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.