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
Snapshots
Details
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.
Permanent Citation