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


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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.



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