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Journal of Mathematical Physics

N-Body Bose System with a Finite Number of States. I. Irreducible Representations

Joseph B. Aviles

Published Apr 1, 1963• 1 citation
Publisher

Abstract

For the purpose of investigating the N-Body Bose System with a finite number of states, we analyze Xij = Ai†Aj, i, j ≤ K. (A and A† are the usual annihilation and creation operators of the second quantization formalism). The Hamiltonian for a fixed number of particles may be expressed in terms of the (finite-dimensional) irreducible representations of the Xij. A set of fundamental equations is defined for the irreducible representations Xij(N, K), i, j ≤ K and is solved for arbitrary N and K. The analysis of the structure of the Xij(N, K) yields a simple, systematic method for listing all possible ways in which n1+ … ni+ … + nK = N may be satisfied for ni positive integral or zero. This leads to a particularly simple method for constructing the Xij(N, K), i, j ≤ K.

DOI
10.1063/1.1703979

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For the purpose of investigating the N-Body Bose System with a finite number of states, we analyze Xij = Ai†Aj, i, j ≤ K. (A and A† are the usual annihilation and creation operators of the second quantization formalism). The Hamiltonian for a fixed number of particles may be expressed in terms of the (finite-dimensional) irreducible representations of the Xij. A set of fundamental equations is defined for the irreducible representations Xij(N, K), i, j ≤ K and is solved for arbitrary N and K. The analysis of the structure of the Xij(N, K) yields a simple, systematic method for listing all possible ways in which n1+ … ni+ … + nK = N may be satisfied for ni positive integral or zero. This leads to a particularly simple method for constructing the Xij(N, K), i, j ≤ K.
The many-body problem for a system of composite particles is formulated in a way which takes explicit account of the composite nature of these particles and allows a clear separation between interatomic and intraatomic interactions. A second-quantization representation, fully equivalent to the conventional representation in which nuclei and electrons appear explicitly, is developed in terms of atomic annihilation and creation operators satisfying elementary Bose or Fermi commutation relations. All effects of the composite nature of the atoms are exactly contained in the interatomic and intraatomic matrix elements and in certain exchange integrals. An application is made to the problem of Bose condensation of fermion pairs.
Summary of the lectures: 1. Creation and annihilation operators for the system of indistinguishable particles 1.1 The permutation group and the states of a system of indistinguishable particles 1.2 Dimension of the Hilbert space of a system of indistinguishable particles 1.3 Definition and properties of the creation and annihilation operators 1.4 TheFockspace 1.5 The representations of state vectors and operators 1.5.1 N-particlewave-functions 1.5.2 The second-quantization representation of operators 1.5.3 Examples 1.6 Evolution of operators in the Heisenberg picture 1.7 Statistical operators of indistinguishable particles 1.7.1 The averages of the s-particle operators 1.7.2 The general structure of the one-particle statistical operator 2 Quadratic Hamiltonian and the diagonalization 2.1 Diagionalization of the Hamiltonian quadratic in the fermion operators 2.2 Diagionalization of the Hamiltonian quadratic in the boson operators 2.2.1 Diagonalization of the quadratic bosonic Hamiltonian possessing a zero mode 2.3 Long range order, condensation and the Bogoliubov spectrum of weakly interacting Bose gas 2.3.1 The excitation spectrum of weakly interacting Bose gas inabox 2.3.2 The excitation spectrum of an interacting Bose gas in an externalpotential 2.4 The Jordan-Wigner transformation: fermionization of interacting 1D spin chains
This is a self-contained and hopefully readable account on the method of creation and annihilation operators (also known as the Fock space representation or the "second quantization" formalism) for non-relativistic quantum mechanics of many particles. Assuming knowledge only on conventional quantum mechanics in the wave function formalism, we define the creation and annihilation operators, discuss their properties, and introduce corresponding representations of states and operators of many-particle systems. As the title of the note suggests, we cover most topics treated in sections 6.7 and 6.8 of Feynman's "Statistical Mechanics: A Set of Lectures". As a preliminary, we also carefully discuss the symmetry of wave functions describing indistinguishable particles. We note that all the contents of the present note are completely standard, and the definitions and the derivations presented here have been known to many. Although the style of the present note may be slightly more mathematical than standard physics literatures, we do not try to achieve full mathematical rigor.(Note to experts: In particular we here DERIVE the (anti)commutation relations of the creation and annihilation operators, rather than simply declaring them. In this sense our approach is quite close to that of Feynman's. But we here focus on the action of creation/annihilation operators on general NNN body wave functions, while Feynman makes a heavy use of Slater-determinant-type states from the beginning. We hope that our presentation provides a better perspective on the formalism.)
A many-boson model is formulated and expressions for its exact eigenstates and energies are obtained for both an arbitrary finite and an infinite number of bosons. The Hamiltonian of the model contains interactions between bosons whose momenta have equal magnitudes but opposite directions. The matrix elements of this interaction are taken to be a constant over a range of momenta surrounding k = 0. The ground state of the 2N-particle system is shown to be a product of N pair-creation operators acting on the vacuum state. Each of these pair-creation operators depends upon one of N parameters which are called pair energies. The N pair energies are shown to satisfy a coupled system of nonlinear algebraic equations. The energy of the state is the sum of the pair energies and the occupation probabilities of the single-particle levels are given as simple functions of the pair energies. Similar results are derived for the excited states of the system and for the states of an odd number of particles. These results are valid for both a repulsive and an attractive interaction, since they only depend upon the form of the interaction. The equations are solved algebraically for two model systems. The first of these is one whose single-particle kinetic energy takes on only one value. The equations for this system are solved for an arbitrary interaction strength and it is shown that the pair energies are proportional to the zeros of certain Laguerre polynomials. The second system is one in which the single-particle kinetic energy can take on two values. The equations for this system are solved in the strong repulsive-interaction limit and it is shown that the pair energies are proportional to the zeros of certain Jacobi polynomials. The excitation energies of this second system are shown to be proportional to 1/n and the occupations of the two single-particle levels in the ground state are shown to be proportional to n, where n is the total number of particles. For a repulsive interaction and an arbitrary single-particle spectrum, the algebraic equations for the pair energies are converted into an approximate integral equation for the density of roots which is accurate to order 1/n. This integral equation is solved for a strong interaction which, in the context of this model, means an interaction whose strength is greater than a constant times 1/V⅔ in the limit of a large volume. From this solution, the following results are obtained: (1) the lowest two single-particle levels have occupations of order n; (2) the excitation spectrum is that of a set of noninteracting quasiparticles; and (3) the quasiparticle spectrum has two zeros corresponding to the lowest two single-particle levels. Apart from the presence of two zeros, the quasiparticle spectrum does not differ significantly from that of the noninteracting particles.
The Hamiltonian for an arbitrary Coulomb three-body system is written in closed form in terms of nine generators of three independent o(2,1) algebras and its six physical parameters (i.e., three masses and three charges). Moreover, it is shown that such a Hamiltonian can also be written in the second quantized form by using six pairs of creation and annihilation (boson) operators. The obtained finite-term representation of the Coulomb three-body Hamiltonian in the second quantized form is exact, i.e., it is not based on any approximation. In particular, this means that the original Coulomb three-body problem can be reduced to the equivalent problem of three interacting, complex (i.e., charged) boson fields. The developed procedure opens new avenues in obtaining analytical and semianalytical (or highly accurate) solutions for various Coulomb three-body problems.