6Quantum mechanics in three dimensions
IB Quantum Mechanics
6.1 Introduction
To begin with, we translate everything we’ve had for the one-dimensional world
into the three-dimensional setting.
A quantum state of a particle in three dimensions is given by a wavefunction
ψ
(
x
) at fixed time, or Ψ(
x, t
) for a state evolving in time. The inner product is
defined as
(ϕ, ψ) =
Z
ϕ(x)
∗
ψ(x) d
3
x.
We adopt the convention that no limits on the integral means integrating over
all space. If ψ is normalized, i.e.
kψk
2
= (ψ, ψ) =
Z
|ψ(x)|
2
d
3
x = 1,
then the probability of measuring the particle to be inside a small volume
δV
(containing x) is
|ψ(x)|
2
δV.
The position and momentum are Hermitian operators
ˆ
x = (ˆx
1
, ˆx
2
, ˆx
3
), ˆx
i
ψ = x
i
ψ
and
ˆ
p = (ˆp
1
, ˆp
2
, ˆp
3
) = −i~∇ = −i~
∂
∂x
1
,
∂
∂x
2
,
∂
∂x
3
We have the canonical commutation relations
[ˆx
i
, ˆp
j
] = −i~δ
ij
, [ˆx
i
, ˆx
j
] = [ˆp
i
, ˆp
j
] = 0.
We see that position and momentum in different directions don’t come into
conflict. We can have a definite position in one direction and a definite momentum
in another direction. The uncertainty principle only kicks in when we are in the
same direction. In particular, we have
(∆x
i
)(∆p
j
) ≥
~
2
δ
ij
.
Similar to what we did in classical mechanics, we assume our particles are
structureless.
Definition
(Structureless particle)
.
A structureless particle is one for which all
observables can be written in terms of position and momentum.
In reality, many particles are not structureless, and (at least) possess an
additional quantity known as “spin”. We will not study these in this course, but
only mention it briefly near the end. Instead, we will just pretend all particles
are structureless for simplicity.
The Hamiltonian for a structureless particle in a potential V is
H =
ˆ
p
2
2m
+ V (
ˆ
x) = −
~
2
2m
∇
2
+ V (x).
The time-dependent Schr¨odinger equation is
i~
∂Ψ
∂t
= HΨ = −
~
2
2m
∇
2
Ψ + V (x)Ψ.
The probability current is defined as
j = −
i~
2m
(Ψ
∗
∇Ψ − Ψ∇Ψ
∗
).
This probability current obeys the conservation equation
∂
∂t
|Ψ(x, t)|
2
= −∇ · j.
This implies that for any fixed volume V ,
d
dt
Z
V
|Ψ(x, t)|
2
d
3
x = −
Z
V
∇ · j d
3
x = −
Z
∂V
j · dS,
and this is true for any fixed volume V with boundary ∂V . So if
|Ψ(x, t)| → 0
sufficiently rapidly as |x| → ∞, then the boundary term disappears and
d
dt
Z
|Ψ(x, t)|
2
d
3
x = 0.
This is the conservation of probability (or normalization).