Page 56 - PRINCIPLES OF QUANTUM MECHANICS as Applied to Chemistry and Chemical Physics

2.4 Time-independent Schro Èdinger equation 47
d÷(t)
i" ˆ E÷(t)
dt
which has the solution
÷(t) ˆ e ÿiEt=" (2:29)
The integration constant in equation (2.29) has arbitrarily been set equal to
unity. The spatial-dependent equation is
2
2
" d ø(x)
ÿ ‡ V(x)ø(x) ˆ Eø(x) (2:30)
2m dx 2
and is called the time-independent Schro Èdinger equation. The solution of this
differential equation depends on the speci®cation of the potential energy V(x).
Note that the separation of equation (2.6) into spatial and temporal parts is
contingent on the potential V(x) being time-independent.
The wave function Ø(x, t) is then
Ø(x, t) ˆ ø(x)e ÿiEt=" (2:31)
2
and the probability density jØ(x, t)j is now given by
2

jØ(x, t)j ˆ Ø (x, t)Ø(x, t) ˆ ø (x)e iEt=" ø(x)e ÿiEt=" ˆjø(x)j 2
Thus, the probability density depends only on the position variable x and does
not change with time. For this reason the wave function Ø(x, t) in equation
(2.31) is called a stationary state.If Ø(x, t) is normalized, then ø(x) is also
normalized

1
2
jø(x)j dx ˆ 1 (2:32)
ÿ1
which is the reason why we set the integration constant in equation (2.29) equal
to unity.
The total energy, when expressed in terms of position and momentum, is
called the Hamiltonian, H, and is given by
p 2
H(x, p) ˆ ‡ V(x)
2m
The expectation value hHi of the Hamiltonian may be obtained by applying
equation (2.22)

2
1 " @ 2

hHiˆ Ø (x, t) ÿ ‡ V(x) Ø(x, t)dx
2m @x 2
ÿ1
For the stationary state (2.31), this expression becomes

2 2
1 " @

hHiˆ ø (x) ÿ ‡ V(x) ø(x)dx
2m @x 2
ÿ1
If we substitute equation (2.30) into the integrand, we obtain