QM in 3D position-space

Spherically symmetric potential

Many applications of quantum mechanics for single particles in 3D space have a spherically symmetric potential, i.e. . Using spherical coördinates and separating variables, we find a complete set of stationary states given by¹

where are spherical harmonics and is a solution of the radial equation

which is solved more easily after the change of variables gives

which is analogous to QM in 1D position-space for an effective potential containing an additional centrifugal term.

Properties

• We are guaranteed at least -fold energy degeneracy due to the spherical symmetry ( does not affect the energy).

Examples

• QM of the coulomb potential

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Footnotes

1. 2018. Introduction to quantum mechanics, §4.1, pp. 131ff. ↩