Solution for Simple Harmonic Oscillator (Xuen Zhen): Difference between revisions
(Added initial content to introduction.) |
No edit summary |
||
| (5 intermediate revisions by the same user not shown) | |||
| Line 1: | Line 1: | ||
<b>Claimed by Lim, Xuen Zhen (Spring 2022)</b> | |||
===Introduction=== | ===Introduction=== | ||
One of the many situations which can be analyzed with the Schrodinger’s Equation is the one-dimensional simple harmonic oscillator. This situation draws an analogy between a quantum-mechanical particle’s oscillating movement and the movement of a common classical oscillator: an object attached to a spring. Like its classical spring counterpart described under Hooke's Law, a quantum harmonic oscillator has the force function <math> F = -k x </math> and the associated potential function <math> U = \frac{1}{2} k x^2 </math>, with <math> k </math> being the force constant (spring constant in classical case). Despite the simplicity of a harmonic oscillator's arbitrary smooth potential, it acts as an important foundation to solving more complicated quantum systems due to it being one of the few quantum-mechanical systems with an exact, known analytical solution. | One of the many situations which can be analyzed with the Schrodinger’s Equation is the one-dimensional simple harmonic oscillator. This situation draws an analogy between a quantum-mechanical particle’s oscillating movement and the movement of a common classical oscillator: an object attached to a spring. Like its classical spring counterpart described under Hooke's Law, a quantum harmonic oscillator has the force function <math> F = -k x </math> and the associated potential function <math> U = \frac{1}{2} k x^2 </math>, with <math> k </math> being the force constant (spring constant in classical case). Despite the simplicity of a harmonic oscillator's arbitrary smooth potential, it acts as an important foundation to solving more complicated quantum systems due to it being one of the few quantum-mechanical systems with an exact, known analytical solution. | ||
===Mathematical | ===Mathematical Setup=== | ||
We may use the time-independent Schrodinger's equation to represent the state of a quantum particle in the harmonic potential by substituting the potential <math> U </math> substituted with <math> \frac{1}{2} k x^2 </math>. | |||
<br> | |||
<math>\frac{-\hbar^2}{2m} \frac{\partial^2 \Psi}{\partial x^2} + \frac{1}{2} k x^2 \Psi = E \Psi</math> | |||
<br> | |||
The solution to this equation are the wave function <math> \Psi </math> and the energy function <math> E </math> that satisfies the above conditions. | |||
===Deriving the Solution=== | |||
===Applications=== | ===Applications=== | ||
Latest revision as of 13:27, 24 April 2022
Claimed by Lim, Xuen Zhen (Spring 2022)
Introduction
One of the many situations which can be analyzed with the Schrodinger’s Equation is the one-dimensional simple harmonic oscillator. This situation draws an analogy between a quantum-mechanical particle’s oscillating movement and the movement of a common classical oscillator: an object attached to a spring. Like its classical spring counterpart described under Hooke's Law, a quantum harmonic oscillator has the force function [math]\displaystyle{ F = -k x }[/math] and the associated potential function [math]\displaystyle{ U = \frac{1}{2} k x^2 }[/math], with [math]\displaystyle{ k }[/math] being the force constant (spring constant in classical case). Despite the simplicity of a harmonic oscillator's arbitrary smooth potential, it acts as an important foundation to solving more complicated quantum systems due to it being one of the few quantum-mechanical systems with an exact, known analytical solution.
Mathematical Setup
We may use the time-independent Schrodinger's equation to represent the state of a quantum particle in the harmonic potential by substituting the potential [math]\displaystyle{ U }[/math] substituted with [math]\displaystyle{ \frac{1}{2} k x^2 }[/math].
[math]\displaystyle{ \frac{-\hbar^2}{2m} \frac{\partial^2 \Psi}{\partial x^2} + \frac{1}{2} k x^2 \Psi = E \Psi }[/math]
The solution to this equation are the wave function [math]\displaystyle{ \Psi }[/math] and the energy function [math]\displaystyle{ E }[/math] that satisfies the above conditions.