In quantum mechanics, the case of a particle in a one-dimensional ring is similar to the particle in a box. The particle follows the path of a semicircle from
0
to
π
where it cannot escape, because the potential from
π
to
2
π
is infinite. Instead there is total reflection, meaning the particle bounces back and forth between
0
to
π
. The Schrödinger equation for a free particle which is restricted to a semicircle (technically, whose configuration space is the circle
S
1
) is
−
ℏ
2
2
m
∇
2
ψ
=
E
ψ
(
1
)
Using cylindrical coordinates on the 1-dimensional semicircle, the wave function depends only on the angular coordinate, and so
∇
2
=
1
s
2
∂
2
∂
ϕ
2
(
2
)
Substituting the Laplacian in cylindrical coordinates, the wave function is therefore expressed as
−
ℏ
2
2
m
s
2
d
2
ψ
d
ϕ
2
=
E
ψ
(
3
)
The moment of inertia for a semicircle, best expressed in cylindrical coordinates, is
I
=
d
e
f
∭
V
r
2
ρ
(
r
,
ϕ
,
z
)
r
d
r
d
ϕ
d
z
. Solving the integral, one finds that the moment of inertia of a semicircle is
I
=
m
s
2
, exactly the same for a hoop of the same radius. The wave function can now be expressed as
−
ℏ
2
2
I
d
2
ψ
d
ϕ
2
=
E
ψ
, which is easily solvable.
Since the particle cannot escape the region from
0
to
π
, the general solution to this differential equation is
ψ
(
ϕ
)
=
A
cos
(
m
ϕ
)
+
B
sin
(
m
ϕ
)
(
4
)
Defining
m
=
2
I
E
ℏ
2
, we can calculate the energy as
E
=
m
2
ℏ
2
2
I
. We then apply the boundary conditions, where
ψ
and
d
ψ
d
ϕ
are continuous and the wave function is normalizable:
∫
0
π
|
ψ
(
ϕ
)
|
2
d
ϕ
=
1
(
5
)
.
Like the infinite square well, the first boundary condition demands that the wave function equals 0 at both
ϕ
=
0
and
ϕ
=
π
. Basically
ψ
(
0
)
=
ψ
(
π
)
=
0
(
6
)
.
Since the wave function
ψ
(
0
)
=
0
, the coefficient A must equal 0 because
cos
(
0
)
=
1
. The wave function also equals 0 at
ϕ
=
π
so we must apply this boundary condition. Discarding the trivial solution where B=0, the wave function
ψ
(
π
)
=
0
=
B
sin
(
m
π
)
only when m is an integer since
sin
(
n
π
)
=
0
. This boundary condition quantizes the energy where the energy equals
E
=
m
2
ℏ
2
2
I
where m is any integer. The condition m=0 is ruled out because
ψ
=
0
everywhere, meaning that the particle is not in the potential at all. Negative integers are also ruled out.
We then normalize the wave function, yielding a result where
B
=
2
π
. The normalized wave function is
ψ
(
ϕ
)
=
2
π
sin
(
m
ϕ
)
(
7
)
.
The ground state energy of the system is
E
=
ℏ
2
2
I
. Like the particle in a box, there exists nodes in the excited states of the system where both
ψ
(
ϕ
)
and
ψ
(
ϕ
)
2
are both 0, which means that the probability of finding the particle at these nodes are 0.
Since the wave function is only dependent on the azimuthal angle
ϕ
, the measurable quantities of the system are the angular position and angular momentum, expressed with the operators
ϕ
and
L
z
respectively.
Using cylindrical coordinates, the operators
ϕ
and
L
z
are expressed as
ϕ
and
−
i
ℏ
d
d
ϕ
respectively, where these observables play a role similar to position and momentum for the particle in a box. The commutation and uncertainty relations for angular position and angular momentum are given as follows:
[
ϕ
,
L
z
]
=
i
ℏ
ψ
(
ϕ
)
(
8
)
(
Δ
ϕ
)
(
Δ
L
z
)
≥
ℏ
2
where
Δ
ψ
ϕ
=
⟨
ϕ
2
⟩
ψ
−
⟨
ϕ
⟩
ψ
2
and
Δ
ψ
L
z
=
⟨
L
z
2
⟩
ψ
−
⟨
L
z
⟩
ψ
2
(
9
)
As with all quantum mechanics problems, if the boundary conditions are changed so does the wave function. If a particle is confined to the motion of an entire ring ranging from 0 to
2
π
, the particle is subject only to a periodic boundary condition (see particle in a ring). If a particle is confined to the motion of
−
π
2
to
π
2
, the issue of even and odd parity becomes important.
The wave equation for such a potential is given as:
ψ
o
(
ϕ
)
=
2
π
cos
(
m
ϕ
)
(
10
)
ψ
e
(
ϕ
)
=
2
π
sin
(
m
ϕ
)
(
11
)
where
ψ
o
(
ϕ
)
and
ψ
e
(
ϕ
)
are for odd and even m respectively.
Similarly, if the semicircular potential well is a finite well, the solution will resemble that of the finite potential well where the angular operators
ϕ
and
L
z
replace the linear operators x and p.
