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Fund Quantum Mechanics Lect &amp; HW Solutions 43

Fund Quantum Mechanics Lect &amp; HW Solutions 43 - 3.5...

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Unformatted text preview: 3.5. A PARTICLE CONFINED INSIDE A PIPE 3.5.7 Discussion of the eigenfunctions 3.5.7.1 25 Solution pipef-a Question: So how does, say, the one-dimensional eigenstate ψ6 look? Answer: As the graph below shows, it has six blobs where the particle is likely to be found, separated by bands where there is vanishing likelihood of ﬁnding the particle. ψ6 x |ψ6 |2 dark dark dark dark dark dark light light light light light light x light Figure 3.1: One-dimensional eigenstate ψ6 . 3.5.7.2 Solution pipef-b Question: Generalizing the results above, for eigenfunction ψn , any n, how many distinct regions are there where the particle may be found? Answer: There are n of them. 3.5.7.3 Solution pipef-c Question: If you are up to a trick question, consider the following. There are no forces inside the pipe, so the particle has to keep moving until it hits an end of the pipe, then reﬂect backward until it hits the other side and so on. So, it has to cross the center of the pipe regularly. But in the energy eigenstate ψ2 , the particle has zero chance of ever being found at the center of the pipe. What gives? ...
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