Physics Book - User contributions [en]

Entropy

2024-12-03T16:52:59Z

Kparthas8: /* Connectedness */

Author: Keshava Parthasarathy (Fall 2024)

The colloquial definition of entropy is "the degree of disorder or randomness in the system" (Merriam). The scientific definition, while significantly more technical, is also significantly less vague. Put simply entropy is a measure of the number of ways to distribute energy to one or more systems, the more ways to distribute the energy the more entropy a system has.

[[File:Energy_wells_visualization.jpg|thumb|left| Dots represent energy quanta and are distributed among 3 wells representing the 3 ways an atom can store energy. Ω = 6]]A system in this case is a particle or group of particles, usually atoms, and the energy is distributed in specific quanta denoted by q (See mathematical model for details calculating q for different atoms). How can energy be distributed to a single atom? The atom can store energy in discrete levels in each of the spatial directions it can move, namely the x, y, and z directions. It is helpful to visualize this phenomena by picturing every atom to have three bottomless wells and each quanta of energy as being a ball that is sorted into these wells. Thus an atom can store one quanta of energy in three ways, with one direction or well having all the energy and the other two having none. Furthermore, there are 6 ways of distributing 2 quanta to one atom, 3 distributions with 1 well having all the energy, as well as 3 where 2 wells each have 1 quanta of energy. (pictured left)

One system can consist of more than one atom. For example, there are 6 possible distributions of 1 quanta of energy to 2 atoms or 21 possible distributions of 2 quanta to 2 atoms. The number of possible distributions rises rapidly with larger energies and more atoms. The quanta available for a system or group of systems is known as the macrostate, and each possible distribution of that energy through the system or group of systems is a microstate. In the example pictured left, q = 2 is the macrostate and each of the 6 possible distributions are the system's microstates.

The number of possible distributions for a certain quanta of energy q, is denoted by Ω (see mathematical model for how to calculate Ω). To find the number of ways energy can be distributed to 2 systems, simply multiply each system's respective value for Ω. Such that, <math> Ω_{Total} = Ω_1*Ω_2 </math>.

The Fundamental Assumption of Statistical Mechanics states over time every single microstate of an isolated system is equally likely. Thus if you were to observe 2 quanta distributed to 2 atoms, you are equally likely to observe any of its 21 possible microstates. While this idea is called an assumption it also agrees with experimental results. This principle is very useful when calculating the probability of different divisions of energy. For example, if 4 quanta of energy are shared between two atoms the distribution is as follows.

[[File:Energy_distribution.png|thumb|left]] There are 36 different microstates in which the 4 quanta of energy are shared evenly between the the two atoms and 126 different microstates total. Given the Fundamental Assumption of Statistical Mechanics the most probably distribution is simply the distribution with the most microstates. As an even split has the most microstates, it is the most probable distribution with 29% of the microstates have an even split, corresponding to a 29% chance of finding the energy distributed this way.

As the total number of quanta and atoms in the two systems increases, the number of microstates balloons in size at a nearly comical rate. When the number of quanta, as well as the number of atoms both systems have reaches the 100's, the total number of microstates is comfortably above <math> 10^{100} </math>. Additionally, as these numbers swell, the difference between the probability of the most likely distribution and any others increases dramatically. This means that at the massive quantities of atoms and quanta typical of macroscopic objects, the most probable distribution is essentially the only possible distribution. (This occurs for systems larger than <math> 10^{20} </math> atoms)

Because of the massive disparity in the number of microstates for different distributions of energy among two systems, it's not convenient to use Ω by itself. For ease of calculation, Ω is placed in a natural logarithm (ln) and additionally multiplied by the constant <math> k_B </math> to measure entropy. Thus entropy (denoted S) of a system consisting of two objects, <math>S_{Total} = k_Bln(Ω_1*Ω_2)</math>.

The Second Law of Thermodynamics states that in closed systems, when not in equilibrium the most probable outcome is entropy will increase. This fact helps predict the behavior and distribution of energy between systems. Imagine a lot of energy is distributed to a system of few atoms (system 1) and little energy is distributed to a very large adjacent system of many atoms (system 2). How will the energy distribute itself over time? Because having many atoms allows for more ways for the energy in system 2 contains to be distributed, energy moving from system 1 to system 2 increases <math> Ω_2 </math> more than <math> Ω_1 </math> decreases. This increases <math> Ω_{Total} </math> and thus increases the total entropy of the two systems. Because this and only this scenario follows the Second Law of Thermodynamics, we can conclude that energy will flow from system 1 to system 2 until maximum entropy can be achieved.

The study of microscopic distributions of energy is extremely useful in explaining macroscopic phenomena, from a bouncing rubber ball to the temperatures of 2 adjacent objects. Temperature can be considered a function of the average energy per molecule of an object. This strongly relates to the concept of entropy and the distribution of energy throughout a system. Temperature (in Kelvin) is defined as being inversely proportional to the rate of change of entropy with respect to its energy. This definition leads to an interesting analysis of heat capacity, or the ratio of internal energy change of an object and the resulting change in temperature. When this analysis is done on single atoms the ratio is called specific heat.

Entropy is a core concept in physics, chemistry, and information theory that quantifies the level of disorder, randomness, or uncertainty within a system. It is a cornerstone of thermodynamics, statistical mechanics, and communication science, and it provides insight into natural processes, energy transformations, and information processing. The study of entropy extends to diverse fields, including cosmology, data science, and quantum mechanics, where it continues to reveal fundamental truths about the universe.

Entropy can be broadly understood in two key contexts:

Thermodynamic Entropy: A measure of energy dispersion or unavailable energy within a system.
Information Entropy: A metric of uncertainty or information content in datasets, introduced by Claude Shannon.

Entropy is a measure of disorder, randomness, or the number of ways a system's energy can be distributed. Scientifically, entropy quantifies the number of possible microstates (specific configurations) that correspond to a macrostate (observable state). Higher entropy corresponds to greater energy dispersal and higher disorder.

[[File:Energy_wells_visualization.jpg|thumb|left|Representation of energy quanta distributed among three wells, depicting the three spatial directions an atom can store energy.
Ω
=
6
Ω=6.]]

A system typically refers to particles (e.g., atoms) that store energy in quantized units, called quanta (<math>q</math>). Each atom has discrete energy levels in three spatial directions: <math>x</math>, <math>y</math>, and <math>z</math>. Imagine an atom as having three "wells," each capable of storing energy balls. For instance:

1 quanta of energy can be stored in three ways (one well holds the ball, while the other two are empty).
2 quanta can be distributed in <math>\Omega = 6</math> ways across the wells.
This idea extends to multiple atoms. For example:

1 quanta shared between 2 atoms has <math>\Omega = 6</math> distributions.
2 quanta shared between 2 atoms has <math>\Omega = 21</math> distributions.
The number of possible configurations for distributing energy is denoted <math>\Omega</math>.

=== Microstates and Macrostates ===

A macrostate represents the total energy (<math>q</math>) and number of particles in a system.
A microstate represents one specific way the energy is distributed.
For a system of two subsystems, the total number of microstates is the product of individual subsystem microstates:
<math>\Omega_{\text{Total}} = \Omega_1 \cdot \Omega_2</math>

=== Fundamental Assumption of Statistical Mechanics ===

The Fundamental Assumption of Statistical Mechanics posits that, over time, all microstates of an isolated system are equally likely. Consequently:

The most probable macrostate is the one with the highest number of microstates.
Consider the case of 4 quanta distributed between 2 atoms:

[[File:Energy_distribution.png|thumb|left|Graph showing possible energy distributions and their probabilities.]]
There are <math>\Omega = 36</math> microstates where energy is evenly split, out of <math>\Omega_{\text{Total}} = 126</math> microstates.
The even split is most probable (<math>29%</math>), aligning with experimental data.
As systems grow larger (e.g., <math>q > 100</math> and <math>n > 100</math>), microstate numbers become astronomical (<math>\Omega > 10^{100}</math>). For macroscopic systems (<math>n > 10^{20}</math>), the most probable distribution overwhelmingly dominates.

=== Mathematical Definition of Entropy ===

Entropy (<math>S</math>) provides a practical way to quantify disorder. It is defined using <math>\Omega</math>: <math>S = k_B \ln(\Omega)</math> Where:

<math>k_B</math> is the Boltzmann constant (<math>1.38 \times 10^{-23} , \text{J/K}</math>).
<math>\ln</math> is the natural logarithm.
For two systems: <math>S_{\text{Total}} = k_B \ln(\Omega_1 \cdot \Omega_2) = k_B \ln(\Omega_1) + k_B \ln(\Omega_2)</math>

=== Second Law of Thermodynamics ===

The Second Law of Thermodynamics states that entropy of a closed system tends to increase over time until it reaches equilibrium. Consider two systems:

System 1: Few atoms, high energy.
System 2: Many atoms, low energy.
Energy flows from System 1 to System 2 because <math>\Delta \Omega_2 \gg \Delta \Omega_1</math>, resulting in <math>\Delta S_{\text{Total}} > 0</math>. Entropy increases, and the system approaches equilibrium.

=== Entropy and Temperature ===

Entropy relates closely to temperature (<math>T</math>). Temperature measures energy distribution across a system's microstates and is defined as: <math>\frac{1}{T} = \frac{\partial S}{\partial U}</math> Where <math>U</math> is the internal energy.

This relation explains why heat capacity (<math>C = \frac{\Delta U}{\Delta T}</math>) varies across materials and phases. For single atoms, specific heat provides insight into their microscopic energy distribution.

== Applications of Entropy ==

Thermodynamics: Predicting heat flow and energy transformations.
Statistical Mechanics: Explaining macroscopic phenomena like gas diffusion.
Information Theory: Measuring uncertainty in data streams.
Cosmology: Understanding entropy's role in black holes and the universe's evolution.
Quantum Mechanics: Entanglement and quantum information processing.
Entropy bridges the microscopic world of particles to the macroscopic phenomena we observe, revealing profound insights into the nature of energy, order, and time.

[[File:entropyapplication.png|thumb|center|Illustration of energy flow between two systems, increasing total entropy.]]

==A Mathematical Model==

1 quanta of energy or q = 1 is different for every different type of atom and is found using:
*<math> ħ\sqrt{\frac{4*k_{s,i}}{m_a}} </math>
** Where ħ is the Planck constant, <math>k_{s,i}</math> is the interatomic spring stiffness, and <math> m_a</math> is the mass of the atom.
** This value is measured in Joules

The number of microstates for a given macrostate is found using:
* <math> Ω = \frac{(q + N - 1)!}{q!*(N - 1)!} </math>
** Where N the number of energy wells in the system or 3 * # of atoms in the system and ! represents the factorial mathematical function

Entropy is found using:
* <math> S = k_B * ln(Ω) </math>
** Where (The Boltzmann constant) <math> k_B = 1.38 * 10^{-23} </math>

Temperature (in Kelvin) is defined as:
* <math> \frac{1}{T} = \frac{\partial S}{\partial E_{Internal}} </math>
** Where <math> E_{Internal} = ħ\sqrt{\frac{k_{s,i}}{m_a}} </math>
*** This is preferred to <math>\frac{\partial S}{\partial q}</math> because two objects of different materials can have the same q value but be storing different quantities of energy. For the direct comparison this relationship is useful for, universality must be maintained.

Specific heat is found using:
* <math> C = \frac{∆E_{Atom}}{∆T} </math>
**For macroscopic bodies use: <math> C = \frac{∆E_{System}}{∆T*N} </math>
*** Where N is the number of atoms in the system.

==A Computational Model==

How to calculate entropy with given n and q values:

https://trinket.io/embed/glowscript/c24ad7936e

==Examples==

===Simple===
1) How much energy in Joules is stored in a cluster of 8 copper atoms containing 7 energy quanta given that the interatomic spring stiffness for copper is 7 N/m?

The energy within is equal to 7 times the joule value of one copper energy quantum:
* <math> E = 7 * q_{Copper} </math>

Plug in given values:
* <math> E = 7 * ħ \sqrt\frac{4*7}{\frac{.063}{6.022 * 10^{23}}} </math>
** Dividing the atomic mass of copper by Avogadro's number yields the mass of one copper atom

Solve:
* <math> E = 1.208*10^{-20} </math> Joules

__________________________________________________________________________________________________________________________________________________

2) Calculate <math> Ω_{Total} </math> for a system of 2 nanoparticles, one containing 5 energy quanta and 4 atoms and the second containing 3 energy quanta and 6 atoms.

Calculate <math> Ω_1 </math>:
* <math> Ω_1 = \frac{16!}{5!11!} = 4,368 </math>
**For <math> Ω_1, </math> N = 12

Calculate <math> Ω_2 </math>:
* <math> Ω_2 = \frac{20!}{3!17!} = 1,140 </math>
**For <math> Ω_2, </math> N = 18

Calculate <math> Ω_{Total} </math>:
* <math> Ω_{Total} = Ω_1 * Ω_2 = 4,979,520 </math>

===Middling===

For a nanoparticle of 5 lead atoms (<math> k_{s,i} = 5 </math> N/m), what is the approximate temperature when 6 quanta of energy are stored within the nanoparticle?

Calculate the joule value of one quantum of energy for lead:
* <math> E = ħ\sqrt{\frac{4*5}{\frac{.207}{6.022*10^{23}}}} = 8.044 * 10^{-22} </math> Joules
<br/>

Calculate the entropy of the nano particle when it contains 5 and 7 quanta of energy:
* <math> Ω_5 = \frac{19!}{5!14!} = 11628 </math>
* <math> S = k_B * ln(11628) = k_B * 9.361 </math>
<br/>
* <math> Ω_7 = \frac{21!}{7!14!} = 116280 </math>
* <math> S = k_B * ln(116280) = k_B * 11.664 </math>
<br/>

Derive an approximate formula for T:
* <math> \frac{1}{T} = \frac{\partial S}{\partial E} </math>
*<math> T = \frac{\partial E}{\partial S} </math>
*<math> T ≈ \frac{∆E}{∆S} </math>
<br/>
Solve for T:
* <math> T ≈ \frac{2E}{k_B * (ln(Ω_7) - ln(Ω_5))} </math>
* <math> T ≈ 50.621 K </math>
** Because the relation is a derivative, to find T we must find the slope of a hypothetical E vs S graph at the value of 6 energy quanta. The easiest way to do this is to find the average slope of a region containing the value of 6 energy quanta at the center. As we use the region between q = 5 and q = 7, the change in E is equal to 2 energy quanta for lead.

===Difficult===
What is the specific heat of a cluster of 3 copper atoms containing 4 quanta of energy?

Calculate the joule value of one quantum of energy for copper:
* <math> E = ħ\sqrt{\frac{4*7}{\frac{.063}{6.022*10^{23}}}} = 1.725 * 10^{-21} </math> Joules
<br/>

Calculate the entropy of the nano particle when it contains 3, 4 and, 5 quanta of energy:
* <math> Ω_3 = \frac{11!}{3!8!} = 165 </math>
* <math> S = k_B * ln(165) = k_B * 9.361 </math>
<br/>
* <math> Ω_4 = \frac{12!}{4!8!} = 495 </math>
* <math> S = k_B * ln(495) = k_B * 11.664 </math>
<br/>
* <math> Ω_5 = \frac{13!}{5!8!} = 1287 </math>
* <math> S = k_B * ln(1287) = k_B * 11.664 </math>
<br/>

Derive an approximate formula for T:
* <math> \frac{1}{T} = \frac{\partial S}{\partial E} </math>
*<math> T = \frac{\partial E}{\partial S} </math>
*<math> T ≈ \frac{∆E}{∆S} </math>
<br/>

Solve for T at 3.5 and 4.5 quanta of energy:
* <math> T_{3.5} ≈ \frac{E}{k_B * (ln(Ω_4) - ln(Ω_3))} </math>
* <math> T_{3.5} ≈ 113.74 K </math>
<br/>
* <math> T_{4.5} ≈ \frac{E}{k_B * (ln(Ω_5) - ln(Ω_4))} </math>
* <math> T_{4.5} ≈ 130.89 K </math>
<br/>

Solve for specific heat:
* <math> C = \frac{1}{N}\frac{∆E}{∆T} = \frac{1}{3}\frac{E}{T_{4.5} - T_{3.5}} </math>
* <math> C = 3.353 * 10^{-23} </math> J/K

==Connectedness==

In my research I read that entropy is known as time's arrow, which in my opinion is one of the most powerful denotations of a physics term. Entropy is a fundamental law that makes the universe tick and it is such a powerful force that it will (possibly) cause the eventual end of the entire universe. Since entropy is always increases, over the expanse of an obscene amount of time the universe due to entropy will eventually suffer a "heat death" and cease to exist entirely. This is merely a scientific hypothesis, and though it may be gloom, an Asimov supercomputer Multivac may finally solve the Last Question and reboot the entire universe again.

The study of entropy is pertinent to my major as an Industrial Engineer as the whole idea of entropy is statistical thermodynamics. This is very similar to Industrial Engineering as it is essentially a statistical business major. Though the odds are unlikely that entropy will be directly used in the day of the life of an Industrial Engineer, the same distributions and concepts of probability are universal and carry over regardless of whether the example is of thermodynamic or business.

My understanding of quantum computers is no more than a couple of wikipedia articles and youtube videos, but I assume anything along the fields of quantum mechanics, which definitely relates to entropy, is important in making the chips to withstand intense heat transfers, etc.

==Entropy as Time’s Arrow==
Entropy is often referred to as time's arrow, symbolizing the unidirectional flow of time. This concept stems from the second law of thermodynamics, which states that the entropy of an isolated system tends to increase over time. This increase reflects the natural progression from order to disorder, fundamentally governing the universe’s evolution.

==Entropy and the Heat Death of the Universe==
One of the profound implications of entropy is the hypothesis of heat death. Over an incomprehensibly vast timeframe, the universe could reach a state where all energy is evenly distributed, leaving no usable energy for processes that sustain motion or life. This grim scenario would mark the cessation of all thermodynamic activity, leading to a universe in stasis.

This concept has captured imaginations in both science and fiction. For instance, Isaac Asimov's short story The Last Question imagines a future where humanity creates a supercomputer, Multivac, to solve the ultimate problem of entropy and potentially reverse it. While speculative, such ideas highlight humanity’s grappling with entropy’s cosmic implications.

==Industrial Engineering and Entropy==
Entropy’s principles extend beyond physics, finding relevance in fields like Industrial Engineering. Statistical thermodynamics, which deals with probability distributions of energy states, mirrors the statistical methods employed in optimizing industrial processes.

While Industrial Engineers may not directly apply entropy in daily operations, its probabilistic nature resonates in areas like:

Process Optimization: Applying distributions to analyze and improve workflow.
Systems Design: Understanding how disorder and inefficiency can propagate through systems.
Risk Analysis: Using statistical tools akin to thermodynamic concepts to evaluate uncertainty.
This connectedness underscores how foundational scientific principles like entropy influence diverse disciplines.

Entropy and Quantum Computing
Quantum computing, rooted in quantum mechanics, is deeply intertwined with entropy. The manipulation of qubits—quantum units of information—requires precise thermodynamic control to minimize errors caused by heat and energy dissipation. Advancements in quantum technology depend on innovations in:

Thermal Management: Designing materials and chips resistant to entropy-driven heat transfer.
Quantum States Stability: Ensuring coherence in quantum systems despite increasing entropy.
These efforts represent a frontier where the understanding of entropy bridges fundamental physics and cutting-edge engineering.

==History==

Entropy was formally named from Greek en + tropē meaning "transformation content" by German physicist Rudolf Clausius in the 1850's. The study of entropy grew from the observation that energy is always lost to friction and dissipation in engines. Entropy was the formal name given to this lost energy by Clausius when he began formulating the first thermodynamical systems.

The concept of entropy was then expanded on by Ludwig Boltzmann who provided the rigorous mathematical definition used today by framing the question in terms of statistical mechanics. Surprisingly, it was not Boltzmann who incorporated the previously found Boltzmann constant (<math> k_B</math>) into the definition, but rather J. Willard Gibbs.

The study of entropy has been used in numerous applications since its inception. Erwin Schrödinger used the concept of entropy to explain the remarkably low replication error of DNA structures in living beings in his book "What is Life?". Entropy also has numerous parallels in the study of informational theory regarding information lost in transmission and broadcasting.

== See also ==

Here is a list of great resources about entropy that make it easier to understand, and also help expound more on the details of the topic.

===External links===

Great TED-ED on the subject:
*https://ed.ted.com/lessons/what-is-entropy-jeff-phillips

==References==

*http://gallica.bnf.fr/ark:/12148/bpt6k152107/f369.table
*http://www.panspermia.org/seconlaw.htm
*https://ed.ted.com/lessons/what-is-entropy-jeff-phillips
*https://www.merriam-webster.com/dictionary/entropy
*https://www.grc.nasa.gov/www/k-12/airplane/entropy.html
*https://oli.cmu.edu
*Chabay, Ruth W., and Bruce A. Sherwood. Matter & Interactions. John Wiley & Sons, 2015.

[[Category:Which Category did you place this in?]]

Entropy

2024-12-02T17:12:29Z

Kparthas8: /* Applications of Entropy */

Author: Keshava Parthasarathy (Fall 2024)

The colloquial definition of entropy is "the degree of disorder or randomness in the system" (Merriam). The scientific definition, while significantly more technical, is also significantly less vague. Put simply entropy is a measure of the number of ways to distribute energy to one or more systems, the more ways to distribute the energy the more entropy a system has.

[[File:Energy_wells_visualization.jpg|thumb|left| Dots represent energy quanta and are distributed among 3 wells representing the 3 ways an atom can store energy. Ω = 6]]A system in this case is a particle or group of particles, usually atoms, and the energy is distributed in specific quanta denoted by q (See mathematical model for details calculating q for different atoms). How can energy be distributed to a single atom? The atom can store energy in discrete levels in each of the spatial directions it can move, namely the x, y, and z directions. It is helpful to visualize this phenomena by picturing every atom to have three bottomless wells and each quanta of energy as being a ball that is sorted into these wells. Thus an atom can store one quanta of energy in three ways, with one direction or well having all the energy and the other two having none. Furthermore, there are 6 ways of distributing 2 quanta to one atom, 3 distributions with 1 well having all the energy, as well as 3 where 2 wells each have 1 quanta of energy. (pictured left)

One system can consist of more than one atom. For example, there are 6 possible distributions of 1 quanta of energy to 2 atoms or 21 possible distributions of 2 quanta to 2 atoms. The number of possible distributions rises rapidly with larger energies and more atoms. The quanta available for a system or group of systems is known as the macrostate, and each possible distribution of that energy through the system or group of systems is a microstate. In the example pictured left, q = 2 is the macrostate and each of the 6 possible distributions are the system's microstates.

The number of possible distributions for a certain quanta of energy q, is denoted by Ω (see mathematical model for how to calculate Ω). To find the number of ways energy can be distributed to 2 systems, simply multiply each system's respective value for Ω. Such that, <math> Ω_{Total} = Ω_1*Ω_2 </math>.

The Fundamental Assumption of Statistical Mechanics states over time every single microstate of an isolated system is equally likely. Thus if you were to observe 2 quanta distributed to 2 atoms, you are equally likely to observe any of its 21 possible microstates. While this idea is called an assumption it also agrees with experimental results. This principle is very useful when calculating the probability of different divisions of energy. For example, if 4 quanta of energy are shared between two atoms the distribution is as follows.

[[File:Energy_distribution.png|thumb|left]] There are 36 different microstates in which the 4 quanta of energy are shared evenly between the the two atoms and 126 different microstates total. Given the Fundamental Assumption of Statistical Mechanics the most probably distribution is simply the distribution with the most microstates. As an even split has the most microstates, it is the most probable distribution with 29% of the microstates have an even split, corresponding to a 29% chance of finding the energy distributed this way.

As the total number of quanta and atoms in the two systems increases, the number of microstates balloons in size at a nearly comical rate. When the number of quanta, as well as the number of atoms both systems have reaches the 100's, the total number of microstates is comfortably above <math> 10^{100} </math>. Additionally, as these numbers swell, the difference between the probability of the most likely distribution and any others increases dramatically. This means that at the massive quantities of atoms and quanta typical of macroscopic objects, the most probable distribution is essentially the only possible distribution. (This occurs for systems larger than <math> 10^{20} </math> atoms)

Because of the massive disparity in the number of microstates for different distributions of energy among two systems, it's not convenient to use Ω by itself. For ease of calculation, Ω is placed in a natural logarithm (ln) and additionally multiplied by the constant <math> k_B </math> to measure entropy. Thus entropy (denoted S) of a system consisting of two objects, <math>S_{Total} = k_Bln(Ω_1*Ω_2)</math>.

The Second Law of Thermodynamics states that in closed systems, when not in equilibrium the most probable outcome is entropy will increase. This fact helps predict the behavior and distribution of energy between systems. Imagine a lot of energy is distributed to a system of few atoms (system 1) and little energy is distributed to a very large adjacent system of many atoms (system 2). How will the energy distribute itself over time? Because having many atoms allows for more ways for the energy in system 2 contains to be distributed, energy moving from system 1 to system 2 increases <math> Ω_2 </math> more than <math> Ω_1 </math> decreases. This increases <math> Ω_{Total} </math> and thus increases the total entropy of the two systems. Because this and only this scenario follows the Second Law of Thermodynamics, we can conclude that energy will flow from system 1 to system 2 until maximum entropy can be achieved.

The study of microscopic distributions of energy is extremely useful in explaining macroscopic phenomena, from a bouncing rubber ball to the temperatures of 2 adjacent objects. Temperature can be considered a function of the average energy per molecule of an object. This strongly relates to the concept of entropy and the distribution of energy throughout a system. Temperature (in Kelvin) is defined as being inversely proportional to the rate of change of entropy with respect to its energy. This definition leads to an interesting analysis of heat capacity, or the ratio of internal energy change of an object and the resulting change in temperature. When this analysis is done on single atoms the ratio is called specific heat.

Entropy is a core concept in physics, chemistry, and information theory that quantifies the level of disorder, randomness, or uncertainty within a system. It is a cornerstone of thermodynamics, statistical mechanics, and communication science, and it provides insight into natural processes, energy transformations, and information processing. The study of entropy extends to diverse fields, including cosmology, data science, and quantum mechanics, where it continues to reveal fundamental truths about the universe.

Entropy can be broadly understood in two key contexts:

Thermodynamic Entropy: A measure of energy dispersion or unavailable energy within a system.
Information Entropy: A metric of uncertainty or information content in datasets, introduced by Claude Shannon.

Entropy is a measure of disorder, randomness, or the number of ways a system's energy can be distributed. Scientifically, entropy quantifies the number of possible microstates (specific configurations) that correspond to a macrostate (observable state). Higher entropy corresponds to greater energy dispersal and higher disorder.

[[File:Energy_wells_visualization.jpg|thumb|left|Representation of energy quanta distributed among three wells, depicting the three spatial directions an atom can store energy.
Ω
=
6
Ω=6.]]

A system typically refers to particles (e.g., atoms) that store energy in quantized units, called quanta (<math>q</math>). Each atom has discrete energy levels in three spatial directions: <math>x</math>, <math>y</math>, and <math>z</math>. Imagine an atom as having three "wells," each capable of storing energy balls. For instance:

1 quanta of energy can be stored in three ways (one well holds the ball, while the other two are empty).
2 quanta can be distributed in <math>\Omega = 6</math> ways across the wells.
This idea extends to multiple atoms. For example:

1 quanta shared between 2 atoms has <math>\Omega = 6</math> distributions.
2 quanta shared between 2 atoms has <math>\Omega = 21</math> distributions.
The number of possible configurations for distributing energy is denoted <math>\Omega</math>.

=== Microstates and Macrostates ===

A macrostate represents the total energy (<math>q</math>) and number of particles in a system.
A microstate represents one specific way the energy is distributed.
For a system of two subsystems, the total number of microstates is the product of individual subsystem microstates:
<math>\Omega_{\text{Total}} = \Omega_1 \cdot \Omega_2</math>

=== Fundamental Assumption of Statistical Mechanics ===

The Fundamental Assumption of Statistical Mechanics posits that, over time, all microstates of an isolated system are equally likely. Consequently:

The most probable macrostate is the one with the highest number of microstates.
Consider the case of 4 quanta distributed between 2 atoms:

[[File:Energy_distribution.png|thumb|left|Graph showing possible energy distributions and their probabilities.]]
There are <math>\Omega = 36</math> microstates where energy is evenly split, out of <math>\Omega_{\text{Total}} = 126</math> microstates.
The even split is most probable (<math>29%</math>), aligning with experimental data.
As systems grow larger (e.g., <math>q > 100</math> and <math>n > 100</math>), microstate numbers become astronomical (<math>\Omega > 10^{100}</math>). For macroscopic systems (<math>n > 10^{20}</math>), the most probable distribution overwhelmingly dominates.

=== Mathematical Definition of Entropy ===

Entropy (<math>S</math>) provides a practical way to quantify disorder. It is defined using <math>\Omega</math>: <math>S = k_B \ln(\Omega)</math> Where:

<math>k_B</math> is the Boltzmann constant (<math>1.38 \times 10^{-23} , \text{J/K}</math>).
<math>\ln</math> is the natural logarithm.
For two systems: <math>S_{\text{Total}} = k_B \ln(\Omega_1 \cdot \Omega_2) = k_B \ln(\Omega_1) + k_B \ln(\Omega_2)</math>

=== Second Law of Thermodynamics ===

The Second Law of Thermodynamics states that entropy of a closed system tends to increase over time until it reaches equilibrium. Consider two systems:

System 1: Few atoms, high energy.
System 2: Many atoms, low energy.
Energy flows from System 1 to System 2 because <math>\Delta \Omega_2 \gg \Delta \Omega_1</math>, resulting in <math>\Delta S_{\text{Total}} > 0</math>. Entropy increases, and the system approaches equilibrium.

=== Entropy and Temperature ===

Entropy relates closely to temperature (<math>T</math>). Temperature measures energy distribution across a system's microstates and is defined as: <math>\frac{1}{T} = \frac{\partial S}{\partial U}</math> Where <math>U</math> is the internal energy.

This relation explains why heat capacity (<math>C = \frac{\Delta U}{\Delta T}</math>) varies across materials and phases. For single atoms, specific heat provides insight into their microscopic energy distribution.

== Applications of Entropy ==

Thermodynamics: Predicting heat flow and energy transformations.
Statistical Mechanics: Explaining macroscopic phenomena like gas diffusion.
Information Theory: Measuring uncertainty in data streams.
Cosmology: Understanding entropy's role in black holes and the universe's evolution.
Quantum Mechanics: Entanglement and quantum information processing.
Entropy bridges the microscopic world of particles to the macroscopic phenomena we observe, revealing profound insights into the nature of energy, order, and time.

[[File:entropyapplication.png|thumb|center|Illustration of energy flow between two systems, increasing total entropy.]]

==A Mathematical Model==

1 quanta of energy or q = 1 is different for every different type of atom and is found using:
*<math> ħ\sqrt{\frac{4*k_{s,i}}{m_a}} </math>
** Where ħ is the Planck constant, <math>k_{s,i}</math> is the interatomic spring stiffness, and <math> m_a</math> is the mass of the atom.
** This value is measured in Joules

The number of microstates for a given macrostate is found using:
* <math> Ω = \frac{(q + N - 1)!}{q!*(N - 1)!} </math>
** Where N the number of energy wells in the system or 3 * # of atoms in the system and ! represents the factorial mathematical function

Entropy is found using:
* <math> S = k_B * ln(Ω) </math>
** Where (The Boltzmann constant) <math> k_B = 1.38 * 10^{-23} </math>

Temperature (in Kelvin) is defined as:
* <math> \frac{1}{T} = \frac{\partial S}{\partial E_{Internal}} </math>
** Where <math> E_{Internal} = ħ\sqrt{\frac{k_{s,i}}{m_a}} </math>
*** This is preferred to <math>\frac{\partial S}{\partial q}</math> because two objects of different materials can have the same q value but be storing different quantities of energy. For the direct comparison this relationship is useful for, universality must be maintained.

Specific heat is found using:
* <math> C = \frac{∆E_{Atom}}{∆T} </math>
**For macroscopic bodies use: <math> C = \frac{∆E_{System}}{∆T*N} </math>
*** Where N is the number of atoms in the system.

==A Computational Model==

How to calculate entropy with given n and q values:

https://trinket.io/embed/glowscript/c24ad7936e

==Examples==

===Simple===
1) How much energy in Joules is stored in a cluster of 8 copper atoms containing 7 energy quanta given that the interatomic spring stiffness for copper is 7 N/m?

The energy within is equal to 7 times the joule value of one copper energy quantum:
* <math> E = 7 * q_{Copper} </math>

Plug in given values:
* <math> E = 7 * ħ \sqrt\frac{4*7}{\frac{.063}{6.022 * 10^{23}}} </math>
** Dividing the atomic mass of copper by Avogadro's number yields the mass of one copper atom

Solve:
* <math> E = 1.208*10^{-20} </math> Joules

__________________________________________________________________________________________________________________________________________________

2) Calculate <math> Ω_{Total} </math> for a system of 2 nanoparticles, one containing 5 energy quanta and 4 atoms and the second containing 3 energy quanta and 6 atoms.

Calculate <math> Ω_1 </math>:
* <math> Ω_1 = \frac{16!}{5!11!} = 4,368 </math>
**For <math> Ω_1, </math> N = 12

Calculate <math> Ω_2 </math>:
* <math> Ω_2 = \frac{20!}{3!17!} = 1,140 </math>
**For <math> Ω_2, </math> N = 18

Calculate <math> Ω_{Total} </math>:
* <math> Ω_{Total} = Ω_1 * Ω_2 = 4,979,520 </math>

===Middling===

For a nanoparticle of 5 lead atoms (<math> k_{s,i} = 5 </math> N/m), what is the approximate temperature when 6 quanta of energy are stored within the nanoparticle?

Calculate the joule value of one quantum of energy for lead:
* <math> E = ħ\sqrt{\frac{4*5}{\frac{.207}{6.022*10^{23}}}} = 8.044 * 10^{-22} </math> Joules
<br/>

Calculate the entropy of the nano particle when it contains 5 and 7 quanta of energy:
* <math> Ω_5 = \frac{19!}{5!14!} = 11628 </math>
* <math> S = k_B * ln(11628) = k_B * 9.361 </math>
<br/>
* <math> Ω_7 = \frac{21!}{7!14!} = 116280 </math>
* <math> S = k_B * ln(116280) = k_B * 11.664 </math>
<br/>

Derive an approximate formula for T:
* <math> \frac{1}{T} = \frac{\partial S}{\partial E} </math>
*<math> T = \frac{\partial E}{\partial S} </math>
*<math> T ≈ \frac{∆E}{∆S} </math>
<br/>
Solve for T:
* <math> T ≈ \frac{2E}{k_B * (ln(Ω_7) - ln(Ω_5))} </math>
* <math> T ≈ 50.621 K </math>
** Because the relation is a derivative, to find T we must find the slope of a hypothetical E vs S graph at the value of 6 energy quanta. The easiest way to do this is to find the average slope of a region containing the value of 6 energy quanta at the center. As we use the region between q = 5 and q = 7, the change in E is equal to 2 energy quanta for lead.

===Difficult===
What is the specific heat of a cluster of 3 copper atoms containing 4 quanta of energy?

Calculate the joule value of one quantum of energy for copper:
* <math> E = ħ\sqrt{\frac{4*7}{\frac{.063}{6.022*10^{23}}}} = 1.725 * 10^{-21} </math> Joules
<br/>

Calculate the entropy of the nano particle when it contains 3, 4 and, 5 quanta of energy:
* <math> Ω_3 = \frac{11!}{3!8!} = 165 </math>
* <math> S = k_B * ln(165) = k_B * 9.361 </math>
<br/>
* <math> Ω_4 = \frac{12!}{4!8!} = 495 </math>
* <math> S = k_B * ln(495) = k_B * 11.664 </math>
<br/>
* <math> Ω_5 = \frac{13!}{5!8!} = 1287 </math>
* <math> S = k_B * ln(1287) = k_B * 11.664 </math>
<br/>

Derive an approximate formula for T:
* <math> \frac{1}{T} = \frac{\partial S}{\partial E} </math>
*<math> T = \frac{\partial E}{\partial S} </math>
*<math> T ≈ \frac{∆E}{∆S} </math>
<br/>

Solve for T at 3.5 and 4.5 quanta of energy:
* <math> T_{3.5} ≈ \frac{E}{k_B * (ln(Ω_4) - ln(Ω_3))} </math>
* <math> T_{3.5} ≈ 113.74 K </math>
<br/>
* <math> T_{4.5} ≈ \frac{E}{k_B * (ln(Ω_5) - ln(Ω_4))} </math>
* <math> T_{4.5} ≈ 130.89 K </math>
<br/>

Solve for specific heat:
* <math> C = \frac{1}{N}\frac{∆E}{∆T} = \frac{1}{3}\frac{E}{T_{4.5} - T_{3.5}} </math>
* <math> C = 3.353 * 10^{-23} </math> J/K

==Connectedness==

In my research I read that entropy is known as time's arrow, which in my opinion is one of the most powerful denotations of a physics term. Entropy is a fundamental law that makes the universe tick and it is such a powerful force that it will (possibly) cause the eventual end of the entire universe. Since entropy is always increases, over the expanse of an obscene amount of time the universe due to entropy will eventually suffer a "heat death" and cease to exist entirely. This is merely a scientific hypothesis, and though it may be gloom, an Asimov supercomputer Multivac may finally solve the Last Question and reboot the entire universe again.

The study of entropy is pertinent to my major as an Industrial Engineer as the whole idea of entropy is statistical thermodynamics. This is very similar to Industrial Engineering as it is essentially a statistical business major. Though the odds are unlikely that entropy will be directly used in the day of the life of an Industrial Engineer, the same distributions and concepts of probability are universal and carry over regardless of whether the example is of thermodynamic or business.

My understanding of quantum computers is no more than a couple of wikipedia articles and youtube videos, but I assume anything along the fields of quantum mechanics, which definitely relates to entropy, is important in making the chips to withstand intense heat transfers, etc.

==History==

Entropy was formally named from Greek en + tropē meaning "transformation content" by German physicist Rudolf Clausius in the 1850's. The study of entropy grew from the observation that energy is always lost to friction and dissipation in engines. Entropy was the formal name given to this lost energy by Clausius when he began formulating the first thermodynamical systems.

The concept of entropy was then expanded on by Ludwig Boltzmann who provided the rigorous mathematical definition used today by framing the question in terms of statistical mechanics. Surprisingly, it was not Boltzmann who incorporated the previously found Boltzmann constant (<math> k_B</math>) into the definition, but rather J. Willard Gibbs.

The study of entropy has been used in numerous applications since its inception. Erwin Schrödinger used the concept of entropy to explain the remarkably low replication error of DNA structures in living beings in his book "What is Life?". Entropy also has numerous parallels in the study of informational theory regarding information lost in transmission and broadcasting.

== See also ==

Here is a list of great resources about entropy that make it easier to understand, and also help expound more on the details of the topic.

===External links===

Great TED-ED on the subject:
*https://ed.ted.com/lessons/what-is-entropy-jeff-phillips

==References==

*http://gallica.bnf.fr/ark:/12148/bpt6k152107/f369.table
*http://www.panspermia.org/seconlaw.htm
*https://ed.ted.com/lessons/what-is-entropy-jeff-phillips
*https://www.merriam-webster.com/dictionary/entropy
*https://www.grc.nasa.gov/www/k-12/airplane/entropy.html
*https://oli.cmu.edu
*Chabay, Ruth W., and Bruce A. Sherwood. Matter & Interactions. John Wiley & Sons, 2015.

[[Category:Which Category did you place this in?]]

File:Entropyapplication.png

2024-12-02T17:11:50Z

Kparthas8:

2024-12-02T16:07:36Z

Kparthas8: /* Introduction to Quantum Concepts */

__NOTOC__
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* A physics resource written by experts for an expert audience [https://en.wikipedia.org/wiki/Portal:Physics Physics Portal]
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* A collection of 26 volumes of lecture notes by Prof. Wheeler of Reed College [https://rdc.reed.edu/c/wheeler/home/]
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* 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]
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* 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]
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* 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">
====Isabel Hollhumer F24====
*[[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">
====Jeet Bhatkar====

====Energy Principle====
The Energy Principle is a fundamental concept in physics that describes the relationship between different forms of energy and their conservation within a system. Understanding the Energy Principle is crucial for analyzing the motion and interactions of objects in various physical scenarios.
<div class="mw-collapsible-content">

*[[Kinetic Energy]]
Kinetic energy is the energy an object possesses due to its motion.
*[[Work/Energy]]
Potential energy arises from the position of an object relative to its surroundings. Common forms of potential energy include gravitational potential energy and elastic potential energy.
*[[The Energy Principle]]
Work and energy are closely related concepts. Work (
𝑊) done on an object is defined as the force (
𝐹) applied to the object multiplied by the displacement (
𝑑) of the object in the direction of the force:
The Energy Principle states that the total mechanical energy of a system remains constant if only conservative forces (forces that depend only on the positions of the objects) are acting on the system.
*[[Conservation of Energy]]
The principle of conservation of energy states that the total energy of an isolated system remains constant over time. In other words, energy cannot be created or destroyed, only transformed from one form to another. This principle is a fundamental concept in physics and has wide-ranging applications in mechanics, thermodynamics, and other branches of science.
</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]]
*[[Magnetic Field of a Curved 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">

====Circuitry Basics====
<div class="mw-collapsible-content">
*[[Understanding Fundamentals of Current, Voltage, and Resistance]]
</div>
</div>

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

====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">
*[[Classical Physics]]
</div>
</div>

[[Category:Which Category did you place this in?]]

===Weeks 2 and 3===
<div class="toccolours mw-collapsible mw-collapsed">
====Special Relativity and the Lorentz Transformation====
<div class="mw-collapsible-content">
*[[Frame of Reference]]

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

===Week 4===
<div class="toccolours mw-collapsible mw-collapsed">
====Photons and the Photoelectric Effect====
<div class="mw-collapsible-content">
*[[Spontaneous Photon Emission]]
*[[Light Scattering]]
*[[Lasers]]
*[[Electronic Energy Levels and Photons]]
*[[Quantum Properties of Light]]
*[[The Photoelectric Effect]]
</div>
</div>

===Weeks 5 and 6===
<div class="toccolours mw-collapsible mw-collapsed">
====Matter Waves and Wave-Particle Duality====
<div class="mw-collapsible-content">
*[[Wave-Particle Duality]]
*[[Particle in a 1-Dimensional box]]
*[[Heisenberg Uncertainty Principle]]
</div>
</div>

===Week 7===
<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]]
*[[Fourier Series and Transform]]
</div>
</div>

===Week 8===
<div class="toccolours mw-collapsible mw-collapsed">
====Schrödinger Equation====
<div class="mw-collapsible-content">
*[[The Born Rule]]
*[[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]]
*[[Quantum Harmonic Oscillator]]
*[[Solution for Simple Harmonic Oscillator]]
</div>
</div>

===Week 9===
<div class="toccolours mw-collapsible mw-collapsed">
====Quantum Mechanics====
<div class="mw-collapsible-content">
*[[Quantum Tunneling through Potential Barriers]]
</div>
</div>

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

===Week 10===
<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 11===
<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 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">
====Molecules====
<div class="mw-collapsible-content">
*[[Molecules]]
*[[Covalent Bonds]]
</div>
</div>

===Week 14===
<div class="toccolours mw-collapsible mw-collapsed">
====Statistical Physics====
<div class="mw-collapsible-content">
*[[Application of Statistics in Physics]]
</div>
</div>

===Week 15===
<div class="toccolours mw-collapsible mw-collapsed">
====Statistical Physics====
<div class="mw-collapsible-content">
*[[Temperature & Entropy]]
</div>
</div>

===Additional Topics===
<div class="toccolours mw-collapsible mw-collapsed">
====Thermodynamics====
<div class="mw-collapsible-content">
*[[Maxwell Relations]]
</div>
</div>
<div class="toccolours mw-collapsible mw-collapsed">
====Nuclear Physics====
<div class="mw-collapsible-content">
*[[Nuclear Fission]]
*[[Nuclear Energy from Fission and Fusion]]
</div>
</div>
<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

2024-12-02T16:07:04Z

Kparthas8: /* Introduction to Quantum Concepts */

__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]
* A collection of 26 volumes of lecture notes by Prof. Wheeler of Reed College [https://rdc.reed.edu/c/wheeler/home/]
* 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">
====Isabel Hollhumer F24====
*[[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">
====Jeet Bhatkar====

====Energy Principle====
The Energy Principle is a fundamental concept in physics that describes the relationship between different forms of energy and their conservation within a system. Understanding the Energy Principle is crucial for analyzing the motion and interactions of objects in various physical scenarios.
<div class="mw-collapsible-content">

*[[Kinetic Energy]]
Kinetic energy is the energy an object possesses due to its motion.
*[[Work/Energy]]
Potential energy arises from the position of an object relative to its surroundings. Common forms of potential energy include gravitational potential energy and elastic potential energy.
*[[The Energy Principle]]
Work and energy are closely related concepts. Work (
𝑊) done on an object is defined as the force (
𝐹) applied to the object multiplied by the displacement (
𝑑) of the object in the direction of the force:
The Energy Principle states that the total mechanical energy of a system remains constant if only conservative forces (forces that depend only on the positions of the objects) are acting on the system.
*[[Conservation of Energy]]
The principle of conservation of energy states that the total energy of an isolated system remains constant over time. In other words, energy cannot be created or destroyed, only transformed from one form to another. This principle is a fundamental concept in physics and has wide-ranging applications in mechanics, thermodynamics, and other branches of science.
</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%;">

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<body>
<header>
<h1>Introduction to Quantum Concepts</h1>
<p>A beginner-friendly guide to the essentials of quantum mechanics.</p>
</header>

<div class="container">
<h2>Table of Contents</h2>
<ul>
<li><a href="#bohr-model">The Bohr Model</a></li>
<li><a href="#energy-graphs">Energy Graphs and the Bohr Model</a></li>
<li><a href="#quantized-levels">Quantized Energy Levels</a></li>
<li><a href="#electron-transitions">Electron Transitions</a></li>
<li><a href="#entropy">Entropy</a></li>
</ul>


<div class="topic" id="bohr-model">
<h2>The Bohr Model</h2>
<p>
In 1913, Niels Bohr proposed a model for the atom that was revolutionary at the time. Unlike previous models,
Bohr suggested that electrons orbit the nucleus in distinct, quantized orbits, rather than moving freely.
</p>
<h3>Key Features:</h3>
<ul>
<li><strong>Quantized Orbits:</strong> Electrons occupy specific energy levels, preventing them from spiraling into the nucleus.</li>
<li><strong>Energy Absorption and Emission:</strong> Electrons absorb energy to jump to a higher level or emit energy to drop to a lower one.</li>
</ul>
<img src="bohr-model-diagram.png" alt="Bohr Model Diagram">
<p><i>Figure 1: Visualization of the quantized orbits of electrons in the Bohr Model.</i></p>
</div>


<div class="topic" id="energy-graphs">
<h2>Energy Graphs and the Bohr Model</h2>
<p>
Energy levels in atoms can be visualized using graphs where the y-axis represents energy and the x-axis is arbitrary.
Each level corresponds to a quantized energy state, with the ground state being the lowest.
</p>
<h3>Interactive Exploration:</h3>
<p>Use the following computational model to observe electron transitions between energy levels dynamically:</p>
<p><i>[Placeholder for an interactive energy graph model]</i></p>
</div>


<div class="topic" id="quantized-levels">
<h2>Quantized Energy Levels</h2>
<p>
One of the most groundbreaking discoveries in quantum mechanics is the idea of quantization. Electrons can
only exist at specific energy levels, described by a principal quantum number \( n \).
</p>
<p><strong>Formula for Energy Levels (Hydrogen Atom):</strong></p>
<p>\[
E_n = -\frac{13.6 \, \text{eV}}{n^2}
\]</p>
<p>where \( n \) is the principal quantum number. This formula shows that energy decreases as \( n \) increases.</p>
</div>


<div class="topic" id="electron-transitions">
<h2>Electron Transitions</h2>
<p>
Electron transitions between energy levels result in the absorption or emission of light. This phenomenon is
the basis of spectroscopy, a key tool in understanding atomic structures.
</p>
<h3>Types of Transitions:</h3>
<ul>
<li><strong>Absorption:</strong> Electrons absorb photons to move to a higher energy level.</li>
<li><strong>Emission:</strong> Electrons release photons when falling to a lower energy level.</li>
</ul>
<img src="electron-transition.png" alt="Electron Transitions">
<p><i>Figure 2: Diagram showing the absorption and emission of photons during electron transitions.</i></p>
</div>


<div class="topic" id="entropy">
<h2>Entropy</h2>
<p>
In quantum mechanics, entropy is a measure of the uncertainty or randomness in a system. Higher entropy
generally corresponds to a system with more possible configurations.
</p>
<p>
For example, consider a particle in a box. The more energy levels available to the particle, the higher its
entropy, as there are more possible states it can occupy.
</p>
<p>
Entropy also plays a role in understanding the distribution of particles in quantum systems, particularly in
thermodynamics and statistical mechanics.
</p>
</div>
</div>

<footer>
<p>© 2024 Quantum Mechanics 101. Created for Physics 1 Mechanics students.</p>
</footer>
</body>

</html>

==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]]
*[[Magnetic Field of a Curved 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">

====Circuitry Basics====
<div class="mw-collapsible-content">
*[[Understanding Fundamentals of Current, Voltage, and Resistance]]
</div>
</div>

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

====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">
*[[Classical Physics]]
</div>
</div>

[[Category:Which Category did you place this in?]]

===Weeks 2 and 3===
<div class="toccolours mw-collapsible mw-collapsed">
====Special Relativity and the Lorentz Transformation====
<div class="mw-collapsible-content">
*[[Frame of Reference]]

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

===Week 4===
<div class="toccolours mw-collapsible mw-collapsed">
====Photons and the Photoelectric Effect====
<div class="mw-collapsible-content">
*[[Spontaneous Photon Emission]]
*[[Light Scattering]]
*[[Lasers]]
*[[Electronic Energy Levels and Photons]]
*[[Quantum Properties of Light]]
*[[The Photoelectric Effect]]
</div>
</div>

===Weeks 5 and 6===
<div class="toccolours mw-collapsible mw-collapsed">
====Matter Waves and Wave-Particle Duality====
<div class="mw-collapsible-content">
*[[Wave-Particle Duality]]
*[[Particle in a 1-Dimensional box]]
*[[Heisenberg Uncertainty Principle]]
</div>
</div>

===Week 7===
<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]]
*[[Fourier Series and Transform]]
</div>
</div>

===Week 8===
<div class="toccolours mw-collapsible mw-collapsed">
====Schrödinger Equation====
<div class="mw-collapsible-content">
*[[The Born Rule]]
*[[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]]
*[[Quantum Harmonic Oscillator]]
*[[Solution for Simple Harmonic Oscillator]]
</div>
</div>

===Week 9===
<div class="toccolours mw-collapsible mw-collapsed">
====Quantum Mechanics====
<div class="mw-collapsible-content">
*[[Quantum Tunneling through Potential Barriers]]
</div>
</div>

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

===Week 10===
<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 11===
<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 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">
====Molecules====
<div class="mw-collapsible-content">
*[[Molecules]]
*[[Covalent Bonds]]
</div>
</div>

===Week 14===
<div class="toccolours mw-collapsible mw-collapsed">
====Statistical Physics====
<div class="mw-collapsible-content">
*[[Application of Statistics in Physics]]
</div>
</div>

===Week 15===
<div class="toccolours mw-collapsible mw-collapsed">
====Statistical Physics====
<div class="mw-collapsible-content">
*[[Temperature & Entropy]]
</div>
</div>

===Additional Topics===
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====Thermodynamics====
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*[[Maxwell Relations]]
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====Nuclear Physics====
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*[[Nuclear Fission]]
*[[Nuclear Energy from Fission and Fusion]]
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====Particle Physics====
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*[[Elementary Particles and Particle Physics Theory]]
*[[String Theory]]
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