Three Dimensions and Angular Momentum

Charles E. Burkhardt, Jacob J. Leventhal

Bipin R. Desai

Marios C. Tsatsos

In this Dissertation the physics of trapped, attractive Bose-Einstein condensates in two and three spatial dimensions are examined. Particular emphasis is put on the collapse of the gas, when the attractive interparticle interaction is raised above a critical value. The states of the system that possess good angular momentum quantum numbers are scrutinized and connections between the angular momentum and the fragmentation of the states are investigated. In two spatial dimensions mean-field states can describe states of good angular momentum. Using a mean-field ansatz, analytical results for the energy, the occupation numbers and the stability of the ground state carrying L quanta of angular momentum are obtained. In three dimensions, however, a mean-field theory does not suffice for the proper description of angular momentum states. The eigenstates of both total angular momentum operators (L^2 and L_z) are derived and it is shown that they are generally many-body states. It is shown, moreover, that angular momentum has a general stabilizing effect on the gas and this connection is expressed quantitatively. Finally, the ground state of the gas is examined, when its container is set into external rotation. It is found that due to the attraction the symmetry of the ground state does not change, angular momentum is not transfered from the rotation to the gas and thus no significant impact in the stability is seen.

Jochen Autschbach

Abstract This chapter treats rotations in three dimensions and the angular momentum quantum mechanically. The angular momentum is a 3-vector that plays the same role in the kinetic energy of rotating particles as the linear momentum 3-vector plays in the kinetic energy for translational motion. The quantum operator for the three components do not commute, i.e. the full vector cannot be known. Each component commutes with the operator for the square-length of the angular momentum vector, and therefore one opts for calculating the square-length and one of the components, z, referred to as the projection. The angular momentum is quantized. The (azimuthal) quantum number ℓ = 0,1,2,3… quantifies the square-length as ℓ(ℓ+1), and the projection is quantified by the (magnbetic) quantum number mℓ = -ℓ…ℓ in integer steps. The eigenfunctions are called spherical harmonics. A ‘rigid rotor’ model is set up to treat the rotational spectrum of a diatomic molecule.

‎Louis Marchildon

D. Dubbers, H.‐J. Stöckmann