Angular Momentum in Spherical Coordinates

Y. F. Bow

Y. B. Band, Y. Avishai

Ingvar Lindgren, John Morrison

Ajoy Kumar Ghatak, S. Lokanathan

Cen Tian-qing

In quantum mechanics textbook, the spherical coordinate forms of both 2 and z are obtained generally by using the Cartesian coordinates, however, the variable substitution calculation is complex. In this paper, the spherical coordinates on the orthogonal basis of coordinate are regarded to variables vectors, and differential nature of the spherical coordinates is used. The derivation for angular momentum operators 2 and z in quantum mechanics is predigested.

Caio Lima Firme

This chapter discusses the classic circular motion as a basis for the circular motion of a particle. The derivation of the bidimensional Laplace operator in polar coordinate system is presented next in order to solve the Schrödinger equation of the particle in a ring (whose wave function is a part of the solution of the hydrogen atom). Subsequently, there appears the derivation of the wave equation for the particle in a ring which leads to an energy equation and its comparison with the classic equation of the angular momentum leads to the corresponding quantum angular momentum equation. Afterwards, there appears the derivation of the angular momentum in spherical polar coordinate which is part of the tridimensional Laplace operator in spherical polar coordinate which is obtained in the next section. Then, the tridimensional Laplace operator in spherical polar coordinate is used in Schrödinger equation of the particle in a sphere (a product of angular equation and azimuthal equation) where the azimuthal equation was already solved (from the particle in a ring given in a previous section) and then this section is devoted to derive the angular or polar equation and the corresponding quantum angular momentum equation. (195 words).