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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it satisfies two criteria
A^(\u1421 ‡ \u1422) ˆ A^\u1421 ‡ A^\u1422 (3:2a)
A^(c\u142) ˆ cA^\u142 (3:2b)
where c is any complex constant. In the three examples given above, the
operators are linear. Some nonlinear operators are \u2018exp\u2019 (take the exponential
of) and [ ]2 (take the square of), since
ex‡ y ˆ exe y 6ˆ ex ‡ e y
e cx 6ˆ cex
[x‡ y]2 ˆ x2 ‡ 2xy‡ y2 6ˆ x2 ‡ y2
[c(x‡ y)]2 6ˆ c[x‡ y]2
The operator C^ is the sum of the operators A^ and B^ if
C^\u142 ˆ (A^‡ B^)\u142 ˆ A^\u142‡ B^\u142
The operator C^ is the product of the operators A^ and B^ if
C^\u142 ˆ A^B^\u142 ˆ A^(B^\u142)
where first B^ operates on \u142 and then A^ operates on the resulting function.
Operators obey the associative law of multiplication, namely
A^(B^C^) ˆ (A^B^)C^
Operators may be combined. Thus, the square A^2 of an operator A^ is just the
product A^A^
A^2\u142 ˆ A^A^\u142 ˆ A^(A^\u142)
Similar definitions apply to higher powers of A^. As another example, the
differential equation
d2 y
‡ k2 y ˆ 0
may be written as (D^2x ‡ k2)y ˆ 0, where the operator (D^2x ‡ k2) is the sum of
the two product operators D^2x and k
In multiplication, the order of A^ and B^ is important because A^B^\u142 is not
necessarily equal to B^A^\u142. For example, if A^ ˆ x and B^ ˆ D^x, then we have
A^B^\u142 ˆ xD^x\u142 ˆ x(d\u142=dx) while, on the other hand, B^A^\u142 ˆ D^x(x\u142) ˆ
\u142‡ x(d\u142=dx). The commutator of A^ and B^, written as [A^, B^], is an operator
defined as
[A^, B^] ˆ A^B^ÿ B^A^ (3:3)
from which it follows that [A^, B^] ˆ ÿ[B^, A^]. If A^B^\u142 ˆ B^A^\u142, then we have
A^B^ ˆ B^A^ and [A^, B^] ˆ 0; in this case we say that A^ and B^ commute. By
expansion of each side of the following expressions, we can readily prove the
[A^, B^C^] ˆ [A^, B^]C^ ‡ B^[A^, C^] (3:4a)
[A^B^, C^] ˆ [A^, C^]B^‡ A^[B^, C^] (3:4b)
The operator A^ is the reciprocal of B^ if A^B^ ˆ B^A^ ˆ 1, where 1 may be
regarded as the unit operator, i.e., \u2018multiply by unity\u2019. We may write A^ ˆ B^ÿ1
66 General principles of quantum theory
and B^ ˆ A^ÿ1. If the operator A^ possesses a reciprocal, it is non-singular, in
which case the expression ö ˆ A^\u142 may be solved for \u142, giving \u142 ˆ A^ÿ1ö. If
A^ possesses no reciprocal, it is singular and the expression ö ˆ A^\u142 may not be
3.2 Eigenfunctions and eigenvalues
Consider a finite set of functions f i and the relationship
c1 f1 ‡ c2 f 2 ‡ \ufffd \ufffd \ufffd ‡ cn f n ˆ 0
where c1, c2, . . . are complex constants. If an equation of this form exists, then
the functions are linearly dependent. However, if no such relationship exists,
except for the trivial one with c1 ˆ c2 ˆ \ufffd \ufffd \ufffd ˆ cn ˆ 0, then the functions are
linearly independent. This definition can be extended to include an infinite set
of functions.
In general, the function ö obtained by the application of the operator A^ on
an arbitrary function \u142, as expressed in equation (3.1), is linearly independent
of \u142. However, for some particular function \u1421, it is possible that
A^\u1421 ˆ Æ1\u1421
where Æ1 is a complex number. In such a case \u1421 is said to be an eigenfunction
of A^ and Æ1 is the corresponding eigenvalue. For a given operator A^, many
eigenfunctions may exist, so that
A^\u142i ˆ Æi\u142i (3:5)
where \u142i are the eigenfunctions, which may even be infinite in number, and Æi
are the corresponding eigenvalues. Each eigenfunction of A^ is unique, that is to
say, is linearly independent of the other eigenfunctions.
Sometimes two or more eigenfunctions have the same eigenvalue. In that
situation the eigenvalue is said to be degenerate. When two, three, . . . , n
eigenfunctions have the same eigenvalue, the eigenvalue is doubly, triply, . . . ,
n-fold degenerate. When an eigenvalue corresponds only to a single eigenfunc-
tion, the eigenvalue is non-degenerate.
A simple example of an eigenvalue equation involves the operator D^x
mentioned in Section 3.1. When D^x operates on e
kx, the result is
kx ˆ d
ekx ˆ kekx
Thus, the exponentials ekx are eigenfunctions of D^x with corresponding
eigenvalues k. Since both the real part and the imaginary part of k can have
any values from ÿ1 to ‡1, there are an infinite number of eigenfunctions
and these eigenfunctions form a continuum of functions.
3.2 Eigenfunctions and eigenvalues 67
Another example is the operator D^2x acting on either sin nx or cos nx, where
n is a positive integer (n > 1), for which we obtain
D^2x sin nx ˆ ÿn2 sin nx
D^2x cos nx ˆ ÿn2 cos nx
The functions sin nx and cos nx are eigenfunctions of D^2x with eigenvalues
ÿn2. Although there are an infinite number of eigenfunctions in this example,
these eigenfunctions form a discrete, rather than a continuous, set.
In order that the eigenfunctions \u142i have physical significance in their
application to quantum theory, they are chosen from a special class of func-
tions, namely, those which are continuous, have continuous derivatives, are
single-valued, and are square integrable. We refer to functions with these
properties as well-behaved functions. Throughout this book we implicitly
assume that all functions are well-behaved.
Scalar product and orthogonality
The scalar product of two functions \u142(x) and ö(x) is defined as…1
ö\ufffd(x)\u142(x) dx
For functions of the three cartesian coordinates x, y, z, the scalar product of
\u142(x, y, z) and ö(x, y, z) is…1
ö\ufffd(x, y, z)\u142(x, y, z) dx dy dz
For the functions \u142(r, \u141, j) and ö(r, \u141, j) of the spherical coordinates r, \u141,
j, the scalar product is…2ð
ö\ufffd(r, \u141, j)\u142(r, \u141, j)r2 sin \u141 dr d\u141 dj
In order to express equations in general terms, we adopt the notation
dô to
indicate integration over the full range of all the coordinates of the system
being considered and write the scalar product in the form…
ö\ufffd\u142 dô
For further convenience we also introduce a notation devised by Dirac and
write the scalar product of \u142 and ö as hö j\u142i, so that
hö j\u142i \ufffd
ö\ufffd\u142 dô
The significance of this notation is discussed in Section 3.6. From the definition
of the scalar product and of the notation hö j\u142i, we note that
68 General principles of quantum theory
hö j\u142i\ufffd ˆ h\u142 jöi
hö j c\u142i ˆ chö j\u142i
hcö j\u142i ˆ c\ufffdhö j\u142i
where c is an arbitrary complex constant. Since the integral h\u142 j\u142i\ufffd equals
h\u142 j\u142i, the scalar product h\u142 j\u142i is real.
If the scalar product of \u142 and ö vanishes, i.e., if hö j\u142i ˆ 0, then \u142 and ö
are said to be orthogonal. If the eigenfunctions \u142i of an operator A^ obey the
h\u142 j j\u142ii ˆ 0 all i, j with i 6ˆ j
the functions \u142i form an orthogonal set. Furthermore, if the scalar product of
\u142i with itself is unity, the function \u142i is said to be normalized. A set of
functions which are both orthogonal to one another and normalized are said to
be orthonormal
h\u142 j j\u142ii ˆ äij (3:6)
where äij is the Kronecker delta function,
äij ˆ 1, i ˆ j
ˆ 0, i 6ˆ j (3:7)
3.3 Hermitian operators
The linear operator A^ is hermitian with respect to the set of functions \u142i of the
variables q1, q2, . . . if it possesses the property that…
\u142\ufffdj A^\u142i dô ˆ
\u142i(A^\u142 j)
\ufffd dô (3:8)
The integration is over the entire range of all the variables. The differential dô
has the form
dô ˆ w(q1, q2, . . .) dq1 dq2 . . .
where w(q1, q2, . . .) is a weighting function that depends on the choice of the
coordinates q1, q2, . . . For cartesian coordinates the weighting function
w(x, y, z) equals unity; for spherical coordinates, w(r, \u141, j) equals r2 sin\u141.
Special variables introduced to simplify specific problems have their own
weighting functions, which may differ from unity (see for example Section
6.3). Equation (3.8) may also be expressed in Dirac notation
h\u142 j j A^\u142ii ˆ hA^\u142 j j\u142ii (3:9)
in which the brackets indicate integration over all the variables using their
weighting function.
3.3 Hermitian operators 69
For illustration, we consider some examples involving only one variable,
namely, the cartesian