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Unformatted text preview: 3.5. A PARTICLE CONFINED INSIDE A PIPE 27 Canceling the terms that both sides have in common:
3¯ 2 π 2
8¯ 2 π 2
and canceling the common factors and rearranging:
So when ℓy /ℓx = 3/8 = 0.61 or more, the third lowest energy state is given by E121 rather
than E311 . Obviously, it will look more like a box than a pipe then, with the y -dimension 61%
of the x-dimension. 220.127.116.11 Solution pipeg-c Question: Shade the regions where the particle is likely to be found in the ψ322 energy
Answer: The wave function is
ψ322 = 8
ℓx ℓy ℓz
ℓz Now the trick is to realize that the wave function is zero when any of the three sines is zero.
Looking along the z -direction, you will see an array of 3 times 2 blobs, or 6 blobs:
|ψx3 |2 ψy2 light
|ψy2 |2 light light x light Figure 3.2: Eigenstate ψ322 .
The white horizontal centerline line along the pipe corresponds to sin(2πy/ℓy ) being zero at
y = 2 ℓy , and the two white vertical white lines correspond to sin(3πx/ℓx ) being zero at x = 3 ℓx
and x = 3 ℓx . The sin(2πz/ℓz ) factor in the wave function will split it further into six blobs
front and 6 blobs rear, but that is not visible when looking along the z -direction; the front
blobs cover the rear ones. Seen from the top, you would again see an array of 3 times 2 blobs,
the top blobs hiding the bottom ones. ...
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