Quantum-Mechanical Particle in a Cylinder
Quantum-Mechanical Particle in a Cylinder
This is one of the three classic particle-in-a-box problems in elementary quantum mechanics, along with the cuboid and the sphere. For a particle of mass in a right circular cylinder of radius and altitude , the Schrödinger equation in cylindrical coordinates , , can be written as
M
R
L
ρ
z
ϕ
-ψ++ψ+ψ=Eψ
2
ℏ
2M
2
∂
∂
2
ρ
1
ρ
∂ψ
∂ρ
2
∂
∂
2
z
1
2
ρ
2
∂
∂
2
ϕ
The equation is separable, as . The and factors are elementary functions: (z)=, with and (ϕ)=, with . The equation for can be reduced to Ρ''(ρ)+ρΡ'(ρ)+-Ρ(ρ)=0 with the boundary condition . The solutions are Bessel functions (ρ) such that R= is the zero of the Bessel function . The total energy is then given by =+.
ψ(ρ,z,ϕ)=Ρ(ρ)Z(z)Φ(ϕ)
z
ϕ
Z
n
2/L
sin(nπz/L)n=1,2,…,
Φ
m
-1/2
(2π)
imϕ
e
m=0,±1,±2,…
Ρ(ρ)
2
ρ
2
k
2
ρ
2
m
Ρ(R)=0
J
m
k
mp
k
mp
j
mp
th
p
J
m
E
nmp
2
ℏ
2M
2
j
mp
2
R
2
π
2
n
2
L
This Demonstration shows contour plots of the wavefunction through horizontal cross sections of the cylinder, representing constant values of between 0 and . The wavefunction is positive in the light blue regions.
z
L