Quantum-mechanical many-particle treatment of sound propagation

Nobuyuki Imoto, John Jeffers, R. Loudon

Ajit Kumar

Toshiyuki Nishiyama

Some collective behaviors of boson assemblies are described in terms of the classical density operator and the classical velocity operator. We introduce these two classical quantities by taking average of the quantized operators with respect to Heisenberg wave packets. Equations of these two quantities are determined and we obtain the transport equations. Thus the assembly is described as a fluid having the quantum mechanical pressure tensor. The collective behavior of the assembly in question is analysed applying the theory of sound waves. Such an approximate treatment enables us to regard the assembly in question as the assembly of oscillation quanta. When a particle is projected into the assembly with the velocity v0 which is larger than the sound speed of the medium, suffering the collective response of the remainder of the bosons, the particle is decelerated and emits sound quanta. These processes are analysed by a classical treatment. Further we consider the interaction of every particle in the assembly with the field of the collective oscillation representing behaviors of the remainder of the assembly. The self-energy of the particle and the Møller interaction between particles are obtained according to the interaction representation of Tomonaga and Schwinger as well as the canonical transformation of Bloch and Nordsieck.

Philippe Nozières, David Pines

The wave propagation might be appropriately called quasi-particle sound, since in the long wave-length limit the only density fluctuations of importance are those produced by exciting a single quasi-particle from the condensate. Quasi-particle sound has essentially the same physical origin as zero sound in a Fermi liquid. The quasi-particle sound velocity will be temperature dependent, since the energy of a given quasi-particle depends on the thermal excitation of other quasi-particles. Superfluid hydrodynamics is richer than its Fermi liquid counterpart because one can have relative motion of the thermal quasi-particles and the condensate. This chapter describes the propagation of first and second sound in the hydrodynamic limit and then go on to a consideration of quasi-particle sound in the collisionless regime. A wide variety of irreversible phenomena which have as their physical origin collisions between the thermally-excited quasi-particles.

Thomas Brennan

Robert Mills