🪗 Simple harmonic oscillator

It’s not terribly difficult to apply a similar approach of the simulation of the infinite square well to the simple harmonic oscillator (SHO). What’s the same? There are still energy eigenstates (stationary states) that have definite energy (frequency). You can still form superpositions of these stationary states to produce more general states that “slosh” in time. These superposition states can be used to model realistic scenarios that might occur in various situations. What’s different? Well, the potential energy function is pretty different, rather than a piecewise constant potential like the infinite square well, or the finite square well, the SHO potential is $V (x) = \frac{1}{2}m\omega^2 x^2$. This results in quantitatively different eigenstate energies and wavefunctions. — Visualizing Quantum Mechanics with Python

📌 Highly recommended background information: Simulating quantum mechanics with Python

Quantum Harmonic Oscillator

🧠 Idea taken from the book Visualizing Quantum Mechanics with Python

Theoretical background

For the quantum harmonic oscillator:

$$\psi_n(x) = \frac{1}{\sqrt{2^n n! \sqrt{\pi}}} H_n(x) e^{-x^2/2}$$

with energy:

$$E_n = \hbar \omega (n + 1/2)$$

it holds that the time evolution is given by:

$$\psi_n(x,t) = \psi_n(x) e^{-i (n+1/2)\omega t}$$

In the code we assume

• dimensionless x
• ℏ = 1
• ω = 1