Thursday, May 9, 2013

Coherent Expectations

OK, so let's say you're assigned the problem of determining the mean value, (the expectation value), for n, (the number state), in a harmonic oscillator with a coherent state.  You go back to your favorite coherent state reference by Nieto and Carruthers[1] and get the probability for finding your coherent oscillator in the nth level almost immediately, (picture 1),


You're looking for the expectation value for n though, so you need to multiply the probabilty by n and sum the whole mess over all possible values of n, (zero to infinity).  Here's what you get, (excuse my sloppiness in picture 2).  Also, the favored notation for coherent states around here happens to be lambda instead of alpha.



So, that looks like a mess. How do you make it more tractable and get down to a single value?  Enter our genius quantum mechanics professor.  He points out that if you just factorize and relabel things a bit, you wind up with (picture 3)


Cool!

Don't forget the fluctuation question though.  In order to work that one through, you need the expectation value for n squared.  That blows everything!  The trick above works for n, but no so much for n squared.  Time to roll up your sleeves and trudge into sums math again, or maybe not.  Our professor also pointed out that (picture 4)


And that was kind of cool, it gets us back to a product of n.  We needed a product with n squared though.  At which point after a little I rolling, you get shown (picture 5)


Oh... Oh Crap!  There's the product of n squared you needed.  And so to wrap up the expectation value for n squared (picture 6)


Pretty cool!

There are bits of this that elude me for the moment.  It's one of those cool math tricks that I can see my way to once I know what I need, but fail to see how I might have arrived at it without prompting from an outside source.

Another interesting point, for me anyway.  It's very reminiscent of the Hermitian recursion relation involving differentiation and multiplication by x that you run into when calculating expectation values for position and momentum in a non-forced harmonics oscillator.

Does anyone know if there's there a reason for the correspondence?

References:
1.  Nieto and Carruthers on coherent states
http://dx.doi.org/10.1119%2F1.1971895
Carruthers P. (1965). Coherent States and the Forced Quantum Oscillator, American Journal of Physics, 33 (7) 537. DOI:

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