4More results in one dimensions
IB Quantum Mechanics
4.2 Scattering
Consider the time-dependent Schr¨odinger equation with a potential barrier. We
would like to send a wavepacket towards the barrier and see what happens.
u
Ψ
Classically, we would expect the particle to either pass through the barrier or get
reflected. However, in quantum mechanics, we would expect it to “partly” pass
through and “partly” get reflected. So the resultant wave is something like this:
AΨ
ref
BΨ
tr
Here Ψ, Ψ
ref
and Ψ
tr
are normalized wavefunctions, and
P
ref
= |A|
2
, P
tr
= |B|
2
.
are the probabilities of reflection and transmission respectively.
This is generally hard to solve. Scattering problems are much simpler to
solve for momentum eigenstates of the form
e
ikx
. However, these are not
normalizable wavefunctions, and despite being mathematically convenient, we
are not allowed to use them directly, since they do not make sense physically.
These, in some sense, represent particles that are “infinitely spread out” and
can appear anywhere in the universe with equal probability, which doesn’t really
make sense.
There are two ways we can get around this problem. We know that we can
construct normalized momentum eigenstates for a single particle confined in a
box −
`
2
≤ x ≤
`
2
, namely
ψ(x) =
1
√
`
e
ikx
,
where the periodic boundary conditions require
ψ
(
x
+
`
) =
ψ
(
x
), i.e.
k
=
2πn
`
for some integer
n
. After calculations have been done, the box can be removed
by taking the limit ` → ∞.
Identical results are obtained more conveniently by allowing Ψ(
x, t
) to repre-
sent beams of infinitely many particles, with
|
Ψ(
x, t
)
|
2
being the density of the
number of particles (per unit length) at
x, t
. When we do this, instead of having
one particle and watching it evolve, we constantly send in particles so that the
system does not appear to change with time. This allows us to find steady
states. Mathematically, this corresponds to finding solutions to the Schr¨odinger
equation that do not change with time. To determine, say, the probability of
reflection, roughly speaking, we look at the proportion of particles moving left
compared to the proportion of particles moving right in this steady state.
In principle, this interpretation is obtained by considering a constant stream
of wavepackets and using some limiting/averaging procedure, but we usually
don’t care about these formalities.
For these particle beams, Ψ(
x, t
) is bounded, but no longer normalizable.
Recall that for a single particle, the probability current was defined as
j(x, t) = −
i~
2m
(Ψ
∗
Ψ
0
− ΨΨ
0∗
).
If we have a particle beam instead of a particle, and Ψ is the particle density
instead of the probability distribution, j now represents the flux of particles at
x, t, i.e. the number of particles passing the point x in unit time.
Recall that a stationary state of energy
E
is of the form Ψ(
x, t
) =
ψ
(
x
)
e
iEt/~
.
We have
|Ψ(x, t)|
2
= |ψ(x)|
2
,
and
j(x, t) = −
i~
2m
(ψ
∗
ψ
0
− ψψ
0∗
).
Often, when solving a scattering problem, the solution will involve sums of
momentum eigenstates. So it helps to understand these better.
Our momentum eigenstates are
ψ(x) = Ce
ikx
,
which are solutions to the time-independent Schr¨odinger equation with
V
= 0
with E =
~
2
k
2
2m
.
Applying the momentum operator, we find that
p
=
~k
is the momentum of
each particle in the beam, and
|ψ
(
x
)
|
2
=
|C|
2
is the density of particles in the
beam. We can also evaluate the current to be
j =
~k
m
|C|
2
.
This makes sense.
~k
m
=
p
m
is the velocity of the particles, and
|C|
2
is how many
particles we have. So this still roughly corresponds to what we used to have
classically.
In scattering problems, we will seek the transmitted and reflected flux
j
tr
,
j
ref
in terms of the incident flux
j
inc
, and the probabilities for transmission and
reflection are then given by
P
tr
=
|j
tr
|
|j
inc
|
, P
ref
=
|j
ref
|
|j
inc
|
.