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arXiv:1211.1895 · Advances in Theoretical and Mathematical Physics

A new approach to the NNN-particle problem in QM

Joachim Schröter

Published Jan 1, 2014• 0 citations
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Abstract

In this paper the old problem of determining the discrete spectrum of a multi-particle Hamiltonian is reconsidered.The aim is to bring a fermionic Hamiltonian for arbitrary numbers N of particles by analytical means into a shape such that modern numerical methods can successfully be applied.For this purpose the Cook-Schroeck Formalism is taken as starting point.This includes the use of the occupation number representation.It is shown that the N -particle Hamiltonian is determined in a canonical way by a fictional 2particle Hamiltonian.A special approximation of this 2-particle operator delivers an approximation of the N -particle Hamiltonian, which is the orthogonal sum of finite dimensional operators.A complete classification of the matrices of these operators is given.Finally the method presented here is formulated as a work program for practical applications.The connection with other methods for solving the same problem is discussed.

DOI
10.4310/atmp.2014.v18.n6.a3

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In this paper the old problem of determining the discrete spectrum of a multi-particle Hamiltonian is reconsidered.The aim is to bring a fermionic Hamiltonian for arbitrary numbers N of particles by analytical means into a shape such that modern numerical methods can successfully be applied.For this purpose the Cook-Schroeck Formalism is taken as starting point.This includes the use of the occupation number representation.It is shown that the N -particle Hamiltonian is determined in a canonical way by a fictional 2particle Hamiltonian.A special approximation of this 2-particle operator delivers an approximation of the N -particle Hamiltonian, which is the orthogonal sum of finite dimensional operators.A complete classification of the matrices of these operators is given.Finally the method presented here is formulated as a work program for practical applications.The connection with other methods for solving the same problem is discussed.
In this paper the old problem of determining the discrete spectrum of a multi-particle Hamiltonian is reconsidered.The aim is to bring a fermionic Hamiltonian for arbitrary numbers N of particles by analytical means into a shape such that modern numerical methods can successfully be applied.For this purpose the Cook-Schroeck Formalism is taken as starting point.This includes the use of the occupation number representation.It is shown that the N -particle Hamiltonian is determined in a canonical way by a fictional 2particle Hamiltonian.A special approximation of this 2-particle operator delivers an approximation of the N -particle Hamiltonian, which is the orthogonal sum of finite dimensional operators.A complete classification of the matrices of these operators is given.Finally the method presented here is formulated as a work program for practical applications.The connection with other methods for solving the same problem is discussed.
The problem of finding the spectrum of the Fock-space Hamiltonian \ifmmode \hat{H}\else \^{H}\fi{} for a system of many fermions is analyzed. The quasiparticle formalism is employed, with ``holes'' and ``particles'' defined with respect to some N-particle determinantal wave function (the model vacuum). The basic idea behind the proposed approach is to perform a similarity transformation of Hamiltonian \ifmmode \hat{H}\else \^{H}\fi{}, such that the resulting effective Hamiltonian \ifmmode \hat{G}\else \^{G}\fi{} is, unlike \ifmmode \hat{H}\else \^{H}\fi{}, a quasiparticle-number--conserving operator. It is shown that eigenvalues of \ifmmode \hat{G}\else \^{G}\fi{}, corresponding to small numbers of quasiparticles (0,1,2) can be easily calculated. This is equivalent to finding eigenvalues of \ifmmode \hat{H}\else \^{H}\fi{} for certain states of N, N\ifmmode\pm\else\textpm\fi{}1, and N\ifmmode\pm\else\textpm\fi{}2 particles. The construction of the operator transforming \ifmmode \hat{H}\else \^{H}\fi{} into \ifmmode \hat{G}\else \^{G}\fi{} (the wave operator) stems from an analysis of the structure of the algebra of linear operators acting in a (finite-dimensional) Fock space. The exponential Ansatz for the wave operator is used, resulting in a generalization of the coupled-cluster (CC) method of Coester [Nucl. Phys. 7, 421 (1958)]. The generalized CC equations determining the wave operator, and equations determining the effective Hamiltonian \ifmmode \hat{G}\else \^{G}\fi{}, are presented in a diagrammatic form. An effort has been made to obtain a concise notation for expressing these equations in an algebraic form. Approximation schemes, necessary for practical applications of the proposed method, are also studied.
The limits of direct unitary transformation of many-fermion Hamiltonians are explored. Practical application of such transformations requires that effective many-body interactions be discarded over the course of a calculation. The truncation of the Hamiltonian leads to finite errors and in some cases divergences. A new formalism is proposed to manage errors and avoid divergences. Removing all interactions from a many-fermion Hamiltonian reduces it to fermion number operators allowing for direct calculation of eigenvalues. If the same transformations are applied to the bare fermions, eigenfermions are produced whose Slater determinants form eigenstates. This enables a hierarchy of diagonalization methods of increasing accuracy as fewer interactions are discarded from the Hamiltonian.
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.
Quadratic programming over orthogonal matrices encompasses a broad class of hard optimization problems that do not have an efficient quantum representation. Such problems are instances of the little noncommutative Grothendieck problem (LNCG), a generalization of binary quadratic programs to continuous, noncommutative variables. In this work, we establish a natural embedding for this class of LNCG problems onto a fermionic Hamiltonian, thereby enabling the study of this classical problem with the tools of quantum information. This embedding is accomplished by a new representation of orthogonal matrices as fermionic quantum states, which we achieve through the well-known double covering of the orthogonal group. Correspondingly, the embedded LNCG Hamiltonian is a two-body fermion model. Determining extremal states of this Hamiltonian provides an outer approximation to the original problem, a quantum analogue to classical semidefinite relaxations. In particular, when optimizing over the special orthogonal group our quantum relaxation obeys additional, powerful constraints based on the convex hull of rotation matrices. The classical size of this convex-hull representation is exponential in matrix dimension, whereas our quantum representation requires only a linear number of qubits. Finally, to project the relaxed solution back into the feasible space, we propose rounding procedures which return orthogonal matrices from appropriate measurements of the quantum state. Through numerical experiments we provide evidence that this rounded quantum relaxation can produce high-quality approximations.