Principles - Methods of Molecular Quantum Mechanics - page 16

where integration is extended over all space and A acts always on

Y and

. The last expression above shows that A is weighted with the
(normalized) probability density

not on Y

.

YY

1.3.3 Time Evolution of the State Function

Y

The state function is obtained by solving the time-dependent

Schroedinger equation:

h @Yð x ; t Þ
@ t

H Yð x ; t Þ¼ i

ð 1 :

76 Þ

a partial differential equation which is second order in the space
coordinate x and first order in the time t. This equation involves the
Hamiltonian of the system H,sothatthetotalenergyE is seen to play
a fundamental role among all physical observables.
If the Hamiltonian H does not depend explicitly on t (the case of
stationary states), then, following the usual mathematical techniques, the
variables in Equation (1.76) can be separated by writing Y as the product

of a space function cð x Þ and a time function g(t):

Yð x ; t Þ¼cð x Þ g ð t Þ ð 1 : 77 Þ

giving upon substitution

H cð x Þ¼ E cð x Þ
g ð t Þ¼ g 0 exp ð i

ð 1 : 78 Þ

v t Þ

where E is the separation constant, g 0 is an integration constant, and
v ¼ E = h. The first part of Equation (1.78) is the eigenvalue equation for
the total energy operator (the Hamiltonian) of the system, and

cð x Þ is
called the amplitude function. This is the Schroedinger equation that we
must solve or approximate for the physical description of our systems. The
second equation gives the time dependence of the stationary state, while
general time dependence is fundamental in spectroscopy. It is immediately
evident that, for the stationary state, the probability

dx is indepen-

YY

dent of time:

2 dx ¼jcð x Þj 2 2 dx

Yð x ; t ÞY ð x ; t Þ dx ¼jYð x ; t Þj j g 0 j ð 1 :

79 Þ