# 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