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Unformatted text preview: π 2
h
8¯ 2 π 2
h
=
2
2mℓy
2mℓ2
x
and canceling the common factors and rearranging:
ℓ2
3
y
=.
2
ℓx
8
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 xdimension. 3.5.8.3 Solution pipegc Question: Shade the regions where the particle is likely to be found in the ψ322 energy
eigenstate.
Answer: The wave function is
ψ322 = 8
2π
3π
2π
sin
x sin
y sin
z
ℓx ℓy ℓz
ℓx
ℓy
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
x
ψx3 2 ψy2 light
y light
light
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
1
1
y = 2 ℓy , and the two white vertical white lines correspond to sin(3πx/ℓx ) being zero at x = 3 ℓx
2
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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This note was uploaded on 11/13/2011 for the course PHY 4458 taught by Professor Garvin during the Fall '11 term at University of Florida.
 Fall '11
 GARVIN

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