4More results in one dimensions
IB Quantum Mechanics
4.1 Gaussian wavepackets
When we solve Schr¨odinger’s equation, what we get is a “wave” that represents
the probability of finding our thing at each position. However, in real life,
we don’t think of particles as being randomly distributed over different places.
Instead, particles are localized to some small regions of space.
These would be represented by wavefunctions in which most of the distribution
is concentrated in some small region:
These are known as wavepackets.
Definition
(Wavepacket)
.
A wavefunction localised in space (about some point,
on some scale) is usually called a wavepacket.
This is a rather loose definition, and can refer to anything that is localized in
space. Often, we would like to consider a particular kind of wavepacket, known
as a Gaussian wavepacket.
Definition (Gaussian wavepacket). A Gaussian wavepacket is given by
Ψ
0
(x, t) =
α
π
1/4
1
γ(t)
1/2
e
−x
2
/2γ(t)
,
for some γ(t).
These are particularly nice wavefunctions. For example, we can show that for
a Gaussian wavepacket, (∆
x
)
ψ
(∆
p
)
ψ
=
~
2
exactly, and uncertainty is minimized.
The Gaussian wavepacket is a solution of the time-dependent Schr¨odinger
equation (with V = 0) for
γ(t) = α +
i~
m
t.
Substituting this
γ
(
t
) into our equation, we find that the probability density is
P
0
(x, t) = |Ψ
0
(x, t)|
2
=
α
π
1/2
1
|γ(t)|
e
−αx
2
/|γ(t)|
2
,
which is peaked at
x
= 0. This corresponds to a particle at rest at the origin,
spreading out with time.
A related solution to the time-dependent Schr¨odinger equation with
V
= 0 is
a moving particle:
Ψ
u
(x, t) = Ψ
0
(x −ut) exp
i
mu
~
x
exp
−i
mu
2
2~
t
.
The probability density resulting from this solution is
P
u
(x, t) = |Ψ
u
(x, t)|
2
= P
0
(x − ut, t).
So this corresponds to a particle moving with velocity u. Furthermore, we get
hˆpi
Ψ
u
= mu.
This corresponds with the classical momentum, mass × velocity.
We see that wavepackets do indeed behave like particles, in the sense that we
can set them moving and the quantum momentum of these objects do indeed
behave like the classical momentum. In fact, we will soon attempt to send them
to a brick wall and see what happens.
In the limit
α → ∞
, our particle becomes more and more spread out in
space. The uncertainty in the position becomes larger and larger, while the
momentum becomes more and more definite. Then the wavefunction above
resembles something like
Ψ(x, t) = Ce
ikx
e
−iEt/~
,
which is a momentum eigenstate with
~k
=
mu
and energy
E
=
1
2
mu
2
=
~
2
k
2
2m
.
Note, however, that this is not normalizable.