Expectation Values

Yes! I've finally made it to this point. Normally, this is early on in a quantum mechanics course, but I've had to get through some material in order to make this make sense. Expectation values are important. Very important. We talked about the wave function, and that's important. I've told you it contains information about how to get real things we can measure, but how?

One answer, a very important one, is called an expectation value. As it turns out, expectation values do not come from quantum mechanics, the concept is actually much older! Expectation values come from probability theory, which was originally invented to predict the odds in gambling. The basic idea of expectation values are extremely valuable in determining likely outcomes, but we're here to talk about quantum theory.

Anything that a physicist can measure (and some things they can't directly measure) has a mathematical object associated with it called an "operator". When the operator is combined with the wave function in a special way, the answer is the average value of that operator, called the expectation value of the observable. What's an example?

So I've shown you a couple different wave functions - the infinite square well (particle in a box) and the harmonic oscillator (the spring). Well, say I had a particle that had a wave function like that. If I'm in a lab and I measure the position of the particle I get one value. Then I reset the experiment and try again, well quantum mechanics says I'm likely to get a DIFFERENT value, even though I performed the same experiment. We just have to deal with that - it's quantum mechanics! What if I want to perform an experiment many times, is there anything that I can measure? Yes there is! If I AVERAGE the results from those experiments, I'll obtain the expectation value, which, as I said before, we can calculate!

Why is this important? As it turns out, expectation values obey classical laws of physics. Once you've used quantum mechanics to predict the expectation values, that's it.