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. 𝑉(𝐫) =𝑉(π‘Ÿ). Using spherical coΓΆrdinates and separating variables, we find a complete set of stationary states given by1

πœ“(𝐫)=𝑅(π‘Ÿ)π‘Œπ‘šβ„“β„“(πœƒ,πœ™)

where π‘Œπ‘šβ„“β„“ are spherical harmonics and 𝑅(π‘Ÿ) is a solution of the radial equation

π‘‘π‘‘π‘Ÿ(π‘Ÿ2π‘‘π‘…π‘‘π‘Ÿ)βˆ’2π‘šπ‘Ÿ2ℏ2(𝑉(π‘Ÿ)βˆ’πΈ)𝑅=β„“(β„“+1)𝑅

which is solved more easily after the change of variables 𝑒(π‘Ÿ) =π‘Ÿπ‘…(π‘Ÿ) gives

βˆ’β„22π‘šπ‘‘2π‘’π‘‘π‘Ÿ2+(𝑉+ℏ22π‘šβ„“(β„“+1)π‘Ÿ2)𝑒=𝐸𝑒

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

Properties

  • We are guaranteed at least (2β„“ +1)-fold energy degeneracy due to the spherical symmetry (π‘šβ„“ does not affect the energy).

Examples

develop | en | SemBr

Footnotes

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

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