Abstract:
In this project, quantum theory of a weakly interacting, dilute Bose gas is presented using the
formalism of second quantization. First, a calculation is performed to obtain an expression for
the Hamiltonian operator based on the concept of a quantized field. The second quantized
version of the Hamiltonian in the momentum representation is developed, which is appropriate
for a cold and dilute Bose gas. The Hamiltonian is then diagonalized using the thermodynamic
properties of the Bose gas. In this project, we consider the ground-state energy of clouds in a
confining potential. While the scattering lengths for alkali atoms are large compared with atomic
dimensions, they are usually small compared with atomic separations in gas clouds. As a
consequence, the effects of atomic interactions may be calculated very reliably by using an
effective interaction proportional to the scattering length. This provides the basis for a meanfield
description of the condensate, which leads to the Gross–Pitaevskii equation. From this we
calculate the energy using both variational methods and the Thomas–Fermi approximation. The
Thomas–Fermi approximation fails near the surface of a cloud, and we calculate a relation
between the chemical potential μ and the total number of particles N. When the atom–atom
interaction is attractive, the system becomes unstable if the number of particles exceeds a critical
value, which we calculate in terms of the trap parameters and the scattering length.
