Three-Dimensional Isotropic Harmonic Oscillator

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The isotropic three-dimensional harmonic oscillator is described by the Schrödinger equation , in units such that
. The wavefunction is separable in Cartesian coordinates, giving a product of three one-dimensional oscillators with total energies
. More interesting is the solution separable in spherical polar coordinates:
, with the radial function
. Here,
is an associated Laguerre polynomial,
, a spherical harmonic and
, a normalization constant. The energy levels are then given by
, being
-fold degenerate. For a given angular momentum quantum number
, the possible values of
are
. The conventional code is used to label angular momentum states, with
representing
.
Contributed by: S. M. Blinder (March 2011)
Open content licensed under CC BY-NC-SA
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Reference: Wikipedia article Quantum Harmonic Oscillator
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