2D Quantum Problem: Particle in a Disk

The eigenvalue equation for the Hamiltonian reads, in polar coordinates , : . In the quantum-mechanical position basis (-representation), the momentum operator ("nabla" operator), so that the energy eigenvalue equation is transformed into a partial differential equation; is Planck's constant . The unique solutions regular at satisfying the boundary condition are Bessel functions of integer order : , where represent the zero of the Bessel function , . The integers give the components of the angular momentum . The wave-mechanical probability densities are oscillating functions and show analogous behavior to those of corresponding rectangular potential problems in one, two, and three dimensions, respectively.

References

[1] J. J. Sakurai, Modern Quantum Mechanics, Reading, MA: Addison–Wesley Publishing Company, 1995.

[2] L. D. Landau and E. M. Lifschitz, Quantum Mechanics, Reading, MA: Addison-Wesley Publishing Company, 1958.