Ladder Operators and Fourier Transforms

I sometimes help my son's friend with his physics homework and it drives me crazy. He's completing Honors Physics at U of W and right now we're working through a course in mathematical physics. The thing that drives me crazy is that it's full of real deep heavy duty physics but they don't explain the physics in class: all they do it the math. We were doing Fourier Transforms last month and he had all these pointless integrations to do. I could recognize some of them as being real physical systems, like when we did the Fourier Transform for 1(/w^2 + 1). That's a perfect example of what's wrong with the course. Just a random function on an assignment that you're expected to integrate with no motivation. We went through the class notes and there was nothing about it. What I figured out is that it's the power spectrum of an exponential decay, a typical impulse response function from electrical engineering. You can sort of see it if you break it into partial fractions and you might recognize the pieces looking similar to the impedance of an RC circuit. Now, working from the time domain to the frequency domain, it's a very simple integration, because you're working with exp(-t) from zero to infinity. The problem on the assignment was much harder because you were in effect working from the frequency domain to the time domain. The super short cut here is to recall that the Fourier Transform applied twice must return you to the original function. So if exp(-t) integrates out to give you 1(/1+jw), then integrating one more time must bring you back to exp(-t). The conplex conjugate gives you the left branch of the function exp(t) from minus infinity to zero, and when you add them together you get the complete Fourier Transform of 1/(1 + w^2). It's pretty cool.

I also figured out another completely different way of doing the problem. It's based on another trick they didn't teach in this course. It's like this: multiplication by jw in the frequency domain corresponds to differentiation in the time domain. You can see that it's obviously true for a simple sine wave, and in fact it's generally true. So if we take the given function and multiply it by -w^2, we've in effect differentiated it twice. Now watch this: working in the frequency domain, subtract this result to the original function: you get 1/(1+w^2) - (-w^2)/(1+w^2) = 1!! Now remember that the Fourier transform of the constant function 1 is just the delta function, and you can translate the whole mathematical statement back to the time domain like so:

"The original function, take-away it's second derivative, gives you the delta function."

This is an ordinary differential equation, and its solution gives you exactly the correct result. It's also a very cool illustration of what's actually happening when you use Laplace Transforms to solve differential equations. Everyone who's taken a course in Laplace Transforms must remember how you're given a table of transforms and a bunch of rules on how to work back and forth, and you solve differential equations by following a bunch of steps without knowing what you're doing. That's how they teach it, and it doesn't have to be that way.

The other annoying thing about this problem is that while the original function, which I described as a power spectrum, is certainly a real function from physics, it's not one for which there is ever any reason to want to evaluate the Fourier Transform. The function exp(-t) to the right of zero is definitely an important impulse response function; and it's Fourier Transform, 1/1+jw, is therefore also significant. But the power in the impulse, is given by it's square, evaluated by multiplying by the complex conjugate. That's where you get 1/1+w^2. It makes no sense to work backwards from this function to the time domain by taking the inverse Fourier transform. It can be done, but the result has no physical significance. Of course, what do you expect from a course in mathematical physics?

I started off by saying how annoying it is trying to help someone get through his coursework when you can see that all this really cool physics is presented as nothing more than a bunch of meaningless mathematical manipulations. But the very last assignment of the year, which we worked on last weekend, was the ultimate. Three days before the end of the term, the prof introduced partial differential equations! On the final assignment, you had things like the quantum harmonic oscillator and the diffusion equation (which deals with the time evolution of a heated metal bar with an initial temperature distribution). But there was nothing in the notes about the physics of these problems: just this assignment, where the equation was written out and the student is asked to "solve the differential equation". It's awful to teach things that way. But one of the equations struck me as being vaguely familiar: we were asked to find the eigenfunctions of the operator d/dx + x. What could that be?

It turns out to be one of the ladder operators of the harmonic oscillator. It's a huge topic of physics that could fill a whole graduate level course, and it's given on the last assignment of the year as a one-line problem. I'll have more to say about this in my next post.