Quantum Hydrodynamics. A Statistical Mechanical Theory of Light Scattering from Simple Non-Polar Fluids

Citation

Zwanzig, Robert Walter (1952) Quantum Hydrodynamics. A Statistical Mechanical Theory of Light Scattering from Simple Non-Polar Fluids. Dissertation (Ph.D.), California Institute of Technology. doi:10.7907/ZJSM-HT46. https://resolver.caltech.edu/CaltechETD:etd-07202004-160131

Abstract

The first part of this thesis is concerned with certain extensions of a formal technique devised by Wigner for handling problems in quantum statistical mechanics, especially to problems in quantum mechanical transport processes. The approach is to find the closest possible analogy between classical and quantum statistical mechanics, so that the extensive work in classical statistical mechanics can be utilized. This analogy is attained with the Wigner distribution function, with which averages of dynamical variables in quantum mechanics may be calculated by integrations in phase space. We will first state some basic properties of distribution functions in classical statistical mechanics, and then state the corresponding properties of the density matrix in quantum mechanics. We will define and discuss the Wigner distribution function, show that it has the desired averaging properties, and obtain the analogue of the Liouville equation satisfied by this function. We will derive the analogue of the Liouville equation in reduced phase space, and then obtain the equations of hydrodynamics from quantum statistical mechanics. This will lead to expressions for the stress tensor and heat current density in terms of singlet and pair distribution functions.

In the second part of this thesis, the statistical mechanical theory of light scattering from fluids is developed. The model used consists of a collection of spherically symmetric, optically isotropic particles, which are capable of interacting both mechanically and electromagnetically. The effects of these interactions are included rigorously. This is done by using the radial distribution function for the spatial configuration of the particles, and the pair moment distribution function, which gives the dipole moment of a particle when it and another particle are fixed at specified positions, and the rest are averaged out. A chain of integral equations is set up, which is capable of giving the local field within the fluid. The index of refraction is then derived, with corrections to the Clausius-Mosotti formula. Finally, the light scattering cross section is obtained. This reduces to the result obtained with the Einstein-Smoluchowski theory in the proper limit, but contains corrections when the wavelength of the light is of the order of interparticle distances.

Thesis Files