More Fun with Wave Functions

Yesterday, I introduced the concept of wave functions, which encode all of the information about a quantum system. Specifically, we can use them to derive the probabilities of different outcomes of a particular experiment.

A friend of mine who was not already familiar with quantum mechanics then asked me a question along the lines of,

 
So, what’s waving?
— JEV
 

That's a good question, and it's one that I tried to explain in yesterday's post, but I feel it's also worth going into more detail on here.

So then, what waves in a wave function? Not necessarily anything!

Wave functions can wave or oscillate, but they do not have to. This fact rather perturbed my friend, who then asked,

 
Well then, why are they called wave functions?
— JEV
 

That is another fair question, and the honest answer is: I do not know.

The first wave functions I ever learned about as a young physics student definitely waved, and I suspect that the first ones ever discovered did the same. For example, solutions to the time-dependent Schrödinger equation tend to oscillate over time. Since then, however, we have encountered wave functions that do not wave, but we still call them "wave functions" simply because that is the convention.

My friend then argued that the term "wave function" is a bad one, as it implies to the layperson that something is waving. I wholeheartedly agree, but unfortunately, that is the term everyone has used throughout the last century, and changing it would be no small task.

I then pointed out that physics is not the only field in which such "mistakes" occur. For example, my friend is an astronomer and he knows full well that a "planetary nebula" has almost nothing to do with planets.

At least wave functions have the potential to wave!

Once that debate was settled, I thought to myself, "well, why not give JEV a waving wave function?" After all, it's really quite simple to do, and I have the perfect starting point: \begin{equation}
\begin{array}{c}
 |\psi \rangle =\alpha |1\rangle +\beta |0\rangle, \\[2ex]
\left| \alpha \right| ^2+\left| \beta \right| ^2=1.
\end{array}\end{equation}

Our old friend \(\lvert\psi\rangle\), the wave function of the quantum light switch, can be made to oscillate with the right choices for \(\alpha\) and \(\beta\)! One simple example is as follows: \begin{equation}
\begin{array}{c}
 \alpha =\sin (\omega  t), \\[2ex]
 \beta =\cos (\omega  t).
\end{array}\end{equation}

In this formula, sine and cosine oscillate over time \(t\) with angular frequency \(\omega\). The total time it takes to complete one oscillation of the wave function is, therefore, \begin{equation}T=\frac{2 \pi }{\omega }.\end{equation}As time runs forward, \(|\psi\rangle\) generates a probability function that looks like this:

So, there you have it, JEV. The wave function is now waving, and I hope you are satisfied!

Thank you, everyone, for keeping up with me—I really appreciate it.

Aaron