Fund Quantum Mechanics Lect & HW Solutions 45

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

Info iconThis preview shows page 1. Sign up to view the full content.

View Full Document Right Arrow Icon
This is the end of the preview. Sign up to access the rest of the document.

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. 3.5.8.3 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. ...
View Full Document

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.

Ask a homework question - tutors are online