Quantum-Mechanical Particle in a Cylinder

cross section

z

25

quantum numbers

n

1

2

3

m

0

1

2

p

1

2

3

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

M

in a right circular cylinder of radius

R

and altitude

L

, the Schrödinger equation in cylindrical coordinates

ρ

,

z

,

ϕ

can be written as

-

2

ℏ

2M

2

∂

ψ

∂

2

ρ

+

1

ρ

∂ψ

∂ρ

+

2

∂

ψ

∂

2

z

+

1

2

ρ

2

∂

ψ

∂

2

ϕ

=Eψ

.

The equation is separable, as

ψ(ρ,z,ϕ)=Ρ(ρ)Z(z)Φ(ϕ)

. The

z

and

ϕ

factors are elementary functions:

Z

n

(z)=

2/L

sin(nπz/L)

, with

n=1,2,…,

and

Φ

m

(ϕ)=

-1/2

(2π)

imϕ

e

, with

m=0,±1,±2,…

. The equation for

Ρ(ρ)

can be reduced to

2

ρ

Ρ''(ρ)+ρΡ'(ρ)+

2

k

2

ρ

-

2

m

Ρ(ρ)=0

with the boundary condition

Ρ(R)=0

. The solutions are Bessel functions

J

m

(

k

mp

ρ)

such that

k

mp

R=

j

mp

is the

th

p

zero of the Bessel function

J

m

. The total energy is then given by

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

z

between 0 and

L

. The wavefunction is positive in the light blue regions.