1
ECE 3060: Introduction to Quantum and Statistical Mechanics
Handout 4
Time-Independent Schrödinger Equation and Time Evolution
Textbook Reading:
Hagelstein, Senturia and Orlando, Chap. 6 (pp. 89-110).
4.1.
Separation of space and time dependence in Schrödinger equation
By applying the principle of the eigenvalue problems to the Schrödinger equation, we can
come up with an important subclass of quantum mechanics when the potential
V(x)
does not
change with time.
We will examine the “time-dependent” Schrödinger equation:
(
)
(
)
(
)
(
)
(
)
t
x
ψ
t
i
t
x
ψ
E
t
x
ψ
x
V
t
x
ψ
x
m
t
x
ψ
H
,
,
ˆ
,
)
(
,
2
,
ˆ
2
2
2
∂
∂
=
=
+
∂
∂
−
=
h
h
(4.1)
We will use the principle of separation of variables and assume that the wave function
can be decoupled as:
)
(
)
(
)
,
(
t
T
x
X
t
x
ψ
=
(4.2)
Notice that this will just include a sub-class of wave functions, as there are many
functions where the separation of space and time is not possible.
Substituting
X(x)T(t)
into the
Schrödinger equation and doing some term rearrangement, we obtain:
E
x
X
x
X
x
V
x
X
dx
d
m
t
T
t
T
dt
d
i
=
⎟
⎟
⎠
⎞
⎜
⎜
⎝
⎛
+
−
=
⎟
⎠
⎞
⎜
⎝
⎛
)
(
)
(
)
(
)
(
2
)
(
)
(
2
2
2
h
h
(4.3)
The first term depends ONLY on
t
, while the second term depends ONLY on
x
.
Therefore, they must be equal to a constant, here we name it,
E
(it will be the energy eigenvalue
indeed).
We will now decouple the Schrödinger equation to two equations that need to be
simultaneously satisfied:
)
(
)
(
t
ET
t
T
dt
d
i
=
h
(4.4)
)
(
)
(
)
(
)
(
2
2
2
2
x
EX
x
X
x
V
x
X
dx
d
m
=
+
−
h
(4.5)
The time equation above is easy to solve, and we have
h
/
)
(
iEt
Ce
t
T
−
=
(4.6)

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