Sunday, February 22, 2015
Time Evolution Preliminary results
I've got a workable form of the time evolution simulation. Below are the 40 frame gif results of two different trials. The first was generated using 2,000 time steps of hbar/I units each, and the second was generated using 200,000 time steps of hbar/(100*I) units each. In both cases I was set to 1 eV, the initial wave was localized to well 500, and the initial matrix and vector was of size 1000. I'm thinking I must have made a mistake in how I treated the the second one though, as it looks drastically different from the first. My thought was that if I reduce the time interval by a factor of 100, then I should increase the number of intervals to be even. Is that idea incorrect?
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The well size actually isn't even taken into account. This is just the absolute squared amplitude of the wave vector.
ReplyDeleteI have an idea as to how to calculate an expectation value related to the spatial extent of the wave-function. sum over:
ReplyDelete\(|c_n|^2 (n-500)^2\) and then take the square root of that. Would that work? Have you tried that?
Not yet. I was considering just taking the standard deviation of the absolute squared amplitudes. Is there a reason that might not work?
DeleteI am not sure. I guess what you really want is the square root of the expectation value of n^2 minus the (expectation value of n) squared.
DeleteNice simulation Christopher! Just want to make sure I'm on the right page: This simulation illustrates what George talked about in his post; that the particle becomes less localized as time proceed?
ReplyDeleteThat would be correct. Over the course of some number of time intervals a particle that starts completely localized to one well disperses out to the wells surrounding it.
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