New Treatment of Systems of Compounded Angular Momentum

TL;DR

This paper develops a novel matrix approach to systems of compounded angular momentum in quantum mechanics, deriving generalized operators and states, and extending Clebsch-Gordan coefficients for joint measurements.

Contribution

It introduces a new matrix treatment for compounded angular momentum systems, generalizes Clebsch-Gordan coefficients, and applies the approach to joint measurements in quantum systems.

Findings

Derived probability amplitudes for compounded systems

Obtained generalized matrix operators and states

Extended Clebsch-Gordan coefficients for new applications

Abstract

The approach to quantum mechanics which we have used to derive the matrix treatment of spin from first principles is now employed to treat systems of compounded angular momentum. A general treatment is first given, which is then applied to the concrete cases of a spin-0 and a spin-1 system obtained by adding the spins of two spin-1/2 systems. Thus the probability amplitudes for measurements on the systems are derived, as well as the matrix vectors and operators corresponding to the systems. The matrix operators and states obtained are different from the standard forms and are much more generalized. The new results are applied to the case of joint measurements on the subsystems of such a system; this is a problem that has been made very topical by the high level of interest in the foundations of quantum mechanics. As a consequence of the insights arising from this treatment, we show that the Clebsch-Gordan coefficients are amenable to generalization, and we give the generalized forms for these cases.