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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integrate over the spatial variablesX
n
"
i
@an(t)
@ t
‡ Enan(t)
\ufffd \ufffd
h\u142k(q)j\u142n(q)i
ˆ
X
n
"
i
@an(t)
@ t
‡ Enan(t)
\ufffd \ufffd
äkn ˆ "
i
@ak(t)
@ t
‡ Ek ak(t) ˆ 0
Replacing the dummy index k by n, we obtain the result
an(t) ˆ cneÿiEn t=" (3:63)
where cn is a constant independent of both q and t. Substitution of equation
(3.63) into (3.60) gives equation (3.59), showing that equation (3.59) is indeed
the most general form for a solution of the time-dependent Schro¨dinger
equation. All solutions may be expressed as the sum over stationary states.
3.8 Parity operator
The parity operator \u2014^ is defined by the relation
\u2014^\u142(q) ˆ \u142(ÿq) (3:64)
Thus, the parity operator reverses the sign of each cartesian coordinate. This
operator is equivalent to an inversion of the coordinate system through the
origin. In one and three dimensions, equation (3.64) takes the form
\u2014^\u142(x) ˆ \u142(ÿx), \u2014^\u142(r) ˆ \u2014^\u142(x, y, z) ˆ \u2014^\u142(ÿx, ÿy, ÿz) ˆ \u142(ÿr)
The operator \u2014^2 is equal to unity since
94 General principles of quantum theory
\u2014^2\u142(q) ˆ \u2014^(\u2014^\u142(q)) ˆ \u2014^\u142(ÿq) ˆ \u142(q)
Further, we see that
\u2014^n\u142(q) ˆ \u142(q), n even
ˆ \u142(ÿq), n odd
or
\u2014^n ˆ 1, n even
ˆ \u2014^, n odd
The operator \u2014^ is linear and hermitian. In the one-dimensional case, the
hermiticity of \u2014^ is demonstrated as follows
höj\u2014^j\u142i ˆ
…1
ÿ1
ö\ufffd(x)\u142(ÿx) dx ˆ ÿ
…ÿ1
1
ö\ufffd(ÿx9)\u142(x9) dx9
ˆ
…1
ÿ1
\u142(x9)\u2014^ö\ufffd(x9) dx9 ˆ h\u2014^öj\u142i
where x in the second integral is replaced by x9 ˆ ÿx to obtain the third
integral. By applying the same procedure to each coordinate, we can show that
\u2014^ is hermitian with respect to multi-dimensional functions.
The eigenvalues º of the parity operator \u2014^ are given by
\u2014^\u142º(q) ˆ º\u142º(q) (3:65)
where \u142º(q) are the corresponding eigenfunctions. If we apply \u2014^ to both sides
of equation (3.65), we obtain
\u2014^2\u142º(q) ˆ º\u2014^\u142º(q) ˆ º2\u142º(q)
Since \u2014^2 ˆ 1, we see that º2 ˆ 1 and that the eigenvalues º, which must be
real because \u2014^ is hermitian, are equal to either ‡1 or ÿ1. To find the
eigenfunctions \u142º(q), we note that equation (3.65) now becomes
\u142º(ÿq) ˆ \ufffd\u142º(q)
For º ˆ 1, the eigenfunctions of \u2014^ are even functions of q, while for º ˆ ÿ1,
they are odd functions of q. An even function of q is said to be of even parity,
while odd parity refers to an odd function of q. Thus, the eigenfunctions of \u2014^
are any well-behaved functions that are either of even or odd parity in their
cartesian variables.
We show next that the parity operator \u2014^ commutes with the Hamiltonian
operator H^ if the potential energy V (q) is an even function of q. The kinetic
energy term in the Hamiltonian operator is given by
ÿ "
2
2m
=2 ˆ ÿ "
2
2m
@2
@q21
‡ @
2
@q22
‡ \ufffd \ufffd \ufffd
 !
3.8 Parity operator 95
and is an even function of each qk . If the potential energy V (q) is also an even
function of each qk , then we have H^(q) ˆ H^(ÿq) and
[ H^ , \u2014^] f (q) ˆ H^(q)\u2014^ f (q)ÿ \u2014^ H^(q) f (q) ˆ H^(q) f (ÿq)ÿ H^(ÿq) f (ÿq) ˆ 0
Since the function f (q) is arbitrary, the commutator of H^ and \u2014^ vanishes.
Thus, these operators have simultaneous eigenfunctions for systems with
V (q) ˆ V (ÿq).
If the potential energy of a system is an even function of the coordinates and
if \u142(q) is a solution of the time-independent Schro¨dinger equation, then the
function \u142(ÿq) is also a solution. When the eigenvalues of the Hamiltonian
operator are non-degenerate, these two solutions are not independent of each
other, but are proportional
\u142(ÿq) ˆ c\u142(q)
These eigenfunctions are also eigenfunctions of the parity operator, leading to
the conclusion that c ˆ \ufffd1. Consequently, some eigenfunctions will be of even
parity while all the others will be of odd parity.
For a degenerate energy eigenvalue, the several corresponding eigenfunc-
tions of H^ may not initially have a definite parity. However, each eigenfunction
may be written as the sum of an even part \u142e(q) and an odd part \u142o(q)
\u142(q) ˆ \u142e(q)‡ \u142o(q)
where
\u142e(q) ˆ 12[\u142(q)‡ \u142(ÿq)] ˆ \u142e(ÿq)
\u142o(q) ˆ 12[\u142(q)ÿ \u142(ÿq)] ˆ ÿ\u142o(ÿq)
Since any linear combination of \u142(q) and \u142(ÿq) satisfies Schro¨dinger\u2019s equa-
tion, the functions \u142e(q) and \u142o(q) are eigenfunctions of H^ . Furthermore, the
functions \u142e(q) and \u142o(q) are also eigenfunctions of the parity operator \u2014^, the
first with eigenvalue ‡1 and the second with eigenvalue ÿ1.
3.9 Hellmann\u2013Feynman theorem
A useful expression for evaluating expectation values is known as the Hell-
mann\u2013Feynman theorem. This theorem is based on the observation that the
Hamiltonian operator for a system depends on at least one parameter º, which
can be considered for mathematical purposes to be a continuous variable. For
example, depending on the particular system, this parameter º may be the mass
of an electron or a nucleus, the electronic charge, the nuclear charge parameter
Z, a constant in the potential energy, a quantum number, or even Planck\u2019s
constant. The eigenfunctions and eigenvalues of H^(º) also depend on this
96 General principles of quantum theory
parameter, so that the time-independent Schro¨dinger equation (3.58) may be
written as
H^(º)\u142n(º) ˆ En(º)\u142n(º) (3:66)
The expectation value of H^(º) is, then
En(º) ˆ h\u142n(º)j H^(º)j\u142n(º)i (3:67)
where we assume that \u142n(º) is normalized
h\u142n(º)j\u142n(º)i ˆ 1 (3:68)
To obtain the Hellmann\u2013Feynman theorem, we differentiate equation (3.67)
with respect to º
d
dº
En(º) ˆ \u142n(º)
\ufffd\ufffd\ufffd\ufffd ddº H^(º)
\ufffd\ufffd\ufffd\ufffd\u142n(º)
* +
‡ d
dº
\u142n(º)
\ufffd\ufffd\ufffd\ufffd H^(º)\ufffd\ufffd\ufffd\ufffd\u142n(º)
* +
‡ \u142n(º)
\ufffd\ufffd\ufffd\ufffd H^(º)\ufffd\ufffd\ufffd\ufffd ddº\u142n(º)
* +
(3:69)
Applying the hermitian property of H^(º) to the third integral on the right-hand
side of equation (3.69) and then applying (3.66) to the second and third terms,
we obtain
d
dº
En(º) ˆ \u142n(º)
\ufffd\ufffd\ufffd\ufffd ddº H^(º)
\ufffd\ufffd\ufffd\ufffd\u142n(º)
* +
‡ En(º) d
dº
\u142n(º)
\ufffd\ufffd\ufffd\ufffd\u142n(º)
* +
‡ \u142n(º)
\ufffd\ufffd\ufffd\ufffd ddº\u142n(º)
* +" #
(3:70)
The derivative of equation (3.68) with respect to º is
d
dº
\u142n(º)
\ufffd\ufffd\ufffd\ufffd\u142n(º)
* +
‡ \u142n(º)
\ufffd\ufffd\ufffd\ufffd ddº\u142n(º)
* +
ˆ 0
showing that the last term on the right-hand side of (3.70) vanishes. We thereby
obtain the Hellmann\u2013Feynman theorem
d
dº
En(º) ˆ \u142n(º)
\ufffd\ufffd\ufffd\ufffd ddº H^(º)
\ufffd\ufffd\ufffd\ufffd\u142n(º)
* +
(3:71)
3.10 Time dependence of the expectation value
The expectation value hAi of the dynamical quantity or observable A is, in
general, a function of the time t. To determine how hAi changes with time, we
take the time derivative of equation (3.46)
3.10 Time dependence of the expectation value 97
dhAi
dt
ˆ d
dt
hØjA^jØi ˆ @Ø
@ t
\ufffd\ufffd\ufffd\ufffd A^ \ufffd\ufffd\ufffd\ufffdØ
* +
‡ Ø
\ufffd\ufffd\ufffd\ufffdA^ \ufffd\ufffd\ufffd\ufffd @Ø@ t
* +
‡ Ø
\ufffd\ufffd\ufffd\ufffd @A^@ t
\ufffd\ufffd\ufffd\ufffdØ
* +
Equation (3.55) may be substituted for the time derivatives of the wave function
to give
dhAi
dt
ˆ i
"
h H^ØjA^jØi ÿ i
"
hØjA^ H^ jØi ‡
*
Ø
\ufffd\ufffd\ufffd\ufffd @A^@ t
\ufffd\ufffd\ufffd\ufffdØ
+
ˆ i
"
hØj H^A^jØi ÿ i
"
hØjA^ H^ jØi ‡ Ø
\ufffd\ufffd\ufffd\ufffd @A^@ t
\ufffd\ufffd\ufffd\ufffdØ
* +
ˆ i
"
hØj[ H^ , A^]jØi ‡ Ø
\ufffd\ufffd\ufffd\ufffd @A^@ t
\ufffd\ufffd\ufffd\ufffdØ
* +
ˆ i
"
h[ H^ , A^]i ‡ @A^
@ t
\ufffd \ufffd
where the hermiticity of H^ and the definition (equation (3.3)) of the commu-
tator have been used. If the operator A^ is not an explicit function of time, then
the last term on the right-hand side vanishes and we have
dhAi
dt
ˆ i
"
h[ H^ , A^]i (3:72)
If we set A^ equal to unity, then the commutator [ H^ , A^] vanishes and equation
(3.72) becomes
dhAi
dt
ˆ 0
or
d
dt
hØjA^jØi ˆ d
dt
hØjØi ˆ 0
We thereby obtain the result in Section 2.2 that if Ø is normalized, it remains
normalized as time progresses.
If the operator A^ in equation (3.72) is set equal to H^ , then again the
commutator vanishes and we have
dhAi
dt
ˆ dhHi
dt
ˆ dE
dt
ˆ 0
Thus, the energy E of the system, which is equal to the expectation value of the
Hamiltonian, is conserved if the Hamiltonian does not depend explicitly on
time.
By setting the operator