Principles of Quantum Mechanics   as Applied to Chemistry and Chemical Physics

Principles of Quantum Mechanics as Applied to Chemistry and Chemical Physics


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this derivation does not apply to a particle in a
potential field.
The position, momentum, and energy are all dynamical quantities and
consequently possess quantum-mechanical operators from which expectation
values at any given time may be determined. Time, on the other hand, has a
unique role in non-relativistic quantum theory as an independent variable;
dynamical quantities are functions of time. Thus, the \u2018uncertainty\u2019 in time
cannot be related to a range of expectation values.
To obtain the energy-time uncertainty principle for a particle in a time-
independent potential field, we set A^ equal to H^ in equation (3.81)
(\u2dcE)(\u2dcB) > 1
2
jh[ H^ , B^]ij
where \u2dcE is the uncertainty in the energy as defined by (3.75) with A^ ˆ H^ .
Substitution of equation (3.72) into this expression gives
(\u2dcE)(\u2dcB) >
"
2
\ufffd\ufffd\ufffd\ufffd dhBidt
\ufffd\ufffd\ufffd\ufffd (3:86)
In a short period of time \u2dct, the change in the expectation value of B is given
by
\u2dcB ˆ dhBi
dt
\u2dct
When this expression is combined with equation (3.86), we obtain the desired
result
(\u2dcE)(\u2dct) >
"
2
(3:87)
We see that the energy and time obey an uncertainty relation when \u2dct is
defined as the period of time required for the expectation value of B to change
by one standard deviation. This definition depends on the choice of the
dynamical variable B so that \u2dct is relatively larger or smaller depending on
that choice. If dhBi=dt is small so that B changes slowly with time, then the
period \u2dct will be long and the uncertainty in the energy will be small.
3.11 Heisenberg uncertainty principle 103
Conversely, if B changes rapidly with time, then the period \u2dct for B to change
by one standard deviation will be short and the uncertainty in the energy of the
system will be large.
Problems
3.1 Which of the following operators are linear?
(a)
p
(b) sin (c) xD^x (d) D^xx
3.2 Demonstrate the validity of the relationships (3.4a) and (3.4b).
3.3 Show that
[A^, [B^, C^]]‡ [B^, [C^, A^]]‡ [C^, [A^, B^]] ˆ 0
where A^, B^, and C^ are arbitrary linear operators.
3.4 Show that (D^x ‡ x)(D^x ÿ x) ˆ D^2x ÿ x2 ÿ 1.
3.5 Show that xeÿx
2
is an eigenfunction of the linear operator (D^2x ÿ 4x2). What is
the eigenvalue?
3.6 Show that the operator D^2x is hermitian. Is the operator iD^
2
x hermitian?
3.7 Show that if the linear operators A^ and B^ do not commute, the operators
(A^B^‡ B^A^) and i[A^, B^] are hermitian.
3.8 If the real normalized functions f (r) and g(r) are not orthogonal, show that their
sum f (r)‡ g(r) and their difference f (r)ÿ g(r) are orthogonal.
3.9 Consider the set of functions \u1421 ˆ eÿx=2, \u1422 ˆ xeÿx=2, \u1423 ˆ x2eÿx=2, \u1424 ˆ
x3eÿx=2, defined over the range 0 < x <1. Use the Schmidt orthogonalization
procedure to construct from the set \u142i an orthogonal set of functions with
w(x) ˆ 1.
3.10 Evaluate the following commutators:
(a) [x, p^x] (b) [x, p^
2
x] (c) [x, H^] (d) [ p^x, H^]
3.11 Evaluate [x, p^3x] and [x
2, p^2x] using equations (3.4).
3.12 Using equation (3.4b), show by iteration that
[x n, p^x] ˆ i&quot;nx nÿ1
where n is a positive integer greater than zero.
3.13 Show that
[ f (x), p^x] ˆ i&quot; d f (x)
dx
3.14 Calculate the expectation values of x, x2, p^, and p^2 for a particle in a one-
dimensional box in state \u142n (see Section 2.5).
3.15 Calculate the expectation value of p^4 for a particle in a one-dimensional box in
state \u142n.
3.16 A hermitian operator A^ has only three normalized eigenfunctions \u1421, \u1422, \u1423,
with corresponding eigenvalues a1 ˆ 1, a2 ˆ 2, a3 ˆ 3, respectively. For a
particular state ö of the system, there is a 50% chance that a measure of A
produces a1 and equal chances for either a2 or a3.
104 General principles of quantum theory
(a) Calculate hAi.
(b) Express the normalized wave function ö of the system in terms of the
eigenfunctions of A^.
3.17 The wave function Ø(x) for a particle in a one-dimensional box of length a is
Ø(x) ˆ C sin7 ðx
a
\ufffd \ufffd
; 0 < x < a
where C is a constant. What are the possible observed values for the energy and
their respective probabilities?
3.18 If j\u142i is an eigenfunction of H^ with eigenvalue E, show that for any operator A^
the expectation value of [ H^ , A^] vanishes, i.e.,
h\u142j[ H^ , A^]j\u142i ˆ 0
3.19 Derive both of the Ehrenfest theorems using equation (3.72).
3.20 Show that
\u2dcH\u2dcx >
&quot;
2m
h p^xi
Problems 105
4
Harmonic oscillator
In this chapter we treat in detail the quantum behavior of the harmonic
oscillator. This physical system serves as an excellent example for illustrating
the basic principles of quantum mechanics that are presented in Chapter 3. The
Schro¨dinger equation for the harmonic oscillator can be solved rigorously and
exactly for the energy eigenvalues and eigenstates. The mathematical process
for the solution is neither trivial, as is the case for the particle in a box, nor
excessively complicated. Moreover, we have the opportunity to introduce the
ladder operator technique for solving the eigenvalue problem.
The harmonic oscillator is an important system in the study of physical
phenomena in both classical and quantum mechanics. Classically, the harmonic
oscillator describes the mechanical behavior of a spring and, by analogy, other
phenomena such as the oscillations of charge flow in an electric circuit, the
vibrations of sound-wave and light-wave generators, and oscillatory chemical
reactions. The quantum-mechanical treatment of the harmonic oscillator may
be applied to the vibrations of molecular bonds and has many other applica-
tions in quantum physics and field theory.
4.1 Classical treatment
The harmonic oscillator is an idealized one-dimensional physical system in
which a single particle of mass m is attracted to the origin by a force F
proportional to the displacement of the particle from the origin
F ˆ ÿkx (4:1)
The proportionality constant k is known as the force constant. The minus sign
in equation (4.1) indicates that the force is in the opposite direction to the
direction of the displacement. The typical experimental representation of the
oscillator consists of a spring with one end stationary and with a mass m
106
attached to the other end. The spring is assumed to obey Hooke\u2019s law, that is to
say, equation (4.1). The constant k is then often called the spring constant.
In classical mechanics the particle obeys Newton\u2019s second law of motion
F ˆ ma ˆ m d
2x
dt2
(4:2)
where a is the acceleration of the particle and t is the time. The combination of
equations (4.1) and (4.2) gives the differential equation
d2x
dt2
ˆ ÿ k
m
x
for which the solution is
x ˆ A sin(2ðít ‡ b) ˆ A sin(øt ‡ b) (4:3)
where the amplitude A of the vibration and the phase b are the two constants
of integration and where the frequency í and the angular frequency ø of
vibration are related to k and m by
ø ˆ 2ðí ˆ

k
m
r
(4:4)
According to equation (4.3), the particle oscillates sinusoidally about the origin
with frequency í and maximum displacement \ufffdA.
The potential energy V of a particle is related to the force F acting on it by
the expression
F ˆ ÿ dV
dx
Thus, from equations (4.1) and (4.4), we see that for a harmonic oscillator the
potential energy is given by
V ˆ 1
2
kx2 ˆ 1
2
mø2x2 (4:5)
The total energy E of the particle undergoing harmonic motion is given by
E ˆ 1
2
mv2 ‡ V ˆ 1
2
mv2 ‡ 1
2
mø2x2 (4:6)
where v is the instantaneous velocity. If the oscillator is undisturbed by outside
forces, the energy E remains fixed at a constant value. When the particle is at
maximum displacement from the origin so that x ˆ \ufffdA, the velocity v is zero
and the potential energy is a maximum. As jxj decreases, the potential
decreases and the velocity increases keeping E constant. As the particle crosses
the origin