Particle in an Infinite Circular Well

plot

density

contour

This Demonstration solves the quantum-mechanical problem of a particle confined to a disk, which can be called an infinite 2D circular well. The probability densities for several energy eigenstates are plotted. The azimuthal quantum number

, equal to the number of angular nodes, determines the angular momentum

mℏ

. The radially excited state energies depend on the zeros of the Bessel function

(kr)

Details

The solutions are of the form

|m|

(kr)sin(mθ)

and

|m|

(kr)cos(mθ)

with the quantized energy levels

n,m

ℏ

n,m

2μ

where

ℏ

is the Planck constant,

is the mass and

n,m

is the

zero of the Besssel function

(k,r)

References

[1] R. W. Robinett, "Visualizing the Solutions for the Circular Infinite Well in Quantum and Classical Mechanics," American Journal of Physics, 64(4), ‎1996 pp. 440–446.

[2] R. W. Robinett, Quantum Mechanics, Classical Results, Modern Systems and Visualized Examples, 2nd ed., Oxford: Oxford University Press, 2006.

[3] R. W. Robinett, "Quantum Mechanics of the Two-Dimensional Circular Billiard Plus Baffle System and Half-Integral Angular Momentum." arxiv.org/pdf/quant-ph/0307035.pdf.

External Links

Bessel Function of the First Kind (Wolfram MathWorld)

Planck's Constant (ScienceWorld)

Quantum Numbers (ScienceWorld)

Magnetic Quantum Number (ScienceWorld)

Schrödinger Equation (ScienceWorld)

Permanent Citation

Enrique Zeleny, Michael Trott

"Particle in an Infinite Circular Well"
http://demonstrations.wolfram.com/ParticleInAnInfiniteCircularWell/
Wolfram Demonstrations Project
Published: August 14, 2013