1Wavefunctions and the Schrodinger equation
IB Quantum Mechanics
1.3 Time evolution of wavefunctions
So far, everything is instantaneous. The wavefunction specifies the state at a
particular time, and the eigenvalues are the properties of the system at that
particular time. However, this is quantum mechanics, or quantum dynamics. We
should be looking at how things change. We want to know how the wavefunction
changes with time. This is what we will get to now.
Time-dependent Schr¨odinger equation
We will write Ψ instead of
ψ
to indicate that we are looking at the time-dependent
wavefunction. The evolution of this Ψ(
x, t
) is described by the time-dependent
Schr¨odinger equation.
Definition
(Time-dependent Schr¨odinger equation)
.
For a time-dependent
wavefunction Ψ(x, t), the time-dependent Schr¨odinger equation is
i~
∂Ψ
∂t
= HΨ. (∗)
For a particle in a potential V (x), this can reads
i~
∂Ψ
∂t
= −
~
2
2m
∂
2
Ψ
∂x
2
+ V (x)Ψ.
While this looks rather scary, it isn’t really that bad. First of all, it is linear. So
the sums and multiples of solutions are also solutions. It is also first-order in
time. So if we know the wavefunction Ψ(
x, t
0
) at a particular time
t
0
, then this
determines the whole function Ψ(x, t).
This is similar to classical dynamics, where knowing the potential
V
(and
hence the Hamiltonian
H
) completely specifies how the system evolves with
time. However, this is in some ways different from classical dynamics. Newton’s
second law is second-order in time, while this is first-order in time. This is
significant since when our equation is first-order in time, then the current state
of the wavefunction completely specifies the evolution of the wavefunction in
time.
Yet, this difference is just an illusion. The wavefunction is the state of the
particle, and not just the “position”. Instead, we can think of it as capturing the
position and momentum. Indeed, if we write the equations of classical dynamics
in terms of position and momentum, it will be first order in time.
Stationary states
It is not the coincidence that the time-independent Schr¨odinger equation and
the time-dependent Schr¨odinger equation are named so similarly (and it is also
not an attempt to confuse students).
We perform separation of variables, and consider a special class of solutions
Ψ(
x, t
) =
T
(
t
)
ψ
(
x
), where Ψ(
x,
0) =
ψ
(
x
) (i.e.
T
(0) = 1). If
ψ
satisfies the
time-independent Schr¨odinger equation
Hψ = Eψ,
then since
H
does not involve time derivatives, we know Ψ is an energy eigenstate
at each fixed t, i.e.
HΨ = EΨ.
So if we want this Ψ to satisfy the Schr¨odinger equation, we must have
i~
˙
T = ET.
The solution is obvious:
T (t) = exp
−
iEt
~
.
We can write our full solution as
Ψ(x, t) = ψ(x) exp
−
iEt
~
.
Note that the frequency is
ω
=
E
~
. So we recover the Energy-frequency relation
we’ve previously had.
Definition (Stationary state). A stationary state is a state of the form
Ψ(x, t) = ψ(x) exp
−
iEt
~
.
where
ψ
(
x
) is an eigenfunction of the Hamiltonian with eigenvalue
E
. This term
is also sometimes applied to ψ instead.
While the stationary states seem to be a rather peculiar class of solutions
that would rarely correspond to an actual physical state in reality, they are in
fact very important in quantum mechanics. The reason is that the stationary
states form a basis of the state space. In other words, every possible state can be
written as a (possibly infinite) linear combination of stationary states. Hence, by
understanding the stationary states, we can understand a lot about a quantum
system.
Conservation of probability
Note that for a stationary state, we have
|Ψ(x, t)|
2
= |ψ(x)|
2
,
which is independent of time. In general, this is true in most cases.
Consider a general Ψ(
x, t
) obeying the time-dependent Schr¨odinger equation.
Proposition. The probability density
P (x, t) = |Ψ(x, t)|
2
obeys a conservation equation
∂P
∂t
= −
∂j
∂x
,
where
j(x, t) = −
i~
2m
Ψ
∗
dΨ
dx
−
dΨ
∗
dx
Ψ
is the probability current.
Since Ψ
∗
Ψ
0
is the complex conjugate of Ψ
0∗
Ψ, we know that Ψ
∗
Ψ
0
−
Ψ
0∗
Ψ is
imaginary. So multiplying by
i
ensures that
j
(
x, t
) is real, which is a good thing
since P is also real.
Proof.
This is straightforward from the Schr¨odinger equation and its complex
conjugate. We have
∂P
∂t
= Ψ
∗
∂Ψ
∂t
+
∂Ψ∗
∂t
Ψ
= Ψ
∗
i~
2m
Ψ
00
−
i~
2m
Ψ
00∗
Ψ
where the two V terms cancel each other out, assuming V is real
= −
∂j
∂x
.
The important thing here is not the specific form of
j
, but that
∂P
∂t
can be
written as the space derivative of some quantity. This implies that the probability
that we find the particle in [a, b] at fixed time t changes as
d
dt
Z
b
a
|Ψ(x, t)|
2
dx =
Z
b
a
−
∂j
∂x
(x, t) dx = j(a, t) − j(b, t).
We can think of the final term as the probability current getting in and out of
the interval at the boundary.
In particular, consider a normalizable state with Ψ
,
Ψ
0
, j →
0 as
x → ±∞
for fixed t. Taking a → −∞ and b → +∞, we have
d
dt
Z
∞
−∞
|Ψ(x, t)|
2
dx = 0.
What does this tell us? This tells us that if Ψ(
x,
0) is normalized, Ψ(
x, t
) is
normalized for all
t
. Hence we know that for each fixed
t
,
|
Ψ(
x, t
)
|
2
is a probability
distribution. So what this really says is that the probability interpretation is
consistent with the time evolution.