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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coordinate x, for which w(x) ˆ 1. An operator that results
in multiplying by a real function f (x) is hermitian, since in this case
f (x)\ufffd ˆ f (x) and equation (3.8) is an identity. Likewise, the momentum
operator p^ ˆ ("=i)(d=dx), which was introduced in Section 2.3, is hermitian
since…1
ÿ1
\u142\ufffdj p^\u142i dx ˆ
…1
ÿ1
\u142\ufffdj
"
i
d\u142i
dx
dx ˆ "
i
\u142\ufffdj \u142i
\ufffd\ufffd\ufffd\ufffd1
ÿ1
ÿ "
i
…1
ÿ1
\u142i
d\u142\ufffdj
dx
dx
The integrated part is zero if the functions \u142i vanish at infinity, which they
must in order to be well-behaved. The remaining integral is
„
\u142i p^\ufffd\u142\ufffdj dx, so
that we have …1
ÿ1
\u142\ufffdj p^\u142i dx ˆ
…1
ÿ1
\u142i( p^\u142 j)
\ufffd dx
The imaginary unit i contained in the operator p^ is essential for the
hermitian character of that operator. The operator D^x ˆ d=dx is not hermitian
because …1
ÿ1
\u142\ufffdj
d\u142i
dx
dx ˆ ÿ
…1
ÿ1
\u142i
d\u142\ufffdj
dx
dx (3:10)
where again the integrated part vanishes. The negative sign on the right-hand
side of equation (3.10) indicates that the operator is not hermitian. The operator
D^2x, however, is hermitian.
The hermitian character of an operator depends not only on the operator
itself, but also on the functions on which it acts and on the range of integration.
An operator may be hermitian with respect to one set of functions, but not with
respect to another set. It may be hermitian with respect to a set of functions
defined over one range of variables, but not with respect to the same set over a
different range. For example, the hermiticity of the momentum operator p^ is
dependent on the vanishing of the functions \u142i at infinity.
The product of two hermitian operators may or may not be hermitian.
Consider the product A^B^ where A^ and B^ are separately hermitian with respect
to a set of functions \u142i, so that
h\u142 j j A^B^\u142ii ˆ hA^\u142 j j B^\u142ii ˆ hB^A^\u142 j j\u142ii (3:11)
where we have assumed that the functions A^\u142i and B^\u142i also lie in the hermitian
domain of A^ and B^. The product A^B^ is hermitian if, and only if, A^ and B^
commute. Using the same procedure, one can easily demonstrate that if A^ and
B^ do not commute, then the operators (A^B^‡ B^A^) and i[A^, B^] are hermitian.
By setting B^ equal to A^ in the product A^B^ in equation (3.11), we see that the
square of a hermitian operator is hermitian. This result can be generalized to
70 General principles of quantum theory
any integral power of A^. Since jA^\u142j2 is always positive, the integral hA^\u142 j A^\u142i
is positive and consequently
h\u142 j A^2\u142i > 0 (3:12)
Eigenvalues
The eigenvalues of a hermitian operator are real. To prove this statement, we
consider the eigenvalue equation
A^\u142 ˆ Æ\u142 (3:13)
where A^ is hermitian, \u142 is an eigenfunction of A^, and Æ is the corresponding
eigenvalue. Multiplying by \u142\ufffd and integrating give
h\u142 j A^\u142i ˆ Æh\u142 j\u142i (3:14)
Multiplication of the complex conjugate of equation (3.13) by \u142 and integrat-
ing give
hA^\u142 j\u142i ˆ Æ\ufffdh\u142 j\u142i (3:15)
Because A^ is hermitian, the left-hand sides of equations (3.14) and (3.15) are
equal, so that
(Æÿ Æ\ufffd)h\u142 j\u142i ˆ 0 (3:16)
Since the integral in equation (3.16) is not equal to zero, we conclude that
Æ ˆ Æ\ufffd and thus Æ is real.
Orthogonality theorem
If \u1421 and \u1422 are eigenfunctions of a hermitian operator A^ with different
eigenvalues Æ1 and Æ2, then \u1421 and \u1422 are orthogonal. To prove this theorem,
we begin with the integral
h\u1422 j A^\u1421i ˆ Æ1h\u1422 j\u1421i (3:17)
Since A^ is hermitian and Æ2 is real, the left-hand side may be written as
h\u1422 j A^\u1421i ˆ hA^\u1422 j\u1421i ˆ Æ2h\u1422 j\u1421i
Thus, equation (3.17) becomes
(Æ2 ÿ Æ1)h\u1422 j\u1421i ˆ 0
Since Æ1 6ˆ Æ2, the functions \u1421 and \u1422 are orthogonal.
Since the Dirac notation suppresses the variables involved in the integration,
we re-express the orthogonality relation in integral notation…
\u142\ufffd2 (q1, q2, . . .)\u1421(q1, q2, . . .)w(q1, q2, . . .) dq1 dq2 . . . ˆ 0
This expression serves as a reminder that, in general, the eigenfunctions of a
3.3 Hermitian operators 71
hermitian operator involve several variables and that the weighting function
must be used. The functions are, therefore, orthogonal with respect to the
weighting function w(q1, q2, . . .).
If the weighting function is real and positive, then we can define ö1 and ö2
as
ö1 ˆ

w
p
\u1421, ö2 ˆ

w
p
\u1422
The functions ö1 and ö2 are then mutually orthogonal with respect to a
weighting function of unity. Moreover, if the operator A^ is hermitian with
respect to \u1421 and \u1422 with a weighting function w, then A^ is hermitian with
respect to ö1 and ö2 with a weighting function equal to unity.
If two or more linearly independent eigenfunctions have the same eigen-
value, so that the eigenvalue is degenerate, the orthogonality theorem does not
apply. However, it is possible to construct eigenfunctions that are mutually
orthogonal. Suppose there are two independent eigenfunctions \u1421 and \u1422 of
the operator A^ with the same eigenvalue Æ. Any linear combination
c1\u1421 ‡ c2\u1422, where c1 and c2 are any pair of complex numbers, is also an
eigenfunction of A^ with the same eigenvalue, so that
A^(c1\u1421 ‡ c2\u1422) ˆ c1A^\u1421 ‡ c2A^\u1422 ˆ Æ(c1\u1421 ‡ c2\u1422)
From any pair \u1421, \u1422 which initially are not orthogonal, we can construct by
selecting appropriate values for c1 and c2 a new pair which are orthogonal. By
selecting different sets of values for c1, c2, we may obtain infinitely many new
pairs of eigenfunctions which are mutually orthogonal.
As an illustration, suppose the members of a set of functions \u1421, \u1422, . . . , \u142n
are not orthogonal. We define a new set of functions ö1, ö2, . . . , ön by the
relations
ö1 ˆ \u1421
ö2 ˆ aö1 ‡ \u1422
ö3 ˆ b1ö1 ‡ b2ö2 ‡ \u1423
..
.
If we require that ö2 be orthogonal to ö1 by setting hö1 jö2i ˆ 0, then the
constant a is given by
a ˆ ÿh\u1421 j\u1422i=h\u1421 j\u1421i ˆ ÿhö1 j\u1422i=hö1 jö1i
and ö2 is determined. We next require ö3 to be orthogonal to ö1 and to ö2,
which gives
72 General principles of quantum theory
b1 ˆ ÿhö1 j\u1423i=hö1 jö1i
b2 ˆ ÿhö2 j\u1423i=hö2 jö2i
In general, we have
ös ˆ \u142s ‡
Xsÿ1
iˆ1
ksiöi
ksi ˆ ÿhöi j\u142si=höi jöii
This construction is known as the Schmidt orthogonalization procedure. Since
the initial selection for ö1 can be any of the original functions \u142i or any linear
combination of them, an infinite number of orthogonal sets öi can be obtained
by the Schmidt procedure.
We conclude that all eigenfunctions of a hermitian operator are either
mutually orthogonal or, if belonging to a degenerate eigenvalue, can be chosen
to be mutually orthogonal. Throughout the remainder of this book, we treat all
the eigenfunctions of a hermitian operator as an orthogonal set.
Extended orthogonality theorem
The orthogonality theorem can also be extended to cover a somewhat more
general form of the eigenvalue equation. For the sake of convenience, we
present in detail the case of a single variable, although the treatment can be
generalized to any number of variables. Suppose that instead of the eigenvalue
equation (3.5), we have for a hermitian operator A^ of one variable
A^\u142i(x) ˆ Æiw(x)\u142i(x) (3:18)
where the function w(x) is real, positive, and the same for all values of i.
Therefore, equation (3.18) can also be written as
A^\ufffd\u142\ufffdj (x) ˆ Æ\ufffdj w(x)\u142\ufffdj (x) (3:19)
Multiplication of equation (3.18) by \u142\ufffdj (x) and integration over x give…
\u142\ufffdj (x)A^\u142i(x) dx ˆ Æi
…
\u142\ufffdj (x)\u142i(x)w(x) dx (3:20)
Now, the operator A^ is hermitian with respect to the functions \u142i with a
weighting function equaling unity, so that the integral on the left-hand side of
equation (3.20) becomes…
\u142\ufffdj (x)A^\u142i(x) dx ˆ
…
\u142i(x)A^
\ufffd\u142\ufffdj (x) dx ˆ Æ\ufffdj
…
\u142\ufffdj (x)\u142i(x)w(x) dx
where equation (3.19) has been used as well. Accordingly, equation (3.20)
becomes
3.3 Hermitian operators 73
(Æi ÿ Æ\ufffdj )
…
\u142\ufffdj (x)\u142i(x)w(x) dx ˆ 0 (3:21)
When j ˆ i, the integral in equation (3.21) cannot vanish because the
product