Quantum Physics Workbook For Dummies Cheat Sheet
When solving quantum physics problems, you need to keep plenty of information in mind. This Cheat Sheet is all about giving you a framework from which to start. Here, you see the essentials for tackling quantum physics problems, including info on the Schrödinger equation, square wells, harmonic oscillators, spherical harmonics, and hydrogen atoms.
The Laws of Quantum Physics: The Schrödinger Equation
The Schrödinger equation is one of the most basic formulas of quantum physics. With the Schrödinger equation, you can solve for the wave functions of particles, and that allows you to say everything you can about the particle — where it is, what its momentum is, and so on.
In the following version of the Schrödinger equation, the first term represents the kinetic energy and the second term represents the potential energy:
Trapping Particles in Square Well Potentials: Wave Functions
One of the most fundamental problems of quantum physics deals with particles trapped on a submicroscopic level in a square well. The square well is always a favorite problem in quantum mechanics classes because the wave function works out so nicely.
The square well has many variations — you can have square wells that are symmetric around the origin, that have infinite walls, that have finite walls, and more. Here’s the square well at its most basic:
This is a one-dimensional well, so you’re concerned only with the x direction; therefore, the Schrödinger equation looks like this:
The wave function looks like this:
where A and B are constants.
Wave Functions for Quantum Harmonic Oscillators
Submicroscopic harmonic oscillators are popular quantum physics problems because harmonic oscillators are relatively simple systems — the force that keeps a particle bound here is proportional to the distance that the particle is from the equilibrium point.
Here’s the harmonic oscillator at its simplest:
And here’s the next-higher state:
In general, you can use the following equation for the wave functions, where Hn is a hermite polynomial:
The Angular Part of the Wave Function: Listing Spherical Harmonics
Many quantum physics problems, such as the hydrogen atom, involve solving problems in spherical coordinates. And when you use spherical coordinates, that almost always means using spherical harmonics.
Spherical harmonics describe the angular part of a particle’s motion when it’s bound in a spherically isotropic potential well. Remembering what the harmonics actually are, sine by sine, can be hard, so here’s a list:
Hydrogen Wave Functions: Single-Electron Atoms in Quantum Physics
One of the triumphs of quantum physics is the solution — to a high degree — of the motion of the electron in the hydrogen atom. But the hydrogen wave functions aren’t easily memorized — you need to remember the radial part of the wave function in addition to the spherical harmonics.
Here’s a list of the first hydrogen atom wave functions: