Unformatted text preview: 3.5. A PARTICLE CONFINED INSIDE A PIPE 3.5.7 Discussion of the eigenfunctions 3.5.7.1 25 Solution pipefa Question: So how does, say, the onedimensional 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: Onedimensional eigenstate ψ6 . 3.5.7.2 Solution pipefb 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. 3.5.7.3 Solution pipefc 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 Threedimensional solution 3.5.8.1 Solution pipega 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 onedimensional ground state energy Ex1 as an approximation to the threedimensional
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 onedimensional 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%. 3.5.8.2 Solution pipegb 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 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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 Fall '11
 GARVIN

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