Principles - Methods of Molecular Quantum Mechanics - page 5

1.2.9 Eigenvalue Equation

The equation

A cð x Þ¼ A cð x Þ

ð 1 : 19 Þ

is called the eigenvalue equation for the linear operator A. When (1.19) is
satisfied, the constant A is called the

the
eigenfunction of the operator A. Often, A is a differential operator, and
there may be a whole spectrum of eigenvalues, each one with its corre-
sponding eigenfunction. The spectrum of the eigenvalues can be either
discrete or continuous. An eigenvalue is said to be n-fold degenerate when
n different independent eigenfunctions belong to it. We shall see later that
the Schroedinger equation for the amplitude

eigenvalue, the function c

(x) is a typical eigenvalue
equation, where A ¼ H ¼ T þ V is the total energy operator (the
Hamiltonian), T being the kinetic energy operator and V the potential
energy characterizing the system (a scalar quantity).

c

1.2.10 Hermitian Operators

A Hermitian operator is a linear operator satisfying the so-called 'turn-
over rule':

<

hcj A wi¼h A cjwi
ð dx c ð dx ð A cð x ÞÞ

ð 1 : 20 Þ

ð x Þð A wð x ÞÞ ¼

:

wð x Þ

The Hermitian operators have the following properties:

(i) real eigenvalues;

(ii) orthogonal (or anyway orthogonalizable) eigenfunctions;

(iii) their eigenfunctions form a complete set.

Completeness also includes the eigenfunctions belonging to the contin-
uous part of the eigenvalue spectrum.
Hermitian operators are i

2 , T ¼ð h 2 2

2

=@ x 2 , r

@=@ x, i r , @ = 2m Þr

H ¼ T þ V, where

(i 2 and i is the imaginary unit ¼ 1),

r¼ i ð@=@ x Þþ j ð@=@ y Þþ k ð@=@ z Þ is the gradient vector operator,

2

=@ z 2 is the Laplacian operator
(in Cartesian coordinates), T is the kinetic energy operator for a particle
of mass m with

2

=@ x 2

2

=@ y 2

2

r ¼rr¼@ þ@ þ@

H is the

h ¼ h = 2 p the reduced Planck constant and

Hamiltonian operator.