Further Developments with Angular Momentum and Multiparticle Systems

Mark S. Swanson

Chapter 7 introduces the concepts of symmetry operations and then focuses on rotation. The general properties of angular momentum and spin are derived. Multiparticle systems are introduced and angular momentum addition is analyzed.

W. H. Klink, S. Wickramasekara

George H. Duffey

Jin-Yan Zeng, Hongbo Zhu, Shouyong Pei

It is pointed out that the possibility of teleporting an arbitrary unknown one-particle spin state is crucially connected with the maximal entanglement of the Einstein-Podolsky-Rosen pair, whose one-particle reduced density matrix is $\ensuremath{\rho}(i)=\frac{1}{2}{\mathbf{I}}_{2}$ $(i=1,2).$ It is shown that, to teleport an arbitrary k-particle spin state, one must prepare an ancillary N-particle $(N>~2k)$ entangled state, whose k-particle reduced density matrix has the form ${(1/2}^{k}){\mathbf{I}}_{{2}^{k}}$ $({\mathbf{I}}_{{2}^{k}}$ is the ${2}^{k}\ifmmode\times\else\texttimes\fi{}{2}^{k}$ identity matrix). An alternative approach to constructing many-particle entangled states is developed by using ${R}_{x}(\ensuremath{\pi}),$ the collective rotation of $\ensuremath{\pi}$ around a given axis (say, x axis). The entangled states constructed by using ${R}_{x}(\ensuremath{\pi})$ operating on the basis of angular momentum uncoupling representation are just the GHZ states, which cannot be used for the teleportation of an arbitrary k $(>~2)$ particle spin state. The entangled states constructed by using ${R}_{x}(\ensuremath{\pi})$ operating on the basis of angular momentum coupling representation turn out to be effective for the teleportation of an arbitrary multiparticle state. A formal extension of the scheme of Bennett et al. to deal with the teleportation of an arbitrary two (or more) particle spin state is discussed.

Jamal Sakhr

For the periodic orbit quantization of a system of particles confined in a two-dimensional disk billiard, the proliferation of multiparticle periodic orbits with length precludes standard length truncation of the trace formula and alternative truncation schemes are necessary. An amplitude ordering of the single-particle periodic orbits reveals a quantum-classical correspondence for the angular momentum of a particle in the disk which is useful for the quantization of all the low energy states of the multiparticle problem. We explicitly show that a simple amplitude truncation of the trace formula better approximates the semiclassical density of states and in particular resolves closely spaced levels that are otherwise unresolved using the standard approach.