Singular angular momentum mappings

Momentum Mappings

The momentum map is essentially due to Lie, [5], pp. 300–343. The modern notion is due to Kostant [3], Souriau [9], and Kirillov [2]. The setting for the moment mapping is a smooth symplectic manifold (M,ω) or even a Poisson manifold (M,P ) with the Poisson bracket on functions {f, g} = P (df, dg) (where P = ω : T M → TM is the Poisson tensor). To each function f there is the associated Hamilto...

Angular Momentum

Angular Momentum in Chemistry

Noting that current chemical theory is based almost exclusively on electronic energy and spin variables the equal importance of orbital angular momentum is explored in this paper. From its classical definition the angular momentum of electrons in an atom is shown to obey Laplace’s equation, which automatically leads to discrete values in terms of spherical harmonics. This analysis assumes a con...

Rotations and Angular Momentum

2 Exercise. Prove it. ◦ In fact, as sophisticates can show you, on a complex space a linear operator which preserves all norms will also preserve all inner products. But the key thing is to remember is that a unitary operator preserves all inner products (and therefore all norms). • This is analogous to the fact that translations and rotations in ordinary three-dimensional space preserve distan...

Angular Momentum Diagrams

Starting from the simplest possible building blocks—a ket, a bra, a time-reversed ket, and a time-reversed bra— a diagrammatic formalism is developed for angular momentum coupling problems. The formalism comprises Clebsch–Gordan coefficients as well as 3jm-symbols. The idea of constructing invariants (internal lines) by contracting contragredient pairs of quantities is emphasized throughout. Th...