A new derivation of quantum-mechanical angular momentum operator L2

Weitzman, James K. Freericks

The derivation of spherical harmonics is the same in nearly every quantum mechanics textbook and classroom.It is found to be difficult to follow, hard to understand, and challenging to reproduce by most students.In this work, we show how one can determine spherical harmonics in a more natural way based on operators and a powerful identity called the exponential disentangling operator identity (known in quantum optics, but little used elsewhere).This new strategy follows naturally after one has introduced Dirac notation, computed the angular momentum algebra, and determined the action of the angular momentum raising and lowering operators on the simultaneous angular momentum eigenstates (under L2 and Lz ).

David V. George

Peter W. Milonni

By considering the allowed values of the z−component of the quantum mechanical angular momentum operator, a simple explanation is given for the term l(l+1). (AIP)

Lawrence C. Biedenharn, James D. Louck, Peter A. Carruthers

Owen L. de Lange, R. E. Raab

Abstract In Chapter 4 we considered a ‘twodimensional’ problem in which L2 and Lz constitute a c.s.c.o. and we derived shift operators for the angular momentum quantum numbers l and m in the eigenkets |lm> of these observables. In Chapter 5 we discussed a ‘one-dimensional’ problem, namely the eigenvalue equation for the radial Hamiltonian (eqn (5.25)). Here we use results from Chapters 4 and 5 to construct shift operators for the eigenkets of the three dimensional problem (5.11)–(5.13).