A Collective Description of Electron Interactions: III. Coulomb Interactions in a Degenerate Electron Gas

TLDR

The study examines a dense electron gas, relating its general properties to classical density‑fluctuation theory and Tomonaga’s one‑dimensional degenerate Fermi‑gas model. The authors analyze the electron behavior in a dense gas using quantum‑mechanical canonical transformations. They reformulate the Coulomb Hamiltonian into collective plasma‑field terms, then apply canonical transformations to decouple field and particle dynamics, yielding independent collective modes and electrons interacting via short‑range screened Coulomb forces. The resulting subsidiary conditions suppress long‑wavelength density fluctuations and reduce the electronic degrees of freedom by the number of collective modes.

Abstract

The behavior of the electrons in a dense electron gas is analyzed quantum-mechanically by a series of canonical transformations. The usual Hamiltonian corresponding to a system of individual electrons with Coulomb interactions is first re-expressed in such a way that the long-range part of the Coulomb interactions between the electrons is described in terms of collective fields, representing organized "plasma" oscillation of the system as a whole. The Hamiltonian then describes these collective fields plus a set of individual electrons which interact with the collective fields and with one another via short-range screened Coulomb interactions. There is, in addition, a set of subsidiary conditions on the system wave function which relate the field and particle variables. The field-particle interaction is eliminated to a high degree of approximation by a further canonical transformation to a new representation in which the Hamiltonian describes independent collective fields, with ${n}^{\ensuremath{'}}$ degrees of freedom, plus the system of electrons interacting via screened Coulomb forces with a range of the order of the inter electronic distance. The new subsidiary conditions act only on the electronic wave functions; they strongly inhibit long wavelength electronic density fluctuations and act to reduce the number of individual electronic degrees of freedom by ${n}^{\ensuremath{'}}$. The general properties of this system are discussed, and the methods and results obtained are related to the classical density fluctuation approach and Tomonaga's one-dimensional treatment of the degenerate Fermi gas.