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