A New Kind of Shift Operators for Infinite Circular and Spherical Wells

A new kind of shift operators for infinite circular and spherical wells is identified. These shift operators depend on all spatial variables of quantum systems and connect some eigenstates of confined systems of different radiiRsharing energy levels with a common eigenvalue. In circular well, the momentum operatorsP±=Px±iPyplay the role of shift operators. ThePxandPyoperators, the third projection of the orbital angular momentum operatorLz, and the HamiltonianHform a complete set of commuting operators with the SO(2) symmetry. In spherical well, the shift operators establish a novel relation betweenψlm(r)andψ(l±1)(m±1)(r).

Orbital angular momentum eigenfunctions for many-particle systems

Methods are presented for constructing eigenfunctions of the total orbital angular momentum operator of a many-particle system without the use of the Clebsch–Gordan coefficients. One of the equations derived in this paper is analogous to Dirac's identity for total spin; and through this equation, a connection is established between eigenfunctions of L2 and irreducible representations of the symmetric group Sn.