Fund Quantum Mechanics Lect & HW Solutions 45

Fund Quantum Mechanics Lect & HW Solutions 45 - 3.5. A...

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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 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 x-dimension. Solution pipeg-c 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 01/06/2012 for the course PHY 3604 taught by Professor Dr.danielarenas during the Fall '11 term at UNF.

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