Page 102 - PRINCIPLES OF QUANTUM MECHANICS as Applied to Chemistry and Chemical Physics
P. 102
3.7 Postulates of quantum mechanics 93
@Ø
^
i" HØ (3:55)
@t
^
where H is the Hamiltonian operator of the system and, in general, changes
with time. However, in this book we only consider systems for which the
Hamiltonian operator is time-independent. To solve the time-dependent SchroÈ-
dinger equation, we express the state function Ø(q, t) as the product of two
functions
Ø(q, t) ø(q)÷(t) (3:56)
where ø(q) depends only on the spatial variables and ÷(t) depends only on the
time. In Section 2.4 we discuss the procedure for separating the partial
differential equation (3.55) into two differential equations, one involving only
the spatial variables and the other only the time. The state function Ø(q, t)is
then shown to be
Ø(q, t) ø(q)e ÿiEt=" (3:57)
where E is the separation constant. Since it follows from equation (3.57) that
2 2
jØ(q, t)j jø(q)j
the probability density is independent of the time t and Ø(q, t)isa stationary
state.
The spatial differential equation, known as the time-independent Schrodin-
È
ger equation, is
^
Hø(q) Eø(q)
Thus, the spatial function ø(q) is actually a set of eigenfunctions ø n (q) of the
^
Hamiltonian operator H with eigenvalues E n . The time-independent SchroÈdin-
ger equation takes the form
^
Hø n (q) E n ø n (q) (3:58)
and the general solution of the time-dependent Schrodinger equation is
È
X
Ø(q, t) c n ø n (q)e ÿiE n t=" (3:59)
n
where c n are arbitrary complex constants.
The appearance of the Hamiltonian operator in equation (3.55) as stipulated
by postulate 5 gives that operator a special status in quantum mechanics.
Knowledge of the eigenfunctions and eigenvalues of the Hamiltonian operator
for a given system is suf®cient to determine the stationary states of the system
and the expectation values of any other dynamical variables.
We next address the question as to whether equation (3.59) is actually the
È
most general solution of the time-dependent Schrodinger equation. Are there
other solutions that are not expressible in the form of equation (3.59)? To
