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An investigation of a class of soluble many body problems

Gregg Stanley Edwards

Published Jan 1, 1963• 0 citations
OpenAlex

Abstract

This thesis is a report of results so far obtained in an investigation of the properties and characteristics of simple interaction systems whose Hamiltonians are quadratic in their generalized coordinates. The exact solutions of two system types are presented. The first is composed of a simple harmonic oscillator coupled to a uniform lattice of coupled simple harmonic oscillators, or, in the small amplitude approximation, to a beaded string. The second is composed of an arbitrary distribution of oscillators coupled to string. The quantization of linear systems such as those described above is discussed with the result that the solution to the equivalent quantum system in terms of operators is just that of the corresponding classical solution as a function of its initial conditions when these are interpreted as quantum operators. The energy flow along the linear chain is examined in the discussion of the non-exponential decay in dispersive systems. This is compared with the original discussion by Trammell (1956) who pointed out the non-exponential decay of quantum systems and the corresponding behavior of dispersive classical systems.

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Journal of Mathematical Physics•Dec 1, 1979•22 citations
This thesis is a report of results so far obtained in an investigation of the properties and characteristics of simple interaction systems whose Hamiltonians are quadratic in their generalized coordinates. The exact solutions of two system types are presented. The first is composed of a simple harmonic oscillator coupled to a uniform lattice of coupled simple harmonic oscillators, or, in the small amplitude approximation, to a beaded string. The second is composed of an arbitrary distribution of oscillators coupled to string. The quantization of linear systems such as those described above is discussed with the result that the solution to the equivalent quantum system in terms of operators is just that of the corresponding classical solution as a function of its initial conditions when these are interpreted as quantum operators. The energy flow along the linear chain is examined in the discussion of the non-exponential decay in dispersive systems. This is compared with the original discussion by Trammell (1956) who pointed out the non-exponential decay of quantum systems and the corresponding behavior of dispersive classical systems.
The Quantum Harmonic Oscillator is one of the most important models in Quantum Mechanics. Analogous to the classical mass vibrating back and forth on a spring, the quantum oscillator system has attracted substantial attention over the years because of its importance in many advanced and difficult quantum problems. This dissertation deals with solving generalized models of the time-dependent Schrodinger equation which are called generalized quantum harmonic oscillators, and these are characterized by an arbitrary quadratic Hamiltonian of linear momentum and position operators. The primary challenge in this work is that most quantum models with timedependence are not solvable explicitly, yet this challenge became the driving motivation for this work. In this dissertation, the methods used to solve the time-dependent Schrodinger equation are the fundamental singularity (or Green's function) and the Fourier (eigenfunction expansion) methods. Certain Riccati- and Ermakov-type systems arise, and these systems are highlighted and investigated. The overall aims of this dissertation are to show that quadratic Hamiltonian systems are completely integrable systems, and to provide explicit approaches to solving the time-dependent SchrDodinger equation governed by an arbitrary quadratic Hamiltonian operator. The methods and results established in the dissertation are not yet well recognized in the literature, yet hold for high promise for further future research. Finally, the most recent results in the dissertation correspond to the harmonic oscillator group and its symmetries. A simple derivation of the maximum kinematical invariance groups of the free particle and quantum harmonic oscillator is constructed from the view point of the Riccati- and Ermakov-type systems, which shows an alternative to the traditional Lie Algebra approach. To conclude, a missing class of solutions of the time-dependent Schrodinger equation for the simple harmonic oscillator in one dimension is constructed. Probability distributions of the particle linear position and momentum, are emphasized with Mathematica animations. The eigenfunctions qualitatively differ from the traditional standing waves of the one-dimensional Schrodinger equation. The physical relevance of these dynamic states is still questionable, and in order to investigate their physical meaning, animations could also be created for the squeezed coherent states. This will be addressed in future work.
We describe a quantum system consisting of a one-dimensional linear chain of n identical harmonic oscillators coupled by a nearest neighbor interaction. Two boundary conditions are taken into account: periodic boundary conditions (where the nth oscillator is coupled back to the first oscillator) and fixed wall boundary conditions (where the first oscillator and the nth oscillator are coupled to a fixed wall). The two systems are characterized by their Hamiltonian. For their quantization, we treat these systems as Wigner quantum systems (WQSs), allowing more solutions than just the canonical quantization solution. In this WQS approach, one is led to certain algebraic relations for operators (which are linear combinations of position and momentum operators) that should satisfy triple relations involving commutators and anti-commutators. These triple relations have a solution in terms of the Lie superalgebra gl(1|n). We study a particular class of gl(1|n) representations V(p), the so-called ladder representations. For these representations, we determine the spectrum of the Hamiltonian and of the position operators (for both types of boundary conditions). Furthermore, we compute the eigenvectors of the position operators in terms of stationary states. This leads to explicit expressions for position probabilities of the n oscillators in the chain. An analysis of the plots of such position probability distributions gives rise to some interesting observations. In particular, the physical behavior of the system as a WQS is very much in agreement with what one would expect from the classical case, except that all physical quantities (energy, position, and momentum of each oscillator) have a finite spectrum.
Some of the most enduring questions in physics--including the quantum measurement problem and the quantization of gravity--involve the interaction of a quantum system with a classical environment. Two linearly coupled harmonic oscillators provide a simple, exactly soluble model for exploring such interaction. Even the ground state of a pair of identical oscillators exhibits effects on the quantum nature of one oscillator, e.g., a diminution of position uncertainty, and an increase in momentum uncertainty and uncertainty product, from their unperturbed values. Interaction between quantum and classical oscillators is simulated by constructing a quantum state with one oscillator initially in its ground state, the other in a coherent or Glauber state. The subsequent wave function for this state is calculated exactly, both for identical and distinct oscillators. The reduced probability distribution for the quantum oscillator, and its position and momentum expectation values and uncertainties, are obtained from this wave function. The oscillator acquires an oscillation amplitude corresponding to a beating between the normal modes of the system; the behavior of the position and momentum uncertainties can become quite complicated. For oscillators with equal unperturbed frequencies, i.e., at resonance, the uncertainties exhibit a time-dependent quantum squeezing which can be extreme.
This thesis is devoted to the study of various examples of exactly solved quantum many-body systems in one-dimension. It is divided into two parts: the first provides background and complementary results to the second, which consists of three scientific papers. The first paper concerns a particu- lar extension, corresponding to the root system CN, of the delta-interaction model. We prove by construction that its exact solution, even in the gen- eral case of distinguishable particles, can be obtained by the coordinate Bethe ansatz. We also elaborate on the physical interpretation of this model. It is well known that the delta-interaction is included in a four parameter family of local interactions. In the second paper we interpret these parameters as cou- pling constants of certain momentum dependent interactions and determine all cases leading to a many-body system of distinguishable particles which can be solved by the coordinate Bethe ansatz. In the third paper we consider the so-called rational Calogero-Sutherland model, describing an arbitrary number of particles on the real line, confined by a harmonic oscillator potential and interacting via a two-body interaction proportional to the inverse square of the inter-particle distance. We construct a novel solution algorithm for this model which enables us to obtain explicit formulas for its eigenfunctions. We also show that our algorithm applies, with minor changes, to all extensions of the rational Calogero-Sutherland model which correspond to a classical root system.