h1>
[[File:Motional EMF Example.jpg]]

*[[Motional Emf using Faraday's Law]]
</div>
</div>http://www.physicsbook.gatech.edu/Special:RecentChangesLinked/Main_Page

<div class="toccolours mw-collapsible mw-collapsed">

If you have a bar attached to two rails, and the rails are connected by a resistor, you have effectively created a circuit. As the bar moves, it creates an "electromotive force"

[[File:MotEMFCR.jpg]]

====Magnetic force====
<div class="mw-collapsible-content">
*[[Magnetic Force]]
*[[Lorentz Force]]
</div>
</div>

<div class="toccolours mw-collapsible mw-collapsed">
====Magnetic torque====
<div class="mw-collapsible-content">
*[[Magnetic Torque]]
*[[Right-Hand Rule]]
</div>
</div>

===Week 12===

<div class="toccolours mw-collapsible mw-collapsed">
====Gauss's Law====
<div class="mw-collapsible-content">
*[[Gauss's Flux Theorem]]
*[[Gauss's Law]]
*[[Magnetic Flux]]
</div>
</div>

<div class="toccolours mw-collapsible mw-collapsed">
====Ampere's Law====
<div class="mw-collapsible-content">
*[[Ampere's Law]]
*[[Ampere-Maxwell Law]]
*[[Magnetic Field of Coaxial Cable Using Ampere's Law]]
*[[Magnetic Field of a Long Thick Wire Using Ampere's Law]]
*[[Magnetic Field of a Toroid Using Ampere's Law]]
*[[Magnetic Field of a Solenoid Using Ampere's Law]]
*[[The Differential Form of Ampere's Law]]
</div>
</div>

===Week 13===
<div class="toccolours mw-collapsible mw-collapsed">
====Semiconductors====
<div class="mw-collapsible-content">
*[[Semiconductor Devices]]
</div>
</div>

<div class="toccolours mw-collapsible mw-collapsed">
====Faraday's Law====
<div class="mw-collapsible-content">
*[[Faraday's Law]]
*[[Motional Emf using Faraday's Law]]
*[[Lenz's Law]]

</div>
</div>

<div class="toccolours mw-collapsible mw-collapsed">

====Maxwell's equations====
<div class="mw-collapsible-content">
*[[Gauss's Law]]
*[[Magnetic Flux]]
*[[Ampere's Law]]
*[[Faraday's Law]]
*[[Maxwell's Electromagnetic Theory]]
</div>
</div>

===Week 14===
<div class="toccolours mw-collapsible mw-collapsed">
====Circuits revisited====
<div class="mw-collapsible-content">

</div>
</div>

<div class="toccolours mw-collapsible mw-collapsed">

====Inductors====
<div class="mw-collapsible-content">
*[[Inductors]]
*[[Current in an LC Circuit]]
*[[Current in an RL Circuit]]
</div>
</div>

===Week 15===
<div class="toccolours mw-collapsible mw-collapsed">
==== Electromagnetic Radiation ====
<div class="mw-collapsible-content">
*[[Electromagnetic Radiation]]
</div>
</div>

<div class="toccolours mw-collapsible mw-collapsed">
====Sparks in the air====
<div class="mw-collapsible-content">
*[[Sparks in Air]]
*[[Spark Plugs]]
</div>
</div>

<div class="toccolours mw-collapsible mw-collapsed">
====Superconductors====
<div class="mw-collapsible-content">
*[[Superconducters]]
*[[Superconductors]]
*[[Meissner effect]]
</div>
</div>
</div>

<div style="float:left; width:30%; padding:1%;">

==Physics 3==

===Week 1===
<div class="toccolours mw-collapsible mw-collapsed">
====Classical Physics====
<div class="mw-collapsible-content">
</div>
</div>

===Week 2===
<div class="toccolours mw-collapsible mw-collapsed">
====Special Relativity====
<div class="mw-collapsible-content">
*[[Frame of Reference]]

*[[Einstein's Theory of Special Relativity]]
*[[Time Dilation]]
*[[Einstein's Theory of General Relativity]]
*[[Albert A. Micheleson & Edward W. Morley]]
*[[Magnetic Force in a Moving Reference Frame]]
</div>
</div>

===Week 3===
<div class="toccolours mw-collapsible mw-collapsed">
====Photons====
<div class="mw-collapsible-content">
*[[Spontaneous Photon Emission]]
*[[Light Scattering: Why is the Sky Blue]]
*[[Lasers]]
*[[Electronic Energy Levels and Photons]]
*[[Quantum Properties of Light]]
*[[The Photoelectric Effect]]
</div>
</div>

===Week 4===
<div class="toccolours mw-collapsible mw-collapsed">
====Matter Waves====
<div class="mw-collapsible-content">
*[[Wave-Particle Duality]]
*[[Particle in a 1-Dimensional box]]
*[[Heisenberg Uncertainty Principle]]
</div>
</div>

<div class="toccolours mw-collapsible mw-collapsed">

====Schrödinger Equation====
<div class="mw-collapsible-content">
*[[Solution for a Single Free Particle]]
*[[Solution for a Single Particle in an Infinite Quantum Well - Darin]]
*[[Solution for a Single Particle in a Semi-Infinite Quantum Well]]
*[[Solution for Simple Harmonic Oscillator]]
</div>
</div>

===Week 5===
<div class="toccolours mw-collapsible mw-collapsed">
====Wave Mechanics====
<div class="mw-collapsible-content">
*[[Standing Waves]]
*[[Wavelength]]
*[[Wavelength and Frequency]]
*[[Mechanical Waves]]
*[[Transverse and Longitudinal Waves]]
</div>
</div>

<div class="toccolours mw-collapsible mw-collapsed">
====Quantum Mechanics====
<div class="mw-collapsible-content">
*[[Quantum Tunneling through Potential Barriers]]
</div>
</div>

===Week 6===
<div class="toccolours mw-collapsible mw-collapsed">
====Rutherford-Bohr Model====
<div class="mw-collapsible-content">
*[[Rutherford Experiment and Atomic Collisions]]
*[[Bohr Model]]
*[[Quantized energy levels]]
*[[Energy graphs and the Bohr model]]
</div>
</div>

===Week 7===
<div class="toccolours mw-collapsible mw-collapsed">
====The Hydrogen Atom====
<div class="mw-collapsible-content">
*[[Quantum Theory]]
*[[Atomic Theory]]
</div>
</div>

===Week 8===
<div class="toccolours mw-collapsible mw-collapsed">
====Many-Electron Atoms====
<div class="mw-collapsible-content">
*[[Quantum Theory]]
*[[Atomic Theory]]
*[[Pauli exclusion principle]]
</div>
</div>

===Week 9===
<div class="toccolours mw-collapsible mw-collapsed">
====Molecules====
<div class="mw-collapsible-content">
</div>
*[[Molecules]]
*[[sp Molecular Bonds]]
</div>

===Week 10===
<div class="toccolours mw-collapsible mw-collapsed">
====Statistical Physics====
<div class="mw-collapsible-content">
</div>
*[[Application of Statistics in Physics]]
*[[Temperature & Entropy]]
</div>
<div class="mw-collapsible-content">

===Week 11===
<div class="toccolours mw-collapsible mw-collapsed">
====Condensed Matter Physics====
<div class="mw-collapsible-content">
</div>
</div>

===Week 12===
<div class="toccolours mw-collapsible mw-collapsed">
====The Nucleus====
<div class="mw-collapsible-content">
*[[Nucleus]]
</div>
</div>

===Week 13===
<div class="toccolours mw-collapsible mw-collapsed">
====Nuclear Physics====
<div class="mw-collapsible-content">
*[[Nuclear Fission]]
*[[Nuclear Energy from Fission and Fusion]]
</div>
</div>

===Week 14===
<div class="toccolours mw-collapsible mw-collapsed">
====Particle Physics====
<div class="mw-collapsible-content">
*[[Elementary Particles and Particle Physics Theory]]
*[[String Theory]]
</div>
</div>
</div>

Main Page

2022-04-24T18:30:33Z

Lim, Xuen Zhen: Undo revision 40173 by Lim, Xuen Zhen (talk)

__NOTOC__
= '''Georgia Tech Student Wiki for Introductory Physics.''' =

This resource was created so that students can contribute and curate content to help those with limited or no access to a textbook. When reading this website, please correct any errors you may come across. If you read something that isn't clear, please consider revising it for future students!

Looking to make a contribution?
#Pick one of the topics from intro physics listed below
#Add content to that topic or improve the quality of what is already there.
#Need to make a new topic? Edit this page and add it to the list under the appropriate category. Then copy and paste the default [[Template]] into your new page and start editing.

Please remember that this is not a textbook and you are not limited to expressing your ideas with only text and equations. Whenever possible embed: pictures, videos, diagrams, simulations, computational models (e.g. Glowscript), and whatever content you think makes learning physics easier for other students.

== Source Material ==
All of the content added to this resource must be in the public domain or similar free resource. If you are unsure about a source, contact the original author for permission. That said, there is a surprisingly large amount of introductory physics content scattered across the web. Here is an incomplete list of intro physics resources (please update as needed).
* A physics resource written by experts for an expert audience [https://en.wikipedia.org/wiki/Portal:Physics Physics Portal]
* A wiki written for students by a physics expert [http://p3server.pa.msu.edu/coursewiki/doku.php?id=183_notes MSU Physics Wiki]
* A wiki book on modern physics [https://en.wikibooks.org/wiki/Modern_Physics Modern Physics Wiki]
* The MIT open courseware for intro physics [http://ocw.mit.edu/resources/res-8-002-a-wikitextbook-for-introductory-mechanics-fall-2009/index.htm MITOCW Wiki]
* An online concept map of intro physics [http://hyperphysics.phy-astr.gsu.edu/hbase/hph.html HyperPhysics]
* Interactive physics simulations [https://phet.colorado.edu/en/simulations/category/physics PhET]
* OpenStax intro physics textbooks: [https://openstax.org/details/books/university-physics-volume-1 Vol1], [https://openstax.org/details/books/university-physics-volume-2 Vol2], [https://openstax.org/details/books/university-physics-volume-3 Vol3]
* The Open Source Physics project is a collection of online physics resources [http://www.opensourcephysics.org/ OSP]
* A resource guide compiled by the [http://www.aapt.org/ AAPT] for educators [http://www.compadre.org/ ComPADRE]
* The Feynman lectures on physics are free to read [http://www.feynmanlectures.caltech.edu/ Feynman]
* Final Study Guide for Modern Physics II created by a lab TA [https://docs.google.com/document/d/1_6GktDPq5tiNFFYs_ZjgjxBAWVQYaXp_2Imha4_nSyc/edit?usp=sharing Modern Physics II Final Study Guide]

== Resources ==
* Commonly used wiki commands [https://en.wikipedia.org/wiki/Help:Cheatsheet Wiki Cheatsheet]
* A guide to representing equations in math mode [https://en.wikipedia.org/wiki/Help:Displaying_a_formula Wiki Math Mode]
* A page to keep track of all the physics [[Constants]]
* A listing of [[Notable Scientist]] with links to their individual pages

<div style="float:left; width:30%; padding:1%;">

==Physics 1==
===Week 1===
<div class="toccolours mw-collapsible mw-collapsed">
====GlowScript 101====
<div class="mw-collapsible-content">
*[[Python Syntax]]
*[[GlowScript]]
</div>
</div>

<div class="toccolours mw-collapsible mw-collapsed">

====VPython====

<div class="mw-collapsible-content">
*[[VPython]]
*[[VPython basics]]
*[[VPython Common Errors and Troubleshooting]]
*[[VPython Functions]]
*[[VPython Lists]]
*[[VPython Loops]]
*[[VPython Multithreading]]
*[[VPython Animation]]
*[[VPython Objects]]
*[[VPython 3D Objects]]
*[[VPython Reference]]
*[[VPython MapReduceFilter]]
*[[VPython GUIs]]
</div>
</div>

<div class="toccolours mw-collapsible mw-collapsed">

====Vectors and Units====
<div class="mw-collapsible-content">
*[[Vectors]]
*[[SI Units]]
</div>
</div>

<div class="toccolours mw-collapsible mw-collapsed">

====Interactions====
<div class="mw-collapsible-content">
*[[Types of Interactions and How to Detect Them]]
</div>
</div>

<div class="toccolours mw-collapsible mw-collapsed">
====Velocity and Momentum====
<div class="mw-collapsible-content">
*[[Newton's First Law of Motion]]
*[[Mass]]
*[[Velocity]]
*[[Speed]]
*[[Speed vs Velocity]]
*[[Relative Velocity]]
*[[Derivation of Average Velocity]]
*[[2-Dimensional Motion]]
*[[3-Dimensional Position and Motion]]
</div>
</div>

===Week 2===
<div class="toccolours mw-collapsible mw-collapsed">
====Momentum and the Momentum Principle====
<div class="mw-collapsible-content">
*[[Linear Momentum]]
*[[Newton's Second Law: the Momentum Principle]]
*[[Impulse and Momentum]]
*[[Net Force]]
*[[Inertia]]
*[[Acceleration]]
*[[Relativistic Momentum]]

</div>
</div>

<div class="toccolours mw-collapsible mw-collapsed">

====Iterative Prediction with a Constant Force====
<div class="mw-collapsible-content">
*[[Iterative Prediction]]
</div>
</div>

===Week 3===
<div class="toccolours mw-collapsible mw-collapsed">

====Analytic Prediction with a Constant Force====
<div class="mw-collapsible-content">

*[[Kinematics]]
*[[Projectile Motion]]
</div>
</div>

<div class="toccolours mw-collapsible mw-collapsed">

====Iterative Prediction with a Varying Force====
<div class="mw-collapsible-content">
*[[Fundamentals of Iterative Prediction with Varying Force]]
*[[Spring_Force]]
*[[Simple Harmonic Motion]]


*[[Iterative Prediction of Spring-Mass System]]
*[[Terminal Speed]]
*[[Predicting Change in multiple dimensions]]
*[[Two Dimensional Harmonic Motion]]
*[[Determinism]]
</div>
</div>

===Week 4===
<div class="toccolours mw-collapsible mw-collapsed">
====Fundamental Interactions====
<div class="mw-collapsible-content">
*[[Gravitational Force]]
*[[Gravitational Force Near Earth]]
*[[Gravitational Force in Space and Other Applications]]
*[[3 or More Body Interactions]]

*[[Electric Force]]
*[[Introduction to Magnetic Force]]
*[[Strong and Weak Force]]
*[[Reciprocity]]
*[[Conservation of Momentum]]
</div>
</div>

===Week 5===
<div class="toccolours mw-collapsible mw-collapsed">
====Properties of Matter====
<div class="mw-collapsible-content">
*[[Kinds of Matter]]
*[[Ball and Spring Model of Matter]]
*[[Density]]
*[[Length and Stiffness of an Interatomic Bond]]
*[[Young's Modulus]]
*[[Speed of Sound in Solids]]
*[[Malleability]]
*[[Ductility]]
*[[Weight]]
*[[Hardness]]
*[[Boiling Point]]
*[[Melting Point]]
*[[Change of State]]
</div>
</div>

===Week 6===
<div class="toccolours mw-collapsible mw-collapsed">
====Identifying Forces====
<div class="mw-collapsible-content">
*[[Free Body Diagram]]
*[[Inclined Plane]]
*[[Compression or Normal Force]]
*[[Tension]]
</div>
</div>
<div class="toccolours mw-collapsible mw-collapsed">

====Curving Motion====
<div class="mw-collapsible-content">
*[[Curving Motion]]
*[[Centripetal Force and Curving Motion]]
*[[Perpetual Freefall (Orbit)]]
</div>
</div>

===Week 7===
<div class="toccolours mw-collapsible mw-collapsed">
====Energy Principle====
<div class="mw-collapsible-content">
*[[Energy of a Single Particle]]
*[[Kinetic Energy]]
*[[Work/Energy]]
*[[The Energy Principle]]
*[[Conservation of Energy]]
</div>
</div>

===Week 8===
<div class="toccolours mw-collapsible mw-collapsed">
====Work by Non-Constant Forces====
<div class="mw-collapsible-content">
*[[Work Done By A Nonconstant Force]]
</div>
</div>
<div class="toccolours mw-collapsible mw-collapsed">
====Potential Energy====
<div class="mw-collapsible-content">
*[[Potential Energy]]
*[[Potential Energy of Macroscopic Springs]]
*[[Spring Potential Energy]]
*[[Ball and Spring Model]]
*[[Gravitational Potential Energy]]
*[[Energy Graphs]]
*[[Escape Velocity]]
</div>
</div>

===Week 9===
<div class="toccolours mw-collapsible mw-collapsed">
====Multiparticle Systems====
<div class="mw-collapsible-content">
*[[Center of Mass]]
*[[Multi-particle analysis of Momentum]]
*[[Potential Energy of a Multiparticle System]]
*[[Work and Energy for an Extended System]]
*[[Internal Energy]]
**[[Potential Energy of a Pair of Neutral Atoms]]
</div>
</div>

===Week 10===
<div class="toccolours mw-collapsible mw-collapsed">
====Choice of System====
<div class="mw-collapsible-content">
*[[System & Surroundings]]
</div>
</div>
<div class="toccolours mw-collapsible mw-collapsed">
====Thermal Energy, Dissipation, and Transfer of Energy====
<div class="mw-collapsible-content">
*[[Thermal Energy]]
*[[Specific Heat]]
*[[Calorific Value(Heat of combustion)]]
*[[First Law of Thermodynamics]]
*[[Second Law of Thermodynamics and Entropy]]
*[[Temperature]]
*[[Transformation of Energy]]
*[[The Maxwell-Boltzmann Distribution]]
*[[Air Resistance]]
*[[The Third Law of Thermodynamics]]
</div>
</div>
<div class="toccolours mw-collapsible mw-collapsed">

====Rotational and Vibrational Energy====
<div class="mw-collapsible-content">
*[[Translational, Rotational and Vibrational Energy]]
*[[Rolling Motion]]
</div>
</div>

===Week 11===
<div class="toccolours mw-collapsible mw-collapsed">
====Different Models of a System====
<div class="mw-collapsible-content">
*[[Point Particle Systems]]
*[[Real Systems]]
</div>
</div>
<div class="toccolours mw-collapsible mw-collapsed">
====Friction====
<div class="mw-collapsible-content">
*[[Friction]]
*[[Static Friction]]
*[[Kinetic Friction]]
</div>
</div>

===Week 12===
<div class="toccolours mw-collapsible mw-collapsed">
====Conservation of Momentum====
<div class="mw-collapsible-content">
*[[Conservation of Momentum]]
</div>
</div>
<div class="toccolours mw-collapsible mw-collapsed">
====Collisions====
<div class="mw-collapsible-content">
*[[Newton's Third Law of Motion]]
*[[Collisions]]
*[[Elastic Collisions]]
*[[Inelastic Collisions]]
*[[Maximally Inelastic Collision]]
*[[Head-on Collision of Equal Masses]]
*[[Head-on Collision of Unequal Masses]]
*[[Scattering: Collisions in 2D and 3D]]
*[[Rutherford Experiment and Atomic Collisions]]
*[[Coefficient of Restitution]]
</div>
</div>

===Week 13===
<div class="toccolours mw-collapsible mw-collapsed">
====Rotations====
<div class="mw-collapsible-content">
*[[Rotational Kinematics]]
*[[Eulerian Angles]]
</div>
</div>
<div class="toccolours mw-collapsible mw-collapsed">

====Angular Momentum====
<div class="mw-collapsible-content">
*[[Total Angular Momentum]]
*[[Translational Angular Momentum]]
*[[Rotational Angular Momentum]]
*[[The Angular Momentum Principle]]
*[[Angular Impulse]]
*[[Predicting the Position of a Rotating System]]
*[[The Moments of Inertia]]
*[[Right Hand Rule]]
</div>
</div>

===Week 14===
<div class="toccolours mw-collapsible mw-collapsed">
====Analyzing Motion with and without Torque====
<div class="mw-collapsible-content">
*[[Torque]]
*[[Torque 2]]
*[[Systems with Zero Torque]]
*[[Systems with Nonzero Torque]]
*[[Torque vs Work]]
*[[Gyroscopes]]
</div>
</div>

===Week 15===
<div class="toccolours mw-collapsible mw-collapsed">
====Introduction to Quantum Concepts====
<div class="mw-collapsible-content">
*[[Bohr Model]]
*[[Energy graphs and the Bohr model]]
*[[Quantized energy levels]]
*[[Electron transitions]]
*[[Entropy]]
</div>
</div>
</div>

<div style="float:left; width:30%; padding:1%;">

==Physics 2==
===Week 1===
<div class="toccolours mw-collapsible mw-collapsed">
====3D Vectors====

<div class="mw-collapsible-content">
*[[Vectors]]
*[[Right-Hand Rule]]
*[[Right Hand Rule]]
</div>
</div>

<div class="toccolours mw-collapsible mw-collapsed">

====Electric field====
<div class="mw-collapsible-content">
*[[Electric Field]]
*[[Electric Field and Electric Potential]]
</div>
</div>

<div class="toccolours mw-collapsible mw-collapsed">

====Electric force====
<div class="mw-collapsible-content">
*[[Electric Force]]
*[[Lorentz Force]]
</div>
</div>

<div class="toccolours mw-collapsible mw-collapsed">

====Electric field of a point particle====
<div class="mw-collapsible-content">
*[[Point Charge]]
</div>
</div>

<div class="toccolours mw-collapsible mw-collapsed">

====Superposition====
<div class="mw-collapsible-content">
*[[Superposition Principle]]
*[[Superposition principle]]
</div>
</div>

<div class="toccolours mw-collapsible mw-collapsed">
====Dipoles====
<div class="mw-collapsible-content">
*[[Electric Dipole]]
*[[Magnetic Dipole]]
</div>
</div>

===Week 2===
<div class="toccolours mw-collapsible mw-collapsed">
====Interactions of charged objects====
<div class="mw-collapsible-content">
*[[Electric Field]]
*[[Electric Potential]]
*[[Electric Force]]
*[[Lorentz Force]]
</div>
</div>

<div class="toccolours mw-collapsible mw-collapsed">
====Tape experiments====
<div class="mw-collapsible-content">
*[[Polarization]]
*[[Electric Polarization]]
</div>
</div>

<div class="toccolours mw-collapsible mw-collapsed">
====Polarization====
<div class="mw-collapsible-content">
*[[Polarization]]
*[[Electric Polarization]]
*[[Polarization of an Atom]]
</div>
</div>

===Week 3===
<div class="toccolours mw-collapsible mw-collapsed">
====Conductors and Insulators====
<div class="mw-collapsible-content">
*[[Conductivity and Resistivity]]
*[[Insulators]]
*[[Potential Difference in an Insulator]]
*[[Conductors]]
*[[Polarization of a conductor]]
</div>
</div>

<div class="toccolours mw-collapsible mw-collapsed">

====Charging and Discharging====
<div class="mw-collapsible-content">
*[[Charge Transfer]]
*[[Electrostatic Discharge]]
*[[Charged Conductor and Charged Insulator]]
</div>
</div>

===Week 4===
<div class="toccolours mw-collapsible mw-collapsed">
====Field of a charged rod====
<div class="mw-collapsible-content">
*[[Field of a Charged Rod|Charged Rod]]
</div>
</div>

<div class="toccolours mw-collapsible mw-collapsed">

====Field of a charged ring/disk/capacitor====
<div class="mw-collapsible-content">
*[[Charged Ring]]
*[[Charged Disk]]
*[[Charged Capacitor]]
</div>
</div>

<div class="toccolours mw-collapsible mw-collapsed">
====Field of a charged sphere====
<div class="mw-collapsible-content">
*[[Charged Spherical Shell]]
*[[Field of a Charged Ball]]
</div>
</div>

===Week 5===
<div class="toccolours mw-collapsible mw-collapsed">
====Potential energy====
<div class="mw-collapsible-content">
*[[Potential Energy]]
</div>
</div>

<div class="toccolours mw-collapsible mw-collapsed">

====Electric potential====
<div class="mw-collapsible-content">
*[[Electric Potential]]
*[[Path Independence of Electric Potential]]
*[[Potential Difference Path Independence, claimed by Aditya Mohile]]
*[[Potential Difference in a Uniform Field]]
*[[Potential Difference of Point Charge in a Non-Uniform Field]]
</div>
</div>
<div class="toccolours mw-collapsible mw-collapsed">
====Sign of a potential difference====
<div class="mw-collapsible-content">
*[[Sign of a Potential Difference]]
</div>
</div>
<div class="toccolours mw-collapsible mw-collapsed">
====Potential at a single location====
<div class="mw-collapsible-content">
*[[Electric Potential]]
*[[Potential Difference at One Location]]
</div>
</div>

<div class="toccolours mw-collapsible mw-collapsed">

====Path independence and round trip potential====
<div class="mw-collapsible-content">
*[[Path Independence of Electric Potential]]
*[[Potential Difference Path Independence, claimed by Aditya Mohile]]
</div>
</div>

===Week 6===
<div class="toccolours mw-collapsible mw-collapsed">
====Electric field and potential in an insulator====
<div class="mw-collapsible-content">
*[[Potential Difference in an Insulator]]
*[[Electric Field in an Insulator]]
</div>
</div>
<div class="toccolours mw-collapsible mw-collapsed">
====Moving charges in a magnetic field====
<div class="mw-collapsible-content">
*[[Magnetic Field]]
*[[Magnetic Force]]
*[[Lorentz Force]]
</div>
</div>
<div class="toccolours mw-collapsible mw-collapsed">
====Biot-Savart Law====
<div class="mw-collapsible-content">
*[[Biot-Savart Law]]
*[[Biot-Savart Law for Currents]]
</div>
</div>
<div class="toccolours mw-collapsible mw-collapsed">

====Moving charges, electron current, and conventional current====
<div class="mw-collapsible-content">
*[[Moving Point Charge]]
*[[Current]]
</div>
</div>

===Week 7===
<div class="toccolours mw-collapsible mw-collapsed">
====Magnetic field of a wire====
<div class="mw-collapsible-content">
*[[Magnetic Field of a Long Straight Wire]]
</div>
</div>

<div class="toccolours mw-collapsible mw-collapsed">

====Magnetic field of a current-carrying loop====
<div class="mw-collapsible-content">
*[[Magnetic Field of a Loop]]
</div>
</div>

<div class="toccolours mw-collapsible mw-collapsed">
====Magnetic field of a Charged Disk====
<div class="mw-collapsible-content">
*[[Magnetic Field of a Disk]]
</div>
</div>

<div class="toccolours mw-collapsible mw-collapsed">
====Magnetic dipoles====
<div class="mw-collapsible-content">
*[[Magnetic Dipole Moment]]
*[[Bar Magnet]]
</div>
</div>

<div class="toccolours mw-collapsible mw-collapsed">
====Atomic structure of magnets====
<div class="mw-collapsible-content">
*[[Atomic Structure of Magnets]]
</div>
</div>

===Week 8===
<div class="toccolours mw-collapsible mw-collapsed">
====Steady state current====
<div class="mw-collapsible-content">
*[[Steady State]]
*[[Non Steady State]]
</div>
</div>

<div class="toccolours mw-collapsible mw-collapsed">
====Kirchoff's Laws====
<div class="mw-collapsible-content">
*[[Kirchoff's Laws]]
</div>
</div>

<div class="toccolours mw-collapsible mw-collapsed">
====Electric fields and energy in circuits====
<div class="mw-collapsible-content">
*[[Electric Potential Difference]]
</div>
</div>

<div class="toccolours mw-collapsible mw-collapsed">

====Macroscopic analysis of circuits====
<div class="mw-collapsible-content">
*[[Series Circuits]]
*[[Parallel Circuits]]
*[[Parallel Circuits vs. Series Circuits*]]
*[[Loop Rule]]
*[[Node Rule]]
*[[Fundamentals of Resistance]]
*[[Problem Solving]]
</div>
</div>

===Week 9===
<div class="toccolours mw-collapsible mw-collapsed">
====Electric field and potential in circuits with capacitors====
<div class="mw-collapsible-content">
*[[Charging and Discharging a Capacitor]]
*[[RC Circuit]]
*[[R Circuit]]
*[[AC and DC]]
</div>
</div>

<div class="toccolours mw-collapsible mw-collapsed">
====Magnetic forces on charges and currents====
<div class="mw-collapsible-content">
*[[Magnetic Force]]
*[[Lorentz Force]]
*[[Motors and Generators]]
*[[Applying Magnetic Force to Currents]]
*[[Magnetic Force in a Moving Reference Frame]]
*[[Right-Hand Rule]]
*[[Analysis of Railgun vs Coil gun technologies]]
</div>
</div>

<div class="toccolours mw-collapsible mw-collapsed">

====Electric and magnetic forces====
<div class="mw-collapsible-content">
*[[Electric Force]]
*[[Magnetic Force]]
*[[Lorentz Force]]
*[[VPython Modelling of Electric and Magnetic Forces]]
</div>
</div>

<div class="toccolours mw-collapsible mw-collapsed">

====Velocity selector====
<div class="mw-collapsible-content">
*[[Lorentz Force]]
*[[Combining Electric and Magnetic Forces]]
</div>
</div>

===Week 10===
<div class="toccolours mw-collapsible mw-collapsed">

====Hall Effect====
<div class="mw-collapsible-content">
*[[Hall Effect]]
<h1><strong>Alayna Baker Spring 2020</strong></h1>
[[File:Hall Effect 1.jpg]]
[[File:Hall Effect 2.jpg]]

*[[Right-Hand Rule]]
*[[Motional Emf]]
*[[Magnetic Force]]
*[[Magnetic Torque]]
</div>
</div>

<div class="toccolours mw-collapsible mw-collapsed">

]]]====Motional EMF====

<div class="mw-collapsible-content">
*[[Motional Emf]]
<h1><strong>Adeline Boswell Fall 2019</strong></h1>
[[File:Motional EMF Example.jpg]]

*[[Motional Emf using Faraday's Law]]
</div>
</div>http://www.physicsbook.gatech.edu/Special:RecentChangesLinked/Main_Page

<div class="toccolours mw-collapsible mw-collapsed">

If you have a bar attached to two rails, and the rails are connected by a resistor, you have effectively created a circuit. As the bar moves, it creates an "electromotive force"

[[File:MotEMFCR.jpg]]

====Magnetic force====
<div class="mw-collapsible-content">
*[[Magnetic Force]]
*[[Lorentz Force]]
</div>
</div>

<div class="toccolours mw-collapsible mw-collapsed">
====Magnetic torque====
<div class="mw-collapsible-content">
*[[Magnetic Torque]]
*[[Right-Hand Rule]]
</div>
</div>

===Week 12===

<div class="toccolours mw-collapsible mw-collapsed">
====Gauss's Law====
<div class="mw-collapsible-content">
*[[Gauss's Flux Theorem]]
*[[Gauss's Law]]
*[[Magnetic Flux]]
</div>
</div>

<div class="toccolours mw-collapsible mw-collapsed">
====Ampere's Law====
<div class="mw-collapsible-content">
*[[Ampere's Law]]
*[[Ampere-Maxwell Law]]
*[[Magnetic Field of Coaxial Cable Using Ampere's Law]]
*[[Magnetic Field of a Long Thick Wire Using Ampere's Law]]
*[[Magnetic Field of a Toroid Using Ampere's Law]]
*[[Magnetic Field of a Solenoid Using Ampere's Law]]
*[[The Differential Form of Ampere's Law]]
</div>
</div>

===Week 13===
<div class="toccolours mw-collapsible mw-collapsed">
====Semiconductors====
<div class="mw-collapsible-content">
*[[Semiconductor Devices]]
</div>
</div>

<div class="toccolours mw-collapsible mw-collapsed">
====Faraday's Law====
<div class="mw-collapsible-content">
*[[Faraday's Law]]
*[[Motional Emf using Faraday's Law]]
*[[Lenz's Law]]

</div>
</div>

<div class="toccolours mw-collapsible mw-collapsed">

====Maxwell's equations====
<div class="mw-collapsible-content">
*[[Gauss's Law]]
*[[Magnetic Flux]]
*[[Ampere's Law]]
*[[Faraday's Law]]
*[[Maxwell's Electromagnetic Theory]]
</div>
</div>

===Week 14===
<div class="toccolours mw-collapsible mw-collapsed">
====Circuits revisited====
<div class="mw-collapsible-content">

</div>
</div>

<div class="toccolours mw-collapsible mw-collapsed">

====Inductors====
<div class="mw-collapsible-content">
*[[Inductors]]
*[[Current in an LC Circuit]]
*[[Current in an RL Circuit]]
</div>
</div>

===Week 15===
<div class="toccolours mw-collapsible mw-collapsed">
==== Electromagnetic Radiation ====
<div class="mw-collapsible-content">
*[[Electromagnetic Radiation]]
</div>
</div>

<div class="toccolours mw-collapsible mw-collapsed">
====Sparks in the air====
<div class="mw-collapsible-content">
*[[Sparks in Air]]
*[[Spark Plugs]]
</div>
</div>

<div class="toccolours mw-collapsible mw-collapsed">
====Superconductors====
<div class="mw-collapsible-content">
*[[Superconducters]]
*[[Superconductors]]
*[[Meissner effect]]
</div>
</div>
</div>

<div style="float:left; width:30%; padding:1%;">

==Physics 3==

===Week 1===
<div class="toccolours mw-collapsible mw-collapsed">
====Classical Physics====
<div class="mw-collapsible-content">
</div>
</div>

===Week 2===
<div class="toccolours mw-collapsible mw-collapsed">
====Special Relativity====
<div class="mw-collapsible-content">
*[[Frame of Reference]]

*[[Einstein's Theory of Special Relativity]]
*[[Time Dilation]]
*[[Einstein's Theory of General Relativity]]
*[[Albert A. Micheleson & Edward W. Morley]]
*[[Magnetic Force in a Moving Reference Frame]]
</div>
</div>

===Week 3===
<div class="toccolours mw-collapsible mw-collapsed">
====Photons====
<div class="mw-collapsible-content">
*[[Spontaneous Photon Emission]]
*[[Light Scattering: Why is the Sky Blue]]
*[[Lasers]]
*[[Electronic Energy Levels and Photons]]
*[[Quantum Properties of Light]]
*[[The Photoelectric Effect]]
</div>
</div>

===Week 4===
<div class="toccolours mw-collapsible mw-collapsed">
====Matter Waves====
<div class="mw-collapsible-content">
*[[Wave-Particle Duality]]
*[[Particle in a 1-Dimensional box]]
*[[Heisenberg Uncertainty Principle]]
</div>
</div>

<div class="toccolours mw-collapsible mw-collapsed">

====Schrödinger Equation====
<div class="mw-collapsible-content">
*[[Solution for a Single Free Particle]]
*[[Solution for a Single Particle in an Infinite Quantum Well - Darin]]
*[[Solution for a Single Particle in a Semi-Infinite Quantum Well]]
*[[Solution for Simple Harmonic Oscillator (Xuen Zhen)]]
</div>
</div>

===Week 5===
<div class="toccolours mw-collapsible mw-collapsed">
====Wave Mechanics====
<div class="mw-collapsible-content">
*[[Standing Waves]]
*[[Wavelength]]
*[[Wavelength and Frequency]]
*[[Mechanical Waves]]
*[[Transverse and Longitudinal Waves]]
</div>
</div>

<div class="toccolours mw-collapsible mw-collapsed">
====Quantum Mechanics====
<div class="mw-collapsible-content">
*[[Quantum Tunneling through Potential Barriers]]
</div>
</div>

===Week 6===
<div class="toccolours mw-collapsible mw-collapsed">
====Rutherford-Bohr Model====
<div class="mw-collapsible-content">
*[[Rutherford Experiment and Atomic Collisions]]
*[[Bohr Model]]
*[[Quantized energy levels]]
*[[Energy graphs and the Bohr model]]
</div>
</div>

===Week 7===
<div class="toccolours mw-collapsible mw-collapsed">
====The Hydrogen Atom====
<div class="mw-collapsible-content">
*[[Quantum Theory]]
*[[Atomic Theory]]
</div>
</div>

===Week 8===
<div class="toccolours mw-collapsible mw-collapsed">
====Many-Electron Atoms====
<div class="mw-collapsible-content">
*[[Quantum Theory]]
*[[Atomic Theory]]
*[[Pauli exclusion principle]]
</div>
</div>

===Week 9===
<div class="toccolours mw-collapsible mw-collapsed">
====Molecules====
<div class="mw-collapsible-content">
</div>
*[[Molecules]]
*[[sp Molecular Bonds]]
</div>

===Week 10===
<div class="toccolours mw-collapsible mw-collapsed">
====Statistical Physics====
<div class="mw-collapsible-content">
</div>
*[[Application of Statistics in Physics]]
*[[Temperature & Entropy]]
</div>
<div class="mw-collapsible-content">

===Week 11===
<div class="toccolours mw-collapsible mw-collapsed">
====Condensed Matter Physics====
<div class="mw-collapsible-content">
</div>
</div>

===Week 12===
<div class="toccolours mw-collapsible mw-collapsed">
====The Nucleus====
<div class="mw-collapsible-content">
*[[Nucleus]]
</div>
</div>

===Week 13===
<div class="toccolours mw-collapsible mw-collapsed">
====Nuclear Physics====
<div class="mw-collapsible-content">
*[[Nuclear Fission]]
*[[Nuclear Energy from Fission and Fusion]]
</div>
</div>

===Week 14===
<div class="toccolours mw-collapsible mw-collapsed">
====Particle Physics====
<div class="mw-collapsible-content">
*[[Elementary Particles and Particle Physics Theory]]
*[[String Theory]]
</div>
</div>
</div>

Main Page

2022-04-24T18:28:28Z

Lim, Xuen Zhen: Removed author name from title of Solution for Simple Harmonic Oscillator.

Main Page

2022-04-24T01:37:44Z

Lim, Xuen Zhen: /* Schrödinger Equation */

__NOTOC__
= '''Georgia Tech Student Wiki for Introductory Physics.''' =

This resource was created so that students can contribute and curate content to help those with limited or no access to a textbook. When reading this website, please correct any errors you may come across. If you read something that isn't clear, please consider revising it for future students!

Looking to make a contribution?
#Pick one of the topics from intro physics listed below
#Add content to that topic or improve the quality of what is already there.
#Need to make a new topic? Edit this page and add it to the list under the appropriate category. Then copy and paste the default [[Template]] into your new page and start editing.

Please remember that this is not a textbook and you are not limited to expressing your ideas with only text and equations. Whenever possible embed: pictures, videos, diagrams, simulations, computational models (e.g. Glowscript), and whatever content you think makes learning physics easier for other students.

== Source Material ==
All of the content added to this resource must be in the public domain or similar free resource. If you are unsure about a source, contact the original author for permission. That said, there is a surprisingly large amount of introductory physics content scattered across the web. Here is an incomplete list of intro physics resources (please update as needed).
* A physics resource written by experts for an expert audience [https://en.wikipedia.org/wiki/Portal:Physics Physics Portal]
* A wiki written for students by a physics expert [http://p3server.pa.msu.edu/coursewiki/doku.php?id=183_notes MSU Physics Wiki]
* A wiki book on modern physics [https://en.wikibooks.org/wiki/Modern_Physics Modern Physics Wiki]
* The MIT open courseware for intro physics [http://ocw.mit.edu/resources/res-8-002-a-wikitextbook-for-introductory-mechanics-fall-2009/index.htm MITOCW Wiki]
* An online concept map of intro physics [http://hyperphysics.phy-astr.gsu.edu/hbase/hph.html HyperPhysics]
* Interactive physics simulations [https://phet.colorado.edu/en/simulations/category/physics PhET]
* OpenStax intro physics textbooks: [https://openstax.org/details/books/university-physics-volume-1 Vol1], [https://openstax.org/details/books/university-physics-volume-2 Vol2], [https://openstax.org/details/books/university-physics-volume-3 Vol3]
* The Open Source Physics project is a collection of online physics resources [http://www.opensourcephysics.org/ OSP]
* A resource guide compiled by the [http://www.aapt.org/ AAPT] for educators [http://www.compadre.org/ ComPADRE]
* The Feynman lectures on physics are free to read [http://www.feynmanlectures.caltech.edu/ Feynman]
* Final Study Guide for Modern Physics II created by a lab TA [https://docs.google.com/document/d/1_6GktDPq5tiNFFYs_ZjgjxBAWVQYaXp_2Imha4_nSyc/edit?usp=sharing Modern Physics II Final Study Guide]

== Resources ==
* Commonly used wiki commands [https://en.wikipedia.org/wiki/Help:Cheatsheet Wiki Cheatsheet]
* A guide to representing equations in math mode [https://en.wikipedia.org/wiki/Help:Displaying_a_formula Wiki Math Mode]
* A page to keep track of all the physics [[Constants]]
* A listing of [[Notable Scientist]] with links to their individual pages

<div style="float:left; width:30%; padding:1%;">

==Physics 1==
===Week 1===
<div class="toccolours mw-collapsible mw-collapsed">
====GlowScript 101====
<div class="mw-collapsible-content">
*[[Python Syntax]]
*[[GlowScript]]
</div>
</div>

<div class="toccolours mw-collapsible mw-collapsed">

====VPython====

<div class="mw-collapsible-content">
*[[VPython]]
*[[VPython basics]]
*[[VPython Common Errors and Troubleshooting]]
*[[VPython Functions]]
*[[VPython Lists]]
*[[VPython Loops]]
*[[VPython Multithreading]]
*[[VPython Animation]]
*[[VPython Objects]]
*[[VPython 3D Objects]]
*[[VPython Reference]]
*[[VPython MapReduceFilter]]
*[[VPython GUIs]]
</div>
</div>

<div class="toccolours mw-collapsible mw-collapsed">

====Vectors and Units====
<div class="mw-collapsible-content">
*[[Vectors]]
*[[SI Units]]
</div>
</div>

<div class="toccolours mw-collapsible mw-collapsed">

====Interactions====
<div class="mw-collapsible-content">
*[[Types of Interactions and How to Detect Them]]
</div>
</div>

<div class="toccolours mw-collapsible mw-collapsed">
====Velocity and Momentum====
<div class="mw-collapsible-content">
*[[Newton's First Law of Motion]]
*[[Mass]]
*[[Velocity]]
*[[Speed]]
*[[Speed vs Velocity]]
*[[Relative Velocity]]
*[[Derivation of Average Velocity]]
*[[2-Dimensional Motion]]
*[[3-Dimensional Position and Motion]]
</div>
</div>

===Week 2===
<div class="toccolours mw-collapsible mw-collapsed">
====Momentum and the Momentum Principle====
<div class="mw-collapsible-content">
*[[Linear Momentum]]
*[[Newton's Second Law: the Momentum Principle]]
*[[Impulse and Momentum]]
*[[Net Force]]
*[[Inertia]]
*[[Acceleration]]
*[[Relativistic Momentum]]

</div>
</div>

<div class="toccolours mw-collapsible mw-collapsed">

====Iterative Prediction with a Constant Force====
<div class="mw-collapsible-content">
*[[Iterative Prediction]]
</div>
</div>

===Week 3===
<div class="toccolours mw-collapsible mw-collapsed">

====Analytic Prediction with a Constant Force====
<div class="mw-collapsible-content">

*[[Kinematics]]
*[[Projectile Motion]]
</div>
</div>

<div class="toccolours mw-collapsible mw-collapsed">

====Iterative Prediction with a Varying Force====
<div class="mw-collapsible-content">
*[[Fundamentals of Iterative Prediction with Varying Force]]
*[[Spring_Force]]
*[[Simple Harmonic Motion]]


*[[Iterative Prediction of Spring-Mass System]]
*[[Terminal Speed]]
*[[Predicting Change in multiple dimensions]]
*[[Two Dimensional Harmonic Motion]]
*[[Determinism]]
</div>
</div>

===Week 4===
<div class="toccolours mw-collapsible mw-collapsed">
====Fundamental Interactions====
<div class="mw-collapsible-content">
*[[Gravitational Force]]
*[[Gravitational Force Near Earth]]
*[[Gravitational Force in Space and Other Applications]]
*[[3 or More Body Interactions]]

*[[Electric Force]]
*[[Introduction to Magnetic Force]]
*[[Strong and Weak Force]]
*[[Reciprocity]]
*[[Conservation of Momentum]]
</div>
</div>

===Week 5===
<div class="toccolours mw-collapsible mw-collapsed">
====Properties of Matter====
<div class="mw-collapsible-content">
*[[Kinds of Matter]]
*[[Ball and Spring Model of Matter]]
*[[Density]]
*[[Length and Stiffness of an Interatomic Bond]]
*[[Young's Modulus]]
*[[Speed of Sound in Solids]]
*[[Malleability]]
*[[Ductility]]
*[[Weight]]
*[[Hardness]]
*[[Boiling Point]]
*[[Melting Point]]
*[[Change of State]]
</div>
</div>

===Week 6===
<div class="toccolours mw-collapsible mw-collapsed">
====Identifying Forces====
<div class="mw-collapsible-content">
*[[Free Body Diagram]]
*[[Inclined Plane]]
*[[Compression or Normal Force]]
*[[Tension]]
</div>
</div>
<div class="toccolours mw-collapsible mw-collapsed">

====Curving Motion====
<div class="mw-collapsible-content">
*[[Curving Motion]]
*[[Centripetal Force and Curving Motion]]
*[[Perpetual Freefall (Orbit)]]
</div>
</div>

===Week 7===
<div class="toccolours mw-collapsible mw-collapsed">
====Energy Principle====
<div class="mw-collapsible-content">
*[[Energy of a Single Particle]]
*[[Kinetic Energy]]
*[[Work/Energy]]
*[[The Energy Principle]]
*[[Conservation of Energy]]
</div>
</div>

===Week 8===
<div class="toccolours mw-collapsible mw-collapsed">
====Work by Non-Constant Forces====
<div class="mw-collapsible-content">
*[[Work Done By A Nonconstant Force]]
</div>
</div>
<div class="toccolours mw-collapsible mw-collapsed">
====Potential Energy====
<div class="mw-collapsible-content">
*[[Potential Energy]]
*[[Potential Energy of Macroscopic Springs]]
*[[Spring Potential Energy]]
*[[Ball and Spring Model]]
*[[Gravitational Potential Energy]]
*[[Energy Graphs]]
*[[Escape Velocity]]
</div>
</div>

===Week 9===
<div class="toccolours mw-collapsible mw-collapsed">
====Multiparticle Systems====
<div class="mw-collapsible-content">
*[[Center of Mass]]
*[[Multi-particle analysis of Momentum]]
*[[Potential Energy of a Multiparticle System]]
*[[Work and Energy for an Extended System]]
*[[Internal Energy]]
**[[Potential Energy of a Pair of Neutral Atoms]]
</div>
</div>

===Week 10===
<div class="toccolours mw-collapsible mw-collapsed">
====Choice of System====
<div class="mw-collapsible-content">
*[[System & Surroundings]]
</div>
</div>
<div class="toccolours mw-collapsible mw-collapsed">
====Thermal Energy, Dissipation, and Transfer of Energy====
<div class="mw-collapsible-content">
*[[Thermal Energy]]
*[[Specific Heat]]
*[[Calorific Value(Heat of combustion)]]
*[[First Law of Thermodynamics]]
*[[Second Law of Thermodynamics and Entropy]]
*[[Temperature]]
*[[Transformation of Energy]]
*[[The Maxwell-Boltzmann Distribution]]
*[[Air Resistance]]
*[[The Third Law of Thermodynamics]]
</div>
</div>
<div class="toccolours mw-collapsible mw-collapsed">

====Rotational and Vibrational Energy====
<div class="mw-collapsible-content">
*[[Translational, Rotational and Vibrational Energy]]
*[[Rolling Motion]]
</div>
</div>

===Week 11===
<div class="toccolours mw-collapsible mw-collapsed">
====Different Models of a System====
<div class="mw-collapsible-content">
*[[Point Particle Systems]]
*[[Real Systems]]
</div>
</div>
<div class="toccolours mw-collapsible mw-collapsed">
====Friction====
<div class="mw-collapsible-content">
*[[Friction]]
*[[Static Friction]]
*[[Kinetic Friction]]
</div>
</div>

===Week 12===
<div class="toccolours mw-collapsible mw-collapsed">
====Conservation of Momentum====
<div class="mw-collapsible-content">
*[[Conservation of Momentum]]
</div>
</div>
<div class="toccolours mw-collapsible mw-collapsed">
====Collisions====
<div class="mw-collapsible-content">
*[[Newton's Third Law of Motion]]
*[[Collisions]]
*[[Elastic Collisions]]
*[[Inelastic Collisions]]
*[[Maximally Inelastic Collision]]
*[[Head-on Collision of Equal Masses]]
*[[Head-on Collision of Unequal Masses]]
*[[Scattering: Collisions in 2D and 3D]]
*[[Rutherford Experiment and Atomic Collisions]]
*[[Coefficient of Restitution]]
</div>
</div>

===Week 13===
<div class="toccolours mw-collapsible mw-collapsed">
====Rotations====
<div class="mw-collapsible-content">
*[[Rotational Kinematics]]
*[[Eulerian Angles]]
</div>
</div>
<div class="toccolours mw-collapsible mw-collapsed">

====Angular Momentum====
<div class="mw-collapsible-content">
*[[Total Angular Momentum]]
*[[Translational Angular Momentum]]
*[[Rotational Angular Momentum]]
*[[The Angular Momentum Principle]]
*[[Angular Impulse]]
*[[Predicting the Position of a Rotating System]]
*[[The Moments of Inertia]]
*[[Right Hand Rule]]
</div>
</div>

===Week 14===
<div class="toccolours mw-collapsible mw-collapsed">
====Analyzing Motion with and without Torque====
<div class="mw-collapsible-content">
*[[Torque]]
*[[Torque 2]]
*[[Systems with Zero Torque]]
*[[Systems with Nonzero Torque]]
*[[Torque vs Work]]
*[[Gyroscopes]]
</div>
</div>

===Week 15===
<div class="toccolours mw-collapsible mw-collapsed">
====Introduction to Quantum Concepts====
<div class="mw-collapsible-content">
*[[Bohr Model]]
*[[Energy graphs and the Bohr model]]
*[[Quantized energy levels]]
*[[Electron transitions]]
*[[Entropy]]
</div>
</div>
</div>

<div style="float:left; width:30%; padding:1%;">

==Physics 2==
===Week 1===
<div class="toccolours mw-collapsible mw-collapsed">
====3D Vectors====

<div class="mw-collapsible-content">
*[[Vectors]]
*[[Right-Hand Rule]]
*[[Right Hand Rule]]
</div>
</div>

<div class="toccolours mw-collapsible mw-collapsed">

====Electric field====
<div class="mw-collapsible-content">
*[[Electric Field]]
*[[Electric Field and Electric Potential]]
</div>
</div>

<div class="toccolours mw-collapsible mw-collapsed">

====Electric force====
<div class="mw-collapsible-content">
*[[Electric Force]]
*[[Lorentz Force]]
</div>
</div>

<div class="toccolours mw-collapsible mw-collapsed">

====Electric field of a point particle====
<div class="mw-collapsible-content">
*[[Point Charge]]
</div>
</div>

<div class="toccolours mw-collapsible mw-collapsed">

====Superposition====
<div class="mw-collapsible-content">
*[[Superposition Principle]]
*[[Superposition principle]]
</div>
</div>

<div class="toccolours mw-collapsible mw-collapsed">
====Dipoles====
<div class="mw-collapsible-content">
*[[Electric Dipole]]
*[[Magnetic Dipole]]
</div>
</div>

===Week 2===
<div class="toccolours mw-collapsible mw-collapsed">
====Interactions of charged objects====
<div class="mw-collapsible-content">
*[[Electric Field]]
*[[Electric Potential]]
*[[Electric Force]]
*[[Lorentz Force]]
</div>
</div>

<div class="toccolours mw-collapsible mw-collapsed">
====Tape experiments====
<div class="mw-collapsible-content">
*[[Polarization]]
*[[Electric Polarization]]
</div>
</div>

<div class="toccolours mw-collapsible mw-collapsed">
====Polarization====
<div class="mw-collapsible-content">
*[[Polarization]]
*[[Electric Polarization]]
*[[Polarization of an Atom]]
</div>
</div>

===Week 3===
<div class="toccolours mw-collapsible mw-collapsed">
====Conductors and Insulators====
<div class="mw-collapsible-content">
*[[Conductivity and Resistivity]]
*[[Insulators]]
*[[Potential Difference in an Insulator]]
*[[Conductors]]
*[[Polarization of a conductor]]
</div>
</div>

<div class="toccolours mw-collapsible mw-collapsed">

====Charging and Discharging====
<div class="mw-collapsible-content">
*[[Charge Transfer]]
*[[Electrostatic Discharge]]
*[[Charged Conductor and Charged Insulator]]
</div>
</div>

===Week 4===
<div class="toccolours mw-collapsible mw-collapsed">
====Field of a charged rod====
<div class="mw-collapsible-content">
*[[Field of a Charged Rod|Charged Rod]]
</div>
</div>

<div class="toccolours mw-collapsible mw-collapsed">

====Field of a charged ring/disk/capacitor====
<div class="mw-collapsible-content">
*[[Charged Ring]]
*[[Charged Disk]]
*[[Charged Capacitor]]
</div>
</div>

<div class="toccolours mw-collapsible mw-collapsed">
====Field of a charged sphere====
<div class="mw-collapsible-content">
*[[Charged Spherical Shell]]
*[[Field of a Charged Ball]]
</div>
</div>

===Week 5===
<div class="toccolours mw-collapsible mw-collapsed">
====Potential energy====
<div class="mw-collapsible-content">
*[[Potential Energy]]
</div>
</div>

<div class="toccolours mw-collapsible mw-collapsed">

====Electric potential====
<div class="mw-collapsible-content">
*[[Electric Potential]]
*[[Path Independence of Electric Potential]]
*[[Potential Difference Path Independence, claimed by Aditya Mohile]]
*[[Potential Difference in a Uniform Field]]
*[[Potential Difference of Point Charge in a Non-Uniform Field]]
</div>
</div>
<div class="toccolours mw-collapsible mw-collapsed">
====Sign of a potential difference====
<div class="mw-collapsible-content">
*[[Sign of a Potential Difference]]
</div>
</div>
<div class="toccolours mw-collapsible mw-collapsed">
====Potential at a single location====
<div class="mw-collapsible-content">
*[[Electric Potential]]
*[[Potential Difference at One Location]]
</div>
</div>

<div class="toccolours mw-collapsible mw-collapsed">

====Path independence and round trip potential====
<div class="mw-collapsible-content">
*[[Path Independence of Electric Potential]]
*[[Potential Difference Path Independence, claimed by Aditya Mohile]]
</div>
</div>

===Week 6===
<div class="toccolours mw-collapsible mw-collapsed">
====Electric field and potential in an insulator====
<div class="mw-collapsible-content">
*[[Potential Difference in an Insulator]]
*[[Electric Field in an Insulator]]
</div>
</div>
<div class="toccolours mw-collapsible mw-collapsed">
====Moving charges in a magnetic field====
<div class="mw-collapsible-content">
*[[Magnetic Field]]
*[[Magnetic Force]]
*[[Lorentz Force]]
</div>
</div>
<div class="toccolours mw-collapsible mw-collapsed">
====Biot-Savart Law====
<div class="mw-collapsible-content">
*[[Biot-Savart Law]]
*[[Biot-Savart Law for Currents]]
</div>
</div>
<div class="toccolours mw-collapsible mw-collapsed">

====Moving charges, electron current, and conventional current====
<div class="mw-collapsible-content">
*[[Moving Point Charge]]
*[[Current]]
</div>
</div>

===Week 7===
<div class="toccolours mw-collapsible mw-collapsed">
====Magnetic field of a wire====
<div class="mw-collapsible-content">
*[[Magnetic Field of a Long Straight Wire]]
</div>
</div>

<div class="toccolours mw-collapsible mw-collapsed">

====Magnetic field of a current-carrying loop====
<div class="mw-collapsible-content">
*[[Magnetic Field of a Loop]]
</div>
</div>

<div class="toccolours mw-collapsible mw-collapsed">
====Magnetic field of a Charged Disk====
<div class="mw-collapsible-content">
*[[Magnetic Field of a Disk]]
</div>
</div>

<div class="toccolours mw-collapsible mw-collapsed">
====Magnetic dipoles====
<div class="mw-collapsible-content">
*[[Magnetic Dipole Moment]]
*[[Bar Magnet]]
</div>
</div>

<div class="toccolours mw-collapsible mw-collapsed">
====Atomic structure of magnets====
<div class="mw-collapsible-content">
*[[Atomic Structure of Magnets]]
</div>
</div>

===Week 8===
<div class="toccolours mw-collapsible mw-collapsed">
====Steady state current====
<div class="mw-collapsible-content">
*[[Steady State]]
*[[Non Steady State]]
</div>
</div>

<div class="toccolours mw-collapsible mw-collapsed">
====Kirchoff's Laws====
<div class="mw-collapsible-content">
*[[Kirchoff's Laws]]
</div>
</div>

<div class="toccolours mw-collapsible mw-collapsed">
====Electric fields and energy in circuits====
<div class="mw-collapsible-content">
*[[Electric Potential Difference]]
</div>
</div>

<div class="toccolours mw-collapsible mw-collapsed">

====Macroscopic analysis of circuits====
<div class="mw-collapsible-content">
*[[Series Circuits]]
*[[Parallel Circuits]]
*[[Parallel Circuits vs. Series Circuits*]]
*[[Loop Rule]]
*[[Node Rule]]
*[[Fundamentals of Resistance]]
*[[Problem Solving]]
</div>
</div>

===Week 9===
<div class="toccolours mw-collapsible mw-collapsed">
====Electric field and potential in circuits with capacitors====
<div class="mw-collapsible-content">
*[[Charging and Discharging a Capacitor]]
*[[RC Circuit]]
*[[R Circuit]]
*[[AC and DC]]
</div>
</div>

<div class="toccolours mw-collapsible mw-collapsed">
====Magnetic forces on charges and currents====
<div class="mw-collapsible-content">
*[[Magnetic Force]]
*[[Lorentz Force]]
*[[Motors and Generators]]
*[[Applying Magnetic Force to Currents]]
*[[Magnetic Force in a Moving Reference Frame]]
*[[Right-Hand Rule]]
*[[Analysis of Railgun vs Coil gun technologies]]
</div>
</div>

<div class="toccolours mw-collapsible mw-collapsed">

====Electric and magnetic forces====
<div class="mw-collapsible-content">
*[[Electric Force]]
*[[Magnetic Force]]
*[[Lorentz Force]]
*[[VPython Modelling of Electric and Magnetic Forces]]
</div>
</div>

<div class="toccolours mw-collapsible mw-collapsed">

====Velocity selector====
<div class="mw-collapsible-content">
*[[Lorentz Force]]
*[[Combining Electric and Magnetic Forces]]
</div>
</div>

===Week 10===
<div class="toccolours mw-collapsible mw-collapsed">

====Hall Effect====
<div class="mw-collapsible-content">
*[[Hall Effect]]
<h1><strong>Alayna Baker Spring 2020</strong></h1>
[[File:Hall Effect 1.jpg]]
[[File:Hall Effect 2.jpg]]

*[[Right-Hand Rule]]
*[[Motional Emf]]
*[[Magnetic Force]]
*[[Magnetic Torque]]
</div>
</div>

<div class="toccolours mw-collapsible mw-collapsed">

]]]====Motional EMF====

<div class="mw-collapsible-content">
*[[Motional Emf]]
<h1><strong>Adeline Boswell Fall 2019</strong></h1>
[[File:Motional EMF Example.jpg]]

*[[Motional Emf using Faraday's Law]]
</div>
</div>http://www.physicsbook.gatech.edu/Special:RecentChangesLinked/Main_Page

<div class="toccolours mw-collapsible mw-collapsed">

If you have a bar attached to two rails, and the rails are connected by a resistor, you have effectively created a circuit. As the bar moves, it creates an "electromotive force"

[[File:MotEMFCR.jpg]]

====Magnetic force====
<div class="mw-collapsible-content">
*[[Magnetic Force]]
*[[Lorentz Force]]
</div>
</div>

<div class="toccolours mw-collapsible mw-collapsed">
====Magnetic torque====
<div class="mw-collapsible-content">
*[[Magnetic Torque]]
*[[Right-Hand Rule]]
</div>
</div>

===Week 12===

<div class="toccolours mw-collapsible mw-collapsed">
====Gauss's Law====
<div class="mw-collapsible-content">
*[[Gauss's Flux Theorem]]
*[[Gauss's Law]]
*[[Magnetic Flux]]
</div>
</div>

<div class="toccolours mw-collapsible mw-collapsed">
====Ampere's Law====
<div class="mw-collapsible-content">
*[[Ampere's Law]]
*[[Ampere-Maxwell Law]]
*[[Magnetic Field of Coaxial Cable Using Ampere's Law]]
*[[Magnetic Field of a Long Thick Wire Using Ampere's Law]]
*[[Magnetic Field of a Toroid Using Ampere's Law]]
*[[Magnetic Field of a Solenoid Using Ampere's Law]]
*[[The Differential Form of Ampere's Law]]
</div>
</div>

===Week 13===
<div class="toccolours mw-collapsible mw-collapsed">
====Semiconductors====
<div class="mw-collapsible-content">
*[[Semiconductor Devices]]
</div>
</div>

<div class="toccolours mw-collapsible mw-collapsed">
====Faraday's Law====
<div class="mw-collapsible-content">
*[[Faraday's Law]]
*[[Motional Emf using Faraday's Law]]
*[[Lenz's Law]]

</div>
</div>

<div class="toccolours mw-collapsible mw-collapsed">

====Maxwell's equations====
<div class="mw-collapsible-content">
*[[Gauss's Law]]
*[[Magnetic Flux]]
*[[Ampere's Law]]
*[[Faraday's Law]]
*[[Maxwell's Electromagnetic Theory]]
</div>
</div>

===Week 14===
<div class="toccolours mw-collapsible mw-collapsed">
====Circuits revisited====
<div class="mw-collapsible-content">

</div>
</div>

<div class="toccolours mw-collapsible mw-collapsed">

====Inductors====
<div class="mw-collapsible-content">
*[[Inductors]]
*[[Current in an LC Circuit]]
*[[Current in an RL Circuit]]
</div>
</div>

===Week 15===
<div class="toccolours mw-collapsible mw-collapsed">
==== Electromagnetic Radiation ====
<div class="mw-collapsible-content">
*[[Electromagnetic Radiation]]
</div>
</div>

<div class="toccolours mw-collapsible mw-collapsed">
====Sparks in the air====
<div class="mw-collapsible-content">
*[[Sparks in Air]]
*[[Spark Plugs]]
</div>
</div>

<div class="toccolours mw-collapsible mw-collapsed">
====Superconductors====
<div class="mw-collapsible-content">
*[[Superconducters]]
*[[Superconductors]]
*[[Meissner effect]]
</div>
</div>
</div>

<div style="float:left; width:30%; padding:1%;">

==Physics 3==

===Week 1===
<div class="toccolours mw-collapsible mw-collapsed">
====Classical Physics====
<div class="mw-collapsible-content">
</div>
</div>

===Week 2===
<div class="toccolours mw-collapsible mw-collapsed">
====Special Relativity====
<div class="mw-collapsible-content">
*[[Frame of Reference]]
*[[Einstein's Theory of Special Relativity]]
*[[Time Dilation]]
*[[Einstein's Theory of General Relativity]]
*[[Albert A. Micheleson & Edward W. Morley]]
*[[Magnetic Force in a Moving Reference Frame]]
</div>
</div>

===Week 3===
<div class="toccolours mw-collapsible mw-collapsed">
====Photons====
<div class="mw-collapsible-content">
*[[Spontaneous Photon Emission]]
*[[Light Scattering: Why is the Sky Blue]]
*[[Lasers]]
*[[Electronic Energy Levels and Photons]]
*[[Quantum Properties of Light]]
*[[The Photoelectric Effect]]
</div>
</div>

===Week 4===
<div class="toccolours mw-collapsible mw-collapsed">
====Matter Waves====
<div class="mw-collapsible-content">
*[[Wave-Particle Duality]]
*[[Particle in a 1-Dimensional box]]
*[[Heisenberg Uncertainty Principle]]
</div>
</div>

<div class="toccolours mw-collapsible mw-collapsed">

====Schrödinger Equation====
<div class="mw-collapsible-content">
*[[Solution for a Single Free Particle]]
<div class="mw-collapsible-content">
*[[Solution for a Single Particle in an Infinite Quantum Well - Darin]]
<div class="mw-collapsible-content"></div>
*[[Solution for a Single Particle in a Semi-Infinite Quantum Well]]
<div class="mw-collapsible-content"></div>
*[[Solution for Simple Harmonic Oscillator (Xuen Zhen)]]

===Week 5===
<div class="toccolours mw-collapsible mw-collapsed">
====Wave Mechanics====
<div class="mw-collapsible-content">
*[[Standing Waves]]
*[[Wavelength]]
*[[Wavelength and Frequency]]
*[[Mechanical Waves]]
*[[Transverse and Longitudinal Waves]]
</div>
</div>

<div class="toccolours mw-collapsible mw-collapsed">
====Quantum Mechanics====
<div class="mw-collapsible-content">
*[[Quantum Tunneling through Potential Barriers]]
</div>
</div>

===Week 6===
<div class="toccolours mw-collapsible mw-collapsed">
====Rutherford-Bohr Model====
<div class="mw-collapsible-content">
*[[Rutherford Experiment and Atomic Collisions]]
*[[Bohr Model]]
*[[Quantized energy levels]]
*[[Energy graphs and the Bohr model]]
</div>
</div>

===Week 7===
<div class="toccolours mw-collapsible mw-collapsed">
====The Hydrogen Atom====
<div class="mw-collapsible-content">
*[[Quantum Theory]]
*[[Atomic Theory]]
</div>
</div>

===Week 8===
<div class="toccolours mw-collapsible mw-collapsed">
====Many-Electron Atoms====
<div class="mw-collapsible-content">
*[[Quantum Theory]]
*[[Atomic Theory]]
*[[Pauli exclusion principle]]
</div>
</div>

===Week 9===
<div class="toccolours mw-collapsible mw-collapsed">
====Molecules====
<div class="mw-collapsible-content">
</div>
*[[Molecules]]
</div>

===Week 10===
<div class="toccolours mw-collapsible mw-collapsed">
====Statistical Physics====
<div class="mw-collapsible-content">
</div>
*[[Application of Statistics in Physics]]
*[[Temperature & Entropy]]
</div>
<div class="mw-collapsible-content">

===Week 11===
<div class="toccolours mw-collapsible mw-collapsed">
====Condensed Matter Physics====
<div class="mw-collapsible-content">
</div>
</div>

===Week 12===
<div class="toccolours mw-collapsible mw-collapsed">
====The Nucleus====
<div class="mw-collapsible-content">
*[[Nucleus]]
</div>
</div>

===Week 13===
<div class="toccolours mw-collapsible mw-collapsed">
====Nuclear Physics====
<div class="mw-collapsible-content">
*[[Nuclear Fission]]
*[[Nuclear Energy from Fission and Fusion]]
</div>
</div>

===Week 14===
<div class="toccolours mw-collapsible mw-collapsed">
====Particle Physics====
<div class="mw-collapsible-content">
*[[Elementary Particles and Particle Physics Theory]]
*[[String Theory]]
</div>
</div>
</div>