Saturday, February 9, 2013

Renormalizing Basis Components in Quantum Wave Functions

I'm studying for our pre-midterm practice exam in quantum mechanics II this weekend, hence I've developed a bit of a fascination for all things, (even trivial things), quantum.  On our last homework, we had to decompose a wave function to its eigenfunction components, (sines and cosines in the infinite potential well that we were assigned), and then write down the time evolution for those components.  It was a bit of an exercise to remember how to normalize the components once I had them, so I'm recording the process here to hopefully make it easier to remember next time.

The wave function that had to be decomposed was




As you can see, a trig identity arose to make simple work of the problem once again.  No need to do a Fourier decomposition to sines and cosines when a simple little power-reduction formula[1] is readily available.

That leaves us with two components and two leading factors, or weights.  The mechanical normalization process is to integrate our wave function over all space, then take the result and divide our original wave function by it so that if we integrate the square of our wave function again we should get one and be normalized.


Now, multiplying back through by 1 over the square of the result we get (picture 3)


Interesting Stuff
An interesting fact about the integration of sine squared over a given region came up while I was doing this.  The result is independent of the frequency of the sine wave.  So, for the following three sine squared functions, you get the same result (pictures 4, 5, and 6)




I believe this has to do with the basis vectors formed by integer multiples in a sine series all having the same unit vector length.  It's nice to see it illustrated in a concrete way though.

References:
1.  http://en.wikipedia.org/wiki/Trigonometric_identity#Power-reduction_formula

Picture of the Day:
Fog on the Organ Mountains out the back door of the old lab

From 2/9/13

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