Biomedical Engineering Reference
In-Depth Information
11.2 The Time-Independent Schrödinger Equation
I am going to use a standard technique from the theory of differential equations called
separation of variables
to simplify the time-dependent equation; I want to show you that
when the potential does not depend on time, we can 'factor out' the time dependence from
the time-dependent Schrödinger equation and just concentrate on the spatial problem. I will
therefore investigate the possibility that we can write Ψ (
r
,
t
) as a product of functions of
the spatial (
r
) and time (
t
) variables, given that the potential only depends on
r
. We write
Ψ (
r
,
t
)
=
ψ (
r
)
T
(
t
)
(11.5)
and substitute into the time-dependent Schrödinger equation to get
−
∂
z
2
U
(
r
,
t
)
ψ (
r
)
T
(
t
)
∂
2
∂
x
2
+
∂ψ (
r
)
T
(
t
)
∂
t
Notice that the differential operators are partial ones and they only operate on functions
that contain them. So we rearrange the above equation
T
(
t
)
h
2
8π
2
m
∂
2
∂
y
2
+
∂
2
h
2π
+
=
j
∂
2
∂
x
2
+
∂
z
2
U
(
r
,
t
)
ψ
(
r
)
h
2
8π
2
m
∂
2
∂
y
2
+
∂
2
h
2π
d
T
(
t
)
d
t
−
+
=
j
ψ
(
r
)
divide by Ψ (
r
,
t
) and simplify to get
1
ψ (
r
)
−
∂
2
∂
x
2
+
U
(
r
,
t
)
ψ (
r
)
d
T
(
t
)
d
t
The left hand side of the equation depends on the spatial variable
r
and the time variable
t
.
Suppose now that the potential depends only on
r
and not on
t
. That is to say
1
ψ (
r
)
h
2
8π
2
m
∂
2
∂
y
2
+
∂
2
∂
z
2
1
T
(
t
)
h
2π
+
=
j
−
∂
2
∂
x
2
+
U
(
r
)
ψ (
r
)
h
2
8π
2
m
∂
2
∂
y
2
+
∂
2
∂
z
2
1
T
(
t
)
h
2π
d
T
(
t
)
d
t
+
=
j
(11.6)
Under these circumstances, the left-hand side of the equation depends only on
r
and the
right-hand side only on
t
. The spatial and the time variables are completely independent
and so the right-hand side and the left-hand side must both equal a constant. In the theory
of differential equations, this constant is called the
separation constant
. In this case it is
equal to the total energy so I will write it as ε:
1
ψ (
r
)
−
∂
2
∂
x
2
+
U
(
r
)
ψ (
r
)
h
2
8π
2
m
∂
2
∂
y
2
+
∂
2
∂
z
2
+
=
ε
1
T
(
t
)
h
2π
d
T
(
t
)
d
t
j
=
ε
The second equation can be solved to give
A
exp
2π
j
ε
t
h
=
−
T
(
t
)
(11.7)
and so the full solution to the time-dependent equation in the special case that the potential
is time independent can be written
ψ (
r
) exp
2π
j
ε
t
h
Ψ (
r
,
t
)
=
−
(11.8)
