4 Spectrum of angular momentum and spin

Lei Zhao, Xu Qiu, Yi LIANG +2 more

By numerically solving the single-particle stationary Schrödinger equation and the Gross-Pitaevskii equation with mean-field interactions at zero temperature, the ground state properties of the rotating spin-orbital-angular-momentum coupled Bose-Einstein condensates in a harmonic trapping potential are investigated in this work. The results show that the rotation lifts the double degeneracy of the single-particle energy spectrum in the angular momentum space, and leads to the vortex state. The angular momentum of the vortex depends on the rotating frequency, the intensity of the laser beam, and the spin-orbital-angular-momentum coupling. In particular, if the rotating frequency is below a critical value, the angular momentum of the ground state vortex remains unaffected by the rotating frequency. When the rotating frequency exceeds the critical value, the angular momentum of the ground state vortex will increase with the rotating frequency increasing. By assuming that the system is confined in a ring trap, the expression of the single-particle energy spectrum in the angular momentum space can be obtained, which clarifies how the rotation frequency affects the angular momentum of the ground state. In the presence of atomic interactions, similar phenomena can also be observed in the mean-field ground state at zero temperature.

Dung Xuan Nguyen, Dam Thanh Son

A low-energy neutral quasiparticle in a fractional quantum Hall system appears in the latter’s energy spectrum on a sphere as a series of many-body excited states labeled by the angular momentum L L and whose energy is a smooth function of L L in the limit of large sphere radius. We argue that the signature of a nonvanishing spin (intrinsic angular momentum) s s of the quasiparticle is the absence, in this series, of states with total angular momentum less than s s . We reinterpret the missing of certain states, observed in an exact-diagonalization calculation of the spectrum of the \nu=7/3 ν = 7 / 3 FQH state in a wide quantum well as well as in many proposed wave functions for the excited states as a consequence of the spin-2 nature of the zero-momentum magnetoroton.

Walter Wilcox

Synopsis: General operator properties of angular momenta are recalled, and a physical motivation for adding them is given. The product and total angular momentum bases are defined. It is shown how to express one basis in terms of the other, and an explicit example of adding two spin 1/2 angular momenta is given. A further example consists of adding a general orbital angular momentum value to spin 1/2. These considerations are applied to perturbations in the hydrogen atom energy spectrum, where the correct choice for calculations is the total angular momentum basis.