4
Time dependence
So far, we have used the timeindependent Schr¨
odinger equation to find stationary
states. These stationary states do not change with time. We can see this by looking at
the timedependent Schr¨
odinger equation
H
Ψ(
r, t
) =
−
2
2
m
∇
2
Ψ(
r, t
) +
V
(
r
)Ψ(
r, t
) =
i
∂
∂t
Ψ(
r, t
)
(4.1)
which relates the Hamiltonian operator to the timederivative of the wavefunction. I’ve
used the spatial variable
r
to indicate that this can be a multidimensional wavefunc
tion.
In the case of a stationary state, the wavefunction is an eigenfunction of the
Hamiltonian,
H
Ψ(
r, t
) =
E
Ψ(
r, t
)
,
(4.2)
and the solutions are separable in time and space (in the same way that the solutions
to a two dimensional box separated in
x
and
y
). Using the same separation of variables
technique that we used for the two dimensional box, we can try a solution of the form
Ψ(
r, t
) =
ψ
(
r
)
T
(
t
)
,
(4.3)
where
ψ
(
r
) satisfies
−
2
2
m
∇
2
ψ
(
r
) +
V
(
r
)
ψ
(
r
) =
E ψ
(
r
)
(4.4)
and the time dependence,
T
(
t
), satisfies
i
∂T
(
t
)
∂t
=
E T
(
t
)
.
(4.5)
The spatial equation is just the time independent Schr¨
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 Fall '06
 henkelman
 Physical chemistry, Space, pH, Spacetime, Fundamental physics concepts, stationary state

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