Perhaps in high school Algebra 1 or 2 you first learned about even and odd functions. The idea seemed simple and pointless at first: an even polynomial function had an even degree, and an odd polynomial function had an odd degree.

f_{even}(x) = x^{2}

f_{odd}(x) = x^{3} + 2x^{2} + 1

Like many students, you may have thought, “Duh. Who cares?” Ah, just like almost everything in math, this is the general feeling FOR YEARS until you have that beautiful, glorious, “aha” moment.

However, even once you get to precalc or calculus, the “aha” moment is still not quite there. You simply learn the fact that for even functions, f(-x) = f(x) and that for odd functions, f(-x) = -f(x). You do a lot of problems showing these equivalencies. If you’re lucky, you see the graphs of these functions and begin to discuss integration “shortcuts” for them. This is the first tiny “aha” moment.

Ok, so we’ve had our tiny “aha” moments. But the real beauty of these facts is revealed in intro college quantum courses. The wavefunction of a quantum object is usually normalized such that it decreases towards 0 at both ends. This, combined with the fact that most quantum operators you work with in an intro course are simple even or odd functions, makes the required integration simple.

Often, beginning physics students labor over these integrals, because there is a lot going on in each one: exponentials, absolute values, imaginary numbers, AND some operator. However, if you remember those discussions of odd and even functions, many terms cancel out or become dominated by the overall even/odd nature.

Additionally, the best part about seeing these integrals beyond the math classroom is that you can actually sanity check your answers based on what you know should be happening physically! For example, the expectation value, <x>, represents where you expect the quantum object to be located with some probability (in one interpretation of QM, but this is not the blog post for that discussion). So given that we’ve chosen the x spatial coordinate arbitrarily, the object has the highest chance of being found at the center of this coordinate. That is, at 0. So as a physics student, you should be very relieved when you get this 0!

Now, these of course are toy examples, but physicists LOVE these types of little tricks, so it will serve you to appreciate them early on.