Quantum Engi Q and A_Part_9

Quantum Engi Q and A_Part_9 - 3.5. A PARTICLE CONFINED...

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Unformatted text preview: 3.5. A PARTICLE CONFINED INSIDE A PIPE 3.5.7 Discussion of the eigenfunctions 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 x |ψ6 |2 dark dark dark dark dark dark light light light light light light x light Figure 3.1: One-dimensional eigenstate ψ6 . 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. 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? 26 CHAPTER 3. BASIC IDEAS OF QUANTUM MECHANICS Answer: Almost every word in the above story is a gross misstatement of what nature really is like when examined on quantum scales. A particle does not have a position, so phrases like “hits an end”, “reflect backward”, and “keep moving” are truly meaningless. On macroscopic scales a particle may have an relatively precisely defined position, but that is only because there is uncertainty in energy. If you could bring a macroscopic particle truly into a single energy eigenstate, it too would have no position. And the smallest thing you might do to figure out where it is would kick it out of that single energy state. 3.5.8 Three-dimensional solution Solution pipeg-a Question: If the cross section dimensions ℓy and ℓz are one tenth the size of the pipe length, how much bigger are the energies Ey1 and Ez1 compared to Ex1 ? So, by what percentage is the one-dimensional ground state energy Ex1 as an approximation to the three-dimensional one, E111 , then in error? Answer: The energies are Ex1 h2 π 2 ¯ = 2mℓ2 x Ey1 h2 π 2 ¯ = 2mℓ2 y Ez1 h2 π 2 ¯ = . 2mℓ2 z If ℓy and ℓz are ten times smaller than ℓx then Ey1 and Ez1 are each 100 times larger than Ex1 . So the one-dimensional ground state energy Ex1 is smaller than the true ground state energy E111 = Ex1 + Ey1 + Ez1 by a factor 201. Which means it is off by 20,000%. Solution pipeg-b Question: At what ratio of ℓy /ℓx does the energy E121 become higher than the energy E311 ? Answer: Using the given expression for Enx ny nz , Enx ny nz = ¯ ¯ n2 h2 π 2 n2 h2 π 2 n2 h2 π 2 x¯ + y 2 + z 2, 2 2mℓx 2mℓy 2mℓz E121 = E311 when 4¯ 2 π 2 h h2 π 2 ¯ 9¯ 2 π 2 h h2 π 2 ¯ h2 π 2 ¯ h2 π 2 ¯ + + = + + 2 2 2 2 2 2mℓx 2mℓy 2mℓz 2mℓx 2mℓy 2mℓ2 z 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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