What Is a Wave Function?

My younger brother Jeff asked me what a wave function is the other day. The following is my response.

What is a wave function? The short answer is that it is a probability amplitude, that also happens to solve Schrodinger’s equation, but I’m sure that won’t help you much.

When physicists started looking at really small things like atoms, they discovered that the rules of classical mechanics no longer apply. That is that dynamical variables like position, velocity, momentum, acceleration, etc. can no longer be computed via Newton’s laws. They found that these dynamical variables come in pairs. For example position and momentum form a pair, so do time and energy.

The really big idea in Quantum Mechanics is Heisenberg’s uncertainty principle which states that the more carefully you measure one variable the more uncertain you will be about its corresponding partner variable. Imagine you were to use very precise lasers to measure the position of an atom, well the more precisely you know the position, the less knowledge you will have of its momentum. This is exactly analogous to ordinary waves. If you create a standing wave on a rope for example, you know exactly its frequency and wavelength, but it doesn’t make sense to even ask where the wave is because it is everywhere. Now imagine sending a single wave pulse along a taut rope, in this case you will have a fairly good idea of its position, but measuring its frequency is now problematic. Waves inherently predict the uncertainty principle.

After Einstein proposed that light behaves like particles with definite or quantized energy (in his famous photoelectric effect paper of 1905), Louis de Broglie posited that perhaps matter can behave like a wave as well. This was the pivotal intuitive leap that paved the way for Schrodinger. He decided that if de Broglie was right then there must be some wave equation that governs matter waves just like there is a wave equation that governs light waves discovered by Maxwell). So he set about trying to figure it out by applying de Broglie’s famous relations (E=hbar*omega and p=hbar*k which relate dynamical quantities usually associated with matter to dynamical quantities related to waves) to Maxwell’s wave equation. He found that if he slightly modified it and allowed the matter waves to be complex-valued, then the equation could be solved. Complex-valued simply means that the function returns two values, or in other words a complex number of the form a+bi where i is the square root of -1.

After physicists had a new equation to play with, they found all sorts of solutions, but for a long time, no one knew how to interpret what the solutions of the equation actually meant physically. Max Born proposed that solutions of Schrodinger’s equation are probability amplitudes.

If you square a probability amplitude you get a probability density which can be used to predict the likelihood of getting a certain value when you measure that dynamical quantity. That’s very abstract so let me make it more concrete. Once you have psi (a function which solves Schrodinger’s equation), you can predict the likelihood that your quantum system will have energy E, or the likelihood that the particle will be found at position x.