41
Quantum Mechanics
CHAPTER OUTLINE
41.1
An Interpretation of Quantum
Mechanics
41.2
The Quantum Particle under
Boundary Conditions
41.3
The Schrödinger Equation
41.4
A Particle in a Well of Finite
Height
41.5
Tunneling Through a Potential
Energy Barrier
41.6
Applications of Tunneling
41.7
The Simple Harmonic Oscillator
ANSWERS TO QUESTIONS
Q41.1
A particle’s wave function represents its state, contain
ing all the information there is about its location and
motion. The squared absolute value of its wave function
tells where we would classically think of the particle as
spending most its time.
Ψ
2
is the probability distribution
function for the position of the particle.
*Q41.2
For the squared wave function to be the probability per
length of ±
nding the particle, we require
ψψ
2
048
74
016
=
−
==
..
nm
nm
nm
and
0.4/
nm
(i) Answer (e). (ii) Answer (e).
*Q41.3 (i)
For a photon a and b are true, c false, d, e, f, and g true, h false, i and j true.
(ii)
For an electron a is true, b false, c, d, e, f true, g false, h, i and j true.
Note that statements a, d, e, f, i, and j are true for both.
*Q41.4
We consider the quantity
h
2
n
2
/8
mL
2
.
In (a) it is
h
2
1/8
m
1
(3 nm)
2
=
h
2
/72
m
1
nm
2
.
In (b) it is
h
2
4/8
m
1
(3 nm)
2
=
h
2
/18
m
1
nm
2
.
In (c) it is
h
2
1/16
m
1
(3 nm)
2
=
h
2
/144
m
1
nm
2
.
In (d) it is
h
2
1/8
m
1
(6 nm)
2
=
h
2
/288
m
1
nm
2
.
In (e) it is 0
2
1/8
m
1
(3 nm)
2
= 0.
The ranking is then b > a > c > d > e.
Q41.5
The motion of the quantum particle does not consist of moving through successive
points. The particle has no de±
nite position. It can sometimes be found on one side of a node and
sometimes on the other side, but never at the node itself. There is no contradiction here, for the
quantum particle is moving as a wave. It is not a classical particle. In particular, the particle does
not speed up to in±
nite speed to cross the node.
463
Note
:
In chapters 39, 40, and 41 we use
u
to represent the speed of a particle with mass, reserving
v
for the
speeds associated with reference frames, wave functions, and photons.
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Chapter 41
Q41.6
Consider a particle bound to a restricted region of space. If its minimum energy were zero,
then the particle could have zero momentum and zero uncertainty in its momentum. At the same
time, the uncertainty in its position would not be inF
nite, but equal to the width of the region. In
such a case, the uncertainty product
∆∆
xp
x
would be zero, violating the uncertainty principle.
This contradiction proves that the minimum energy of the particle is not zero.
*Q41.7
Compare ±igures 41.4 and 41.7 in the text. In the square well with inF
nitely high walls,
the
particle’s simplest wave function has strict nodes separated by the length
L
of the well. The
particle’s wavelength is 2
L
, its momentum
h
L
2
, and its energy
p
m
h
mL
22
2
28
=
. Now in the well with
walls of only F
nite height, the wave function has nonzero amplitude at the walls. In this F
nitedepth
well …
(i) The particle’s wavelength is longer, answer (a).
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 Spring '11
 williams,frank
 Energy, Mass, Photon, dx, Ψ

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