2
You will notice that the evolution in the plane wave manner is actually quite general in
the Schrödinger equation, as general as the separation of variables.
We can also write the space
equation, replacing
X(x)
with
ϕ
(x)
as a more conventional symbol for wave function.
)
(
)
(
)
(
)
(
2
2
2
2
x
E
x
x
V
x
dx
d
m
ψ
φ
=
+
−
h
(
4
.
7
)
h
/
)
(
)
,
(
iEt
e
x
t
x
−
=
(
4
.
8
)
(
Exercise 4.1
) Is the plane wave for
(x,t)=e
i(kx
ω
t)
a necessary or sufficient condition for Eqs.
(4.7) and (4.8)
?
(
Exercise 4.2
) For the procedure that we use to find the time evolution of freeparticle
(x,t)
with
(x,0)=
0
(x)
through
Fourier
transforms
by
∫
∞
∞
−
−
=
dx
e
x
ψ
k
A
ikx
)
(
)
(
0
and
∫
∞
∞
−
−
=
π
dk
e
k
A
t
x
ψ
t
k
ω
kx
i
2
)
(
)
,
(
)
)
(
(
, what is different here by using
h
/
)
(
)
,
(
iEt
e
x
t
x
−
=
?
4.2
Eigenfunctions of the Hamiltonian
Equations (4.7) and (4.8) are called the timeindependent Schrödinger equation.
Although it is still difficult to solve, it is much easier to visualize the physical system:
(x),
as
the eigenfunction for
H
ˆ
, will evolve in time with
h
/
)
(
iEt
e
x
−
, where
E
is the corresponding
eigenvalue.
E
is also the expectation value for the Hamiltonian operator
H
ˆ
with the
eigenfunction
(x),
since
>
=<
=
>
<
>
<
=
>
<
>
<
>=
<

ˆ

)
(

)
(
)
(

ˆ

)
(
)
,
(

)
,
(
)
,
(

ˆ

)
,
(
ˆ
H
E
x
x
x
H
x
t
x
t
x
t
x
H
t
x
H
(4.9)
There is another property for the wave function
h
/
)
(
)
,
(
iEt
e
x
t
x
−
=
.
Notice that
)
(
)
(
)
(
)
(
)
,
(
)
,
(
)
,
(
2
*
*
x
P
x
ψ
x
ψ
x
ψ
t
x
ψ
t
x
ψ
t
x
P
=
=
=
=
(
4
.
1
0
)
The probability density
P(x,t)
does not have a time dependence for the eigenfunctions of
H
ˆ
, and we thus refer the wave function
h
/
)
(
)
,
(
iEt
e
x
t
x
−
=
as a
stationary state
.
Notice that
this is ONLY true for each eigenfunction, but not for its linear superposition.