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2D Quantum Problem: Particle in a Disk

​
number of the energy state m
0
1
2
3
number of the zero j
1
2
3
4
cylinder radius R
The wave functions of a quantum particle of mass
m
0
confined to a disk of radius
R
in the
x
-
y
plane are derived. These functions
f(r,ϕ)
in polar coordinates are two-dimensional solutions of the Schrödinger equation with the potential
V(r)=
0
0≤r<R
∞
otherwise
. There is an infinite number of functions
f(r,ϕ)
that fulfill the boundary condition
f(r=R,ϕ)=0
, depend on two independent integer quantum numbers
m
and
j
. This Demonstration shows the oscillating behavior of the (unnormalized) probability density
p(r)∝f(r,ϕ)
2
|
of a particle with different energy states inside the disk in the interval
r,…,r+dr
,
r≤R
. The ground state is characterized by the quantum number
m=0
; excited states have
m≥1
.
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