2D Quantum Problem: Particle in a Disk
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The wave functions of a quantum particle of mass confined to a disk of radius in the - plane are derived. These functions in polar coordinates are two-dimensional solutions of the Schrödinger equation with the potential . There is an infinite number of functions that fulfill the boundary condition , depend on two independent integer quantum numbers and . This Demonstration shows the oscillating behavior of the (unnormalized) probability density of a particle with different energy states inside the disk in the interval , . The ground state is characterized by the quantum number ; excited states have .
Contributed by: Reinhard Tiebel (March 2011)
Open content licensed under CC BY-NC-SA
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.
 J. J. Sakurai, Modern Quantum Mechanics, Reading, MA: Addison–Wesley Publishing Company, 1995.
 L. D. Landau and E. M. Lifschitz, Quantum Mechanics, Reading, MA: Addison-Wesley Publishing Company, 1958.