Unformatted text preview: 3.5. A PARTICLE CONFINED INSIDE A PIPE 3.5.7 Discussion of the eigenfunctions 220.127.116.11 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
|ψ6 |2 dark dark dark dark dark dark light light light light light light x light Figure 3.1: One-dimensional eigenstate ψ6 . 18.104.22.168 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. 22.214.171.124 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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- Fall '11
- mechanics, Eigenvalue, eigenvector and eigenspace, Eigenfunction, 126.96.36.199, 188.8.131.52, 184.108.40.206, eigenfunction ψn