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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density for every point in space and at all times, the wave function must be
continuous, single-valued, and finite. Since Ø(x, t) satisfies a differential
equation that is second-order in x, its first derivative is also continuous. The
wave function may be multiplied by a phase factor eiÆ, where Æ is real, without
changing its physical significance since
[eiÆØ(x, t)]\ufffd[eiÆØ(x, t)] ˆ Ø\ufffd(x, t)Ø(x, t) ˆ jØ(x, t)j2
Normalization
The particle that is represented by the wave function must be found with
probability equal to unity somewhere in the range ÿ1 < x <1, so that
Ø(x, t) must obey the relation…1
ÿ1
jØ(x, t)j2 dx ˆ 1 (2:9)
38 Schro¨dinger wave mechanics
A function that obeys this equation is said to be normalized. If a function
Ö(x, t) is not normalized, but satisfies the relation…1
ÿ1
Ö\ufffd(x, t)Ö(x, t) dx ˆ N
then the function Ø(x, t) defined by
Ø(x, t) ˆ 1
N
p Ö(x, t)
is normalized.
In order for Ø(x, t) to satisfy equation (2.9), the wave function must be
square-integrable (also called quadratically integrable). Therefore, Ø(x, t)
must go to zero faster than 1=
jxjp as x approaches (\ufffd) infinity. Likewise, the
derivative @Ø=@x must also go to zero as x approaches (\ufffd) infinity.
Once a wave function Ø(x, t) has been normalized, it remains normalized as
time progresses. To prove this assertion, we consider the integral
N ˆ
…1
ÿ1
Ø\ufffdØ dx
and show that N is independent of time for every function Ø that obeys the
Schro¨dinger equation (2.6). The time derivative of N is
dN
dt
ˆ
…1
ÿ1
@
@ t
jØ(x, t)j2 dx (2:10)
where the order of differentiation and integration has been interchanged on the
right-hand side. The derivative of the probability density may be expanded as
follows
@
@ t
jØ(x, t)j2 ˆ @
@ t
(Ø\ufffdØ) ˆ Ø\ufffd @Ø
@ t
‡Ø @Ø
\ufffd
@ t
Equation (2.6) and its complex conjugate may be written in the form
@Ø
@ t
ˆ i&quot;
2m
@2Ø
@x2
ÿ i
&quot;
VØ
@Ø\ufffd
@ t
ˆ ÿ i&quot;
2m
@2Ø\ufffd
@x2
‡ i
&quot;
VØ\ufffd
(2:11)
so that @jØ(x, t)j2=@ t becomes
@
@ t
jØ(x, t)j2 ˆ i&quot;
2m
Ø\ufffd @
2Ø
@x2
ÿØ @
2Ø\ufffd
@x2
\ufffd \ufffd
where the terms containing V cancel. We next note that
@
@x
Ø\ufffd @Ø
@x
ÿØ @Ø
\ufffd
@x
\ufffd \ufffd
ˆ Ø\ufffd @
2Ø
@x2
ÿØ @
2Ø\ufffd
@x2
2.2 The wave function 39
so that
@
@ t
jØ(x, t)j2 ˆ i&quot;
2m
@
@x
Ø\ufffd @Ø
@x
ÿØ @Ø
\ufffd
@x
\ufffd \ufffd
(2:12)
Substitution of equation (2.12) into (2.10) and evaluation of the integral give
dN
dt
ˆ i&quot;
2m
…1
ÿ1
@
@x
Ø\ufffd @Ø
@x
ÿØ @Ø
\ufffd
@x
\ufffd \ufffd
dx ˆ i&quot;
2m
Ø\ufffd @Ø
@x
ÿØ @Ø
\ufffd
@x
\ufffd \ufffd1
ÿ1
Since Ø(x, t) goes to zero as x goes to (\ufffd) infinity, the right-most term
vanishes and we have
dN
dt
ˆ 0
Thus, the integral N is time-independent and the normalization of Ø(x, t) does
not change with time.
Not all wave functions can be normalized. In such cases the quantity
jØ(x, t)j2 may be regarded as the relative probability density, so that the ratio…a2
a1
jØ(x, t)j2 dx…b2
b1
jØ(x, t)j2 dx
represents the probability that the particle will be found between a1 and a2
relative to the probability that it will be found between b1 and b2. As an
example, the plane wave
Ø(x, t) ˆ ei( pxÿEt)=&quot;
does not approach zero as x approaches (\ufffd) infinity and consequently cannot
be normalized. The probability density jØ(x, t)j2 is unity everywhere, so that
the particle is equally likely to be found in any region of a specified width.
Momentum-space wave function
The wave function Ø(x, t) may be represented as a Fourier integral, as shown
in equation (2.7), with its Fourier transform A( p, t) given by equation (2.8).
The transform A( p, t) is uniquely determined by Ø(x, t) and the wave function
Ø(x, t) is uniquely determined by A( p, t). Thus, knowledge of one of these
functions is equivalent to knowledge of the other. Since the wave function
Ø(x, t) completely describes the physical system that it represents, its Fourier
transform A( p, t) also possesses that property. Either function may serve as a
complete description of the state of the system. As a consequence, we may
interpret the quantity jA( p, t)j2 as the probability density for the momentum at
40 Schro¨dinger wave mechanics
time t. By Parseval\u2019s theorem (equation (B.28)), if Ø(x, t) is normalized, then
its Fourier transform A( p, t) is normalized,…1
ÿ1
jØ(x, t)j2 dx ˆ
…1
ÿ1
jA( p, t)j2 d p ˆ 1
The transform A( p, t) is called the momentum-space wave function, while
Ø(x, t) is more accurately known as the coordinate-space wave function.
When there is no confusion, however, Ø(x, t) is usually simply referred to as
the wave function.
2.3 Expectation values of dynamical quantities
Suppose we wish to measure the position of a particle whose wave function is
Ø(x, t). The Born interpretation of jØ(x, t)j2 as the probability density for
finding the associated particle at position x at time t implies that such a
measurement will not yield a unique result. If we have a large number of
particles, each of which is in state Ø(x, t) and we measure the position of each
of these particles in separate experiments all at some time t, then we will obtain
a multitude of different results. We may then calculate the average or mean
value hxi of these measurements. In quantum mechanics, average values of
dynamical quantities are called expectation values. This name is somewhat
misleading, because in an experimental measurement one does not expect to
obtain the expectation value.
By definition, the average or expectation value of x is just the sum over all
possible values of x of the product of x and the probability of obtaining that
value. Since x is a continuous variable, we replace the probability by the
probability density and the sum by an integral to obtain
hxi ˆ
…1
ÿ1
xjØ(x, t)j2 dx (2:13)
More generally, the expectation value h f (x)i of any function f (x) of the
variable x is given by
h f (x)i ˆ
…1
ÿ1
f (x)jØ(x, t)j2 dx (2:14)
Since Ø(x, t) depends on the time t, the expectation values hxi and h f (x)i in
equations (2.13) and (2.14) are functions of t.
The expectation value hpi of the momentum p may be obtained using the
momentum-space wave function A( p, t) in the same way that hxi was obtained
from Ø(x, t). The appropriate expression is
2.3 Expectation values of dynamical quantities 41
hpi ˆ
…1
ÿ1
pjA( p, t)j2 d p ˆ
…1
ÿ1
pA\ufffd( p, t)A( p, t) d p (2:15)
The expectation value h f ( p)i of any function f ( p) of p is given by an
expression analogous to equation (2.14)
h f ( p)i ˆ
…1
ÿ1
f ( p)jA( p, t)j2 d p (2:16)
In general, A( p, t) depends on the time, so that the expectation values hpi and
h f ( p)i are also functions of time.
Both Ø(x, t) and A( p, t) contain the same information about the system,
making it possible to find hpi using the coordinate-space wave function
Ø(x, t) in place of A( p, t). The result of establishing such a procedure will
prove useful when determining expectation values for functions of both
position and momentum. We begin by taking the complex conjugate of A( p, t)
in equation (2.8)
A\ufffd( p, t) ˆ 1
2ð&quot;
p
…1
ÿ1
Ø\ufffd(x, t)ei( pxÿEt)=&quot; dx
Substitution of A\ufffd( p, t) into the integral on the right-hand side of equation
(2.15) gives
hpi ˆ 1
2ð&quot;
p
……1
ÿ1
Ø\ufffd(x, t) pA( p, t)ei( pxÿEt)=&quot; dx d p
ˆ
…1
ÿ1
Ø\ufffd(x, t) 1
2ð&quot;
p
…1
ÿ1
pA( p, t)ei( pxÿEt)=&quot; d p
\ufffd \ufffd
dx (2:17)
In order to evaluate the integral over p, we observe that the derivative of
Ø(x, t) in equation (2.7), with respect to the position variable x, is
@Ø(x, t)
@x
ˆ 1
2ð&quot;
p
…1
ÿ1
i
&quot;
pA( p, t)ei( pxÿEt)=&quot; d p
Substitution of this observation into equation (2.21) gives the final result
hpi ˆ
…1
ÿ1
Ø\ufffd(x, t) &quot;
i
@
@x
\ufffd \ufffd
Ø(x, t) dx (2:18)
Thus,