Angular Momentum Matrix Elements

Wolfgang Witschel

Schwinger's coupled-boson representation of spherical and hyperbolic angular momentum is related to spherical harmonics and is used for the calculation of matrix elements. It is shown that radial matrix elements of the two-dimensional isotropic oscillator and the closely related hydrogenic matrix elements are proportional to 3j-symbols. Matrix elements of exponentials of hyperbolic angular momentum operators which form a formal analogy to the well known rotation matrice d mm' j ( theta ) and which are written as t mm jj' ( theta ) are derived. As only elementary algebra and some operator formulae are used, the algebraic method supplements analytical and group theoretic methods.

Ramandeep S. Johal

We discuss correspondence between the predictions of quantum theories for rotation angle formulated in infinite and finite dimensional Hilbert spaces, taking as example, the calculation of matrix elements of phase-angular momentum commutator. A new derivation of the matrix elements is presented in infinite space, making use of a unitary transformation that maps from the state space of periodic functions to non-periodic functions, over which the spectrum of angular momentum operator is in general, fractional. The approach can be applied to finite dimensional Hilbert space also, for which identical matrix elements are obtained.

Giampiero V. M. Esposito, Giuseppe Marmo, George Sudarshan

Joseph Y. Shapiro

Formulas are obtained which give the matrix representation, relative to a product basis, of the projection operator for the total angular momentum of a system. If the individual angular momenta are not too large, the matrix elements depend upon a small number of parameters, independent of the number of angular momenta coupled. Recurrence relations between elements and symmetry properties of elements are derived. These results enable one to perform the vector coupling of a large number of angular momenta in a relatively simple fashion. The connection between the matrix elements and vector coupling coefficients is discussed. Several important special cases are treated in detail.