# Chapter 4

# Back and Forth with Harmonic Oscillators

In This Chapter

- Hamiltonians: Looking at total energy
- Solving for energy states with creation and annihilation operators
- Understanding the matrix version of harmonic oscillator operators
- Writing computer code to solve the Schrödinger equation

H*armonic oscillators* are physics setups with periodic motion, such as things bouncing on springs or tick-tocking on pendulums. You're probably already familiar with harmonic oscillator problems in the macroscopic arena, but now you're going microscopic. There are many, many physical cases that can be approximated by harmonic oscillators, such as atoms in a crystal structure.

In this chapter, you see both exact solutions to harmonic oscillator problems as well as computational methods for solving them. Knowing how to solve the Schrödinger equation using computers is a useful skill for any quantum physics expert.

## Grappling with the Harmonic Oscillator Hamiltonians

Okay, time to start talking Hamiltonians (and I'm not referring to fans of the U.S. Founding Father Alexander Hamilton). The Hamiltonian will let you find the energy levels of a system.

### Going classical with harmonic oscillation

In classical terms, the force on an object in harmonic oscillation is the following (this is Hooke's law):

F = −*kx*

In this equation, *k* is the spring constant, measured in Newtons/meter, and *x* is displacement. The key point here is that the restoring force on whatever is in harmonic motion is proportional ...