The Simple Harmonic Oscillator
Sunday, February 27, 2022
What is an oscillator?
A classical oscillator is an object of mass attached to a spring of force constant . The spring exerts a restoring force on the object, where is the displacement from the equilibrium position.
Harmonic oscillators have an angular frequency and period . Their amplitude is found by their maximum displacement, , and their maximum kinetic energies occur at the turning points . Therefore, the motion is confined to .
The Quantum Mechanical Version of a Harmonic Oscillator
Although no natural example of a one-dimensional quantum oscillator exist, there are plenty of systems that behave approximately like oscillators. For instance, a vibrating diatomic molecule.
A force has potential energy , meaning
There are no boundaries in this situation, but we know that as and , . The simplest function that satisfies this condition is , where the constant and energy can be found by finding the first and second derivatives of the wave function:
Plugging this into the Schrödinger equation from above,
Instead of solving for , we are trying to make this true for all values of by finding constants that make that true. For this to be true,
Which results in
Using the equation for the angular frequency of a classical harmonic oscillator, .
The coefficient can be found using the normalization condition, and the result for the ground state wave function is . Therefore, the complete wave function is
Note that the wave function penetrates into the forbidden region (beyond ), whereas the classical oscillator does not.
The solution above only works for the ground-state. The more general solution is , where is a polynomial in which the highest power of is . The energies are
Below are a few examples of what probability densities look like for harmonic oscillators:
The resulting uncertainties for this situation are as follows:
And the product of the uncertainties is , meaning the uncertainty is at a minimum (called "compact").