 Principles of Quantum Mechanics as Applied to Chemistry and Chemical Physics

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```interpretation of the wave function 33
ment reinforces the postulate that the wave function for a particle is the sum of
indistinguishable paths and is modified when the paths become distinguishable
by means of a measurement. The nature of the modification is the collapse of
the wave function to one of its components in the sum. Moreover, this new
collapsed wave function may be expressed as the sum of subsequent indis-
tinguishable paths, but remains unchanged if no further interactions with
measuring devices occur.
This statistical interpretation of the significance of the wave function was
postulated by M. Born (1926), although his ideas were based on some
experiments other than the double-slit and Stern\u2013Gerlach experiments. The
concepts that the wave function contains all the information known about the
system it represents and that it collapses to a different state in an experimental
observation were originated by W. Heisenberg (1927). These postulates regard-
ing the meaning of the wave function are part of what has become known as
the Copenhagen interpretation of quantum mechanics. While the Copenhagen
interpretation is disputed by some scientists and philosophers, it is accepted by
the majority of scientists and it provides a consistent theory which agrees with
all experimental observations to date. We adopt the Copenhagen interpretation
of quantum mechanics in this book.3
Problems
1.1 The law of dispersion for surface waves on a sheet of water of uniform depth d is4
ø(k)  (gk tanh dk)1=2
where g is the acceleration due to gravity. What is the group velocity of the
resultant composite wave? What is the limit for deep water (dk > 4)?
1.2 The phase velocity for a particular wave is vph  A=º, where A is a constant. What
is the dispersion relation? What is the group velocity?
1.3 Show that 1
ÿ1
A(k) dk  1
for the gaussian function A(k) in equation (1.19).
3 The historical and philosophical aspects of the Copenhagen interpretation are more extensively discussed
in J. Baggott (1992) The Meaning of Quantum Theory (Oxford University Press, Oxford).
4 For a derivation, see H. Lamb (1932) Hydrodynamics, pp. 363\u201381 (Cambridge University Press, Cam-
bridge).
34 The wave function
1.4 Show that the average value of k is k0 for the gaussian function A(k) in equation
(1.19).
1.5 Show that the gaussian functions A(k) and Ø(x, t) obey Parseval\u2019s theorem (1.18).
1.6 Show that the square pulse A(k) in equation (1.21) and the corresponding function
Ø(x, t) obey Parseval\u2019s theorem.
Problems 35
2
Schro¨dinger wave mechanics
2.1 The Schro¨dinger equation
In the previous chapter we introduced the wave function to represent the
motion of a particle moving in the absence of an external force. In this chapter
we extend the concept of a wave function to make it apply to a particle acted
upon by a non-vanishing force, i.e., a particle moving under the influence of a
potential which depends on position. The force F acting on the particle is
related to the potential or potential energy V (x) by
F  ÿ dV
dx
(2:1)
As in Chapter 1, we initially consider only motion in the x-direction. In Section
2.7, however, we extend the formalism to include three-dimensional motion.
In Chapter 1 we associated the wave packet
Ø(x, t)  1
2ð&quot;
p
1
ÿ1
A( p)ei( pxÿEt)=&quot; d p (2:2)
with the motion in the x-direction of a free particle, where the weighting factor
A( p) is given by
A( p)  1
2ð&quot;
p
1
ÿ1
Ø(x, t)eÿi( pxÿEt)=&quot; dx (2:3)
This wave packet satisfies a partial differential equation, which will be used as
the basis for the further development of a quantum theory. To find this
differential equation, we first differentiate equation (2.2) twice with respect to
the distance variable x to obtain
@2Ø
@x2
 ÿ1
2ð&quot;5
p
1
ÿ1
p2 A( p)ei( pxÿEt)=&quot; d p (2:4)
Differentiation of (2.2) with respect to the time t gives
36
@Ø
@ t
 ÿi
2ð&quot;3
p
1
ÿ1
EA( p)ei( pxÿEt)=&quot; d p (2:5)
The total energy E for a free particle (i.e., for a particle moving in a region of
constant potential energy V ) is given by
E  p
2
2m
 V
which may be combined with equations (2.4) and (2.5) to give
i&quot;
@Ø
@ t
 ÿ &quot;
2
2m
@2Ø
@x2
 VØ
Schro¨dinger (1926) postulated that this differential equation is also valid
when the potential energy is not constant, but is a function of position. In that
case the partial differential equation becomes
i&quot;
@Ø(x, t)
@ t
 ÿ &quot;
2
2m
@2Ø(x, t)
@x2
 V (x)Ø(x, t) (2:6)
which is known as the time-dependent Schro¨dinger equation. The solutions
Ø(x, t) of equation (2.6) are the time-dependent wave functions. An important
goal in wave mechanics is solving equation (2.6) for Ø(x, t) using various
expressions for V (x) that relate to specific physical systems.
When V (x) is not constant, the solutions Ø(x, t) to equation (2.6) may still
be expanded in the form of a wave packet,
Ø(x, t)  1
2ð&quot;
p
1
ÿ1
A( p, t)ei( pxÿEt)=&quot; d p (2:7)
The Fourier transform A( p, t) is then, in general, a function of both p and time
t, and is given by
A( p, t)  1
2ð&quot;
p
1
ÿ1
Ø(x, t)eÿi( pxÿEt)=&quot; dx (2:8)
By way of contrast, recall that in treating the free particle as a wave packet in
Chapter 1, we required that the weighting factor A( p) be independent of time
and we needed to specify a functional form for A( p) in order to study some of
the properties of the wave packet.
2.2 The wave function
Interpretation
Before discussing the methods for solving the Schro¨dinger equation for specific
choices of V (x), we consider the meaning of the wave function. Since the wave
function Ø(x, t) is identified with a particle, we need to establish the connec-
tion between Ø(x, t) and the observable properties of the particle. As in the
2.2 The wave function 37
case of the free particle discussed in Chapter 1, we follow the formulation of
Born (1926).
The fundamental postulate relating the wave function Ø(x, t) to the proper-
ties of the associated particle is that the quantity jØ(x, t)j2  Ø\ufffd(x, t)Ø(x, t)
gives the probability density for finding the particle at point x at time t. Thus,
the probability of finding the particle between x and x dx at time t is
jØ(x, t)j2 dx. The location of a particle, at least within an arbitrarily small
interval, can be determined through a physical measurement. If a series of
measurements are made on a number of particles, each of which has the exact
same wave function, then these particles will be found in many different
locations. Thus, the wave function does not indicate the actual location at
which the particle will be found, but rather provides the probability for finding
the particle in any given interval. More generally, quantum theory provides the
probabilities for the various possible results of an observation rather than a
precise prediction of the result. This feature of quantum theory is in sharp
contrast to the predictive character of classical mechanics.
According to Born\u2019s statistical interpretation, the wave function completely
describes the physical system it represents. There is no information about the
system that is not contained in Ø(x, t). Thus, the state of the system is
determined by its wave function. For this reason the wave function is also
called the state function and is sometimes referred to as the state Ø(x, t).
The product of a function and its complex conjugate is always real and is
positive everywhere. Accordingly, the wave function itself may be a real or a
complex function. At any point x or at any time t, the wave function may be
positive or negative. In order that jØ(x, t)j2 represents a unique probability```