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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the expectation value of the momentum can be obtained by an integration
in coordinate space.
The expectation value of p2 is given by equation (2.16) with f ( p) ˆ p2. The
expression analogous to (2.17) is
hp2i ˆ
…1
ÿ1
Ø\ufffd(x, t) 1
2ð"
p
…1
ÿ1
p2 A( p, t) ei( pxÿEt)=" d p
\ufffd \ufffd
dx
From equation (2.7) it can be seen that the quantity in square brackets equals
42 Schro¨dinger wave mechanics
"
i
\ufffd \ufffd2
@2Ø(x, t)
@x2
so that
hp2i ˆ
…1
ÿ1
Ø\ufffd(x, t) "
i
\ufffd \ufffd2
@2
@x2
Ø(x, t) dx (2:19)
Similarly, the expectation value of pn is given by
hpni ˆ
…1
ÿ1
Ø\ufffd(x, t) "
i
@
@x
\ufffd \ufffdn
Ø(x, t) dx (2:20)
Each of the integrands in equations (2.18), (2.19), and (2.20) is the complex
conjugate of the wave function multiplied by an operator acting on the wave
function. Thus, in the coordinate-space calculation of the expectation value of
the momentum p or the nth power of the momentum, we associate with p the
operator ("=i)(@=@x). We generalize this association to apply to the expectation
value of any function f ( p) of the momentum, so that
h f ( p)i ˆ
…1
ÿ1
Ø\ufffd(x, t) f "
i
@
@x
\ufffd \ufffd
Ø(x, t) dx (2:21)
Equation (2.21) is equivalent to the momentum-space equation (2.16).
We may combine equations (2.14) and (2.21) to find the expectation value of
a function f (x, p) of the position and momentum
h f (x, p)i ˆ
…1
ÿ1
Ø\ufffd(x, t) f x, "
i
@
@x
\ufffd \ufffd
Ø(x, t) dx (2:22)
Ehrenfest\u2019s theorems
According to the correspondence principle as stated by N. Bohr (1928), the
average behavior of a well-defined wave packet should agree with the classical-
mechanical laws of motion for the particle that it represents. Thus, the
expectation values of dynamical variables such as position, velocity, momen-
tum, kinetic energy, potential energy, and force as calculated in quantum
mechanics should obey the same relationships that the dynamical variables
obey in classical theory. This feature of wave mechanics is illustrated by the
derivation of two relationships known as Ehrenfest\u2019s theorems.
The first relationship is obtained by considering the time dependence of the
expectation value of the position coordinate x. The time derivative of hxi in
equation (2.13) is
2.3 Expectation values of dynamical quantities 43
dhxi
dt
ˆ d
dt
…1
ÿ1
xjØ(x, t)j2 dx ˆ
…1
ÿ1
x
@
@ t
jØ(x, t)j2 dx
ˆ i"
2m
…1
ÿ1
x
@
@x
Ø\ufffd @Ø
@x
ÿØ @Ø
\ufffd
@x
\ufffd \ufffd
dx
where equation (2.12) has been used. Integration by parts of the last integral
gives
dhxi
dt
ˆ i"
2m
x Ø\ufffd @Ø
@x
ÿØ @Ø
\ufffd
@x
\ufffd \ufffd1
ÿ1
ÿ i"
2m
…1
ÿ1
Ø\ufffd @Ø
@x
ÿØ @Ø
\ufffd
@x
\ufffd \ufffd
dx
The integrated part vanishes because Ø(x, t) goes to zero as x approaches (\ufffd)
infinity. Another integration by parts of the last term on the right-hand side
yields
dhxi
dt
ˆ 1
m
…1
ÿ1
Ø\ufffd "
i
@
@x
\ufffd \ufffd
Ø dx
According to equation (2.18), the integral on the right-hand side of this
equation is the expectation value of the momentum, so that we have
hpi ˆ m dhxi
dt
(2:23)
Equation (2.23) is the quantum-mechanical analog of the classical definition of
momentum, p ˆ mv ˆ m(dx=dt). This derivation also shows that the associa-
tion in quantum mechanics of the operator ("=i)(@=@x) with the momentum is
consistent with the correspondence principle.
The second relationship is obtained from the time derivative of the expecta-
tion value of the momentum hpi in equation (2.18),
dhpi
dt
ˆ d
dt
…1
ÿ1
Ø\ufffd "
i
@Ø
@x
dx ˆ "
i
…1
ÿ1
@Ø\ufffd
@ t
@Ø
@x
‡Ø\ufffd @
@x
@Ø
@ t
\ufffd \ufffd
dx
We next substitute equations (2.11) for the time derivatives of Ø and Ø\ufffd and
obtain
dhpi
dt
ˆ
…1
ÿ1
ÿ"2
2m
@2Ø\ufffd
@x2
‡ VØ\ufffd
\ufffd \ufffd
@Ø
@x
‡Ø\ufffd @
@x
"2
2m
@2Ø
@x2
ÿ VØ
\ufffd \ufffd" #
dx
ˆ ÿ"
2
2m
…1
ÿ1
@2Ø\ufffd
@x2
@Ø
@x
dx‡ "
2
2m
…1
ÿ1
Ø\ufffd @
3Ø
@x3
dxÿ
…1
ÿ1
Ø\ufffdØ dV
dx
dx
(2:24)
where the terms in V cancel. The first integral on the right-hand side of
equation (2.24) may be integrated by parts twice to give
44 Schro¨dinger wave mechanics
…1
ÿ1
@2Ø\ufffd
@x2
@Ø
@x
dx ˆ @Ø
\ufffd
@x
@Ø
@x
\ufffd \ufffd1
ÿ1
ÿ
…1
ÿ1
@Ø\ufffd
@x
@2Ø
@x2
dx
ˆ @Ø
\ufffd
@x
@Ø
@x
ÿØ\ufffd @
2Ø
@x2
\ufffd \ufffd1
ÿ1
‡
…1
ÿ1
Ø\ufffd @
3Ø
@x3
dx
The integrated part vanishes because Ø and @Ø=@x vanish at (\ufffd) infinity. The
remaining integral cancels the second integral on the right-hand side of
equation (2.24), leaving the final result
dhpi
dt
ˆ ÿ
\ufffd
dV
dx
\ufffd
ˆ hFi (2:25)
where equation (2.1) has been used. Equation (2.25) is the quantum analog of
Newton\u2019s second law of motion, F ˆ ma, and is in agreement with the
correspondence principle.
Heisenberg uncertainty principle
Using expectation values, we can derive the Heisenberg uncertainty principle
introduced in Section 1.5. If we define the uncertainties \u2dcx and \u2dcp as the
standard deviations of x and p, as used in statistics, then we have
\u2dcx ˆ h(xÿ hxi)2i1=2
\u2dcp ˆ h( pÿ hpi)2i1=2
The expectation values of x and of p at a time t are given by equations (2.13)
and (2.18), respectively. For the sake of simplicity in this derivation, we select
the origins of the position and momentum coordinates at time t to be the
centers of the wave packet and its Fourier transform, so that hxi ˆ 0 and
hpi ˆ 0. The squares of the uncertainties \u2dcx and \u2dcp are then given by
(\u2dcx)2 ˆ
…1
ÿ1
x2Ø\ufffdØ dx
(\u2dcp)2 ˆ "
i
\ufffd \ufffd2…1
ÿ1
Ø\ufffd @
2Ø
@x2
dx ˆ "
i
\ufffd \ufffd2
Ø\ufffd @Ø
@x
" #1
ÿ1
ÿ "
i
\ufffd \ufffd2…1
ÿ1
@Ø\ufffd
@x
@Ø
@x
dx
ˆ
…1
ÿ1
ÿ"
i
@Ø\ufffd
@x
\ufffd \ufffd
"
i
@Ø
@x
\ufffd \ufffd
dx
where the integrated term for (\u2dcp)2 vanishes because Ø goes to zero as x
approaches (\ufffd) infinity.
The product (\u2dcx\u2dcp)2 is
2.3 Expectation values of dynamical quantities 45
(\u2dcx\u2dcp)2 ˆ
…1
ÿ1
(xØ\ufffd)(xØ) dx
…1
ÿ1
ÿ"
i
@Ø\ufffd
@x
\ufffd \ufffd
"
i
@Ø
@x
\ufffd \ufffd
dx
Applying Schwarz\u2019s inequality (A.56), we obtain
(\u2dcx\u2dcp)2 >
1
4
\ufffd\ufffd\ufffd\ufffd "i
…1
ÿ1
xØ\ufffd @Ø
@x
‡ xØ @Ø
\ufffd
@x
\ufffd \ufffd
dx
\ufffd\ufffd\ufffd\ufffd2 ˆ "24
\ufffd\ufffd\ufffd\ufffd…1ÿ1x @@x (Ø\ufffdØ) dx
\ufffd\ufffd\ufffd\ufffd2
ˆ "
2
4
\ufffd\ufffd\ufffd\ufffd\ufffdxØ\ufffdØ\ufffd1
ÿ1
ÿ
…1
ÿ1
Ø\ufffdØ dx
\ufffd\ufffd\ufffd\ufffd2
The integrated part vanishes because Ø goes to zero faster than 1=
jxjp , as x
approaches (\ufffd) infinity and the remaining integral is unity by equation (2.9).
Taking the square root, we obtain an explicit form of the Heisenberg uncer-
tainty principle
\u2dcx\u2dcp >
"
2
(2:26)
This expression is consistent with the earlier form, equation (1.44), but relation
(2.26) is based on a precise definition of the uncertainties, whereas relation
(1.44) is not.
2.4 Time-independent Schro¨dinger equation
The first step in the solution of the partial differential equation (2.6) is to
express the wave function Ø(x, t) as the product of two functions
Ø(x, t) ˆ \u142(x)÷(t) (2:27)
where \u142(x) is a function of only the distance x and ÷(t) is a function of only
the time t. Substitution of equation (2.27) into (2.6) and division by the product
\u142(x)÷(t) give
i"
1
÷(t)
d÷(t)
dt
ˆ ÿ "
2
2m
1
\u142(x)
d2\u142(x)
dx2
‡ V (x) (2:28)
The left-hand side of equation (2.28) is a function only of t, while the right-
hand side is a function only of x. Since x and t are independent variables, each
side of equation (2.28) must equal a constant. If this were not true, then the
left-hand side could be changed by varying t while the right-hand side
remained fixed and so the equality would no longer apply. For reasons that will
soon be apparent, we designate this separation constant by E and assume that
it is a real number.
Equation (2.28) is now separable into two independent differential equations,
one for each of the two independent variables x and t. The time-dependent