V.
Time Dependence
Up to this point, the systems we have dealt with have been time
independent. The variable
t
has not appeared in any of our
equations and thus the formalism we have developed is not yet able
to describe any kind of motion. This is at odds with our
understanding of reality, where things clearly
move
. In order to
describe quantum motion, we need one additional rule of quantum
mechanics:
The time evolution of any state is governed by
Schrödinger’s equation
∂
i
Z
ψ
(
t
)
=
H
ˆ
ψ
(
t
)
.
∂
t
Note that in quantum mechanics time,
t
, plays a fundamentally
different role than the position,
q
ˆ
, or momentum,
p
ˆ
. The latter two
are represented by
operators
that act on the states, while the
time is
treated as a parameter
. The state of the system depends on the
parameter,
t
, but it makes no sense to have a state that depends on
an operator like
q
ˆ
. That is to say,
ψ
(
t
)
is welldefined but
ψ
(
q
ˆ
)
is
not.
In most cases, the dependence on
t
is understood and we can write
the shorthand version of Schrödinger’s equation:
ˆ
"
i
Z
ψ
=
H
ψ
.
Time dependent quantum systems are the primary focus of the
second half of this course. However, it is appropriate at this point to
at least introduce the basic principles of quantum dynamics,
especially focusing on how it relates to the time independent
framework we’ve developed.
A. Energy Eigenstates Are Stationary States
First, we want to address the very important question of how
eigenstates of the Hamiltonian (i.e. energy eigenstates) evolve with
time. Applying Schrödinger’s equation,
ˆ
"
i
Z
ψ
=
H
ψ
=
E
ψ
n
n
n
n
This is just a simple first order differential equation for
ψ
(
t
)
and it is
n
easily verified that the general solution is:
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−
iE
n
t
/
Z
ψ
(
t
)
=
e
ψ
(
0
)
n
n
Thus, if the system starts in an energy eigenstate, it will remain in this
eigenstate. The only effect of the time evolution is to multiply the
state by a timedependent phase factor (
e
−
iE
n
t
/
Z
). Since an overall
phase factor cannot influence the outcome of an observation, from an
experimental perspective,
energy eigenstates do not change with
time
. It is therefore termed a “stationary state”.
This motivates our
interest in finding energy eigenstates for arbitrary Hamiltonians; any
other state has the potential to change between observations, but a
stationary state lives forever if we don’t disturb it.
B. The Propagator Governs Time Evolution
So it is trivial to determine
ψ
(
t
)
if the system begins in a stationary
state. But what if the initial state is not an eigenfunction of the
Hamiltonian? How do we evolve an arbitrary
ψ
(
t
)
? As we show
below, time evolution is governed by the propagator,
ˆ
−
t
H
i
/
Z
K
ˆ
(
t
)
≡
e
in terms of which the time evolved state is given by
ψ
(
t
)
=
K
ˆ
(
t
)
ψ
(
0
)
.
In order to prove that this is so, we merely take the derivative of the
propagator ansatz and verify it satisfies Schrödinger’s equation:
∂
ˆ
−
t
H
i
/
Z
i
Z
ψ
(
t
)
=
i
Z
∂
e
ψ
(
0
)
∂
t
∂
t
ˆ
ˆ
ˆ
ˆ
ˆ
ˆ
=
i
Z
∂
(
1
−
i
t
H
−
1
H t
H
2
+
i
HH
t
H
3
+
...
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 Spring '04
 RobertField

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