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arXiv:2108.05719 · Brazilian Journal of Physics

Compact Equations for the Envelope Theory

Lorenzo Cimino, Claude Semay

Published Jan 28, 2022• 4 citations
arXivHTMLPDF

Abstract

The envelope theory is a method to easily obtain approximate, but reliable, solutions for some quantum many-body problems. Quite general Hamiltonians can be considered for systems composed of an arbitrary number of different particles in D dimensions. In the case of identical particles, a compact set of three equations can be written to find the eigensolutions. This set provides also a nice interpretation and a starting point to improve the method. It is shown here that a similar set of seven equations can be determined for a system containing an arbitrary number of two different particles.

DOI
10.1007/s13538-021-01047-7

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The envelope theory is a method to easily obtain approximate, but reliable, solutions for some quantum many-body problems. Quite general Hamiltonians can be considered for systems composed of an arbitrary number of different particles in D dimensions. In the case of identical particles, a compact set of three equations can be written to find the eigensolutions. This set provides also a nice interpretation and a starting point to improve the method. It is shown here that a similar set of seven equations can be determined for a system containing an arbitrary number of two different particles.
The envelope theory, also known as the auxiliary field method, is a technique to compute approximate solutions of a quantum system with N identical particles. It has been generalized for various systems with arbitrary kinematics and one- or two-body potentials. The basic idea is to replace the Hamiltonian H under study by an auxiliary Hamiltonian ˜H which is solvable, the eigenvalues of H˜ being optimized to be as close as possible to those of H. The method is easy to implement since it reduces to find the solution of a transcendental equation. Recently, its accuracy has been tested for eigenvalues and eigenvectors by computing the ground state of various systems containing up to 10 bosons. We show here that the envelope theory can be extended to compute the eigensolutions of a system of identical particles with a type of many-body forces often used in phenomenological models.
A method based on the envelope theory is presented to compute approximate solutions for N-body Hamiltonians with identical particles in D dimensions (D ⩾ 2). In some favorable cases, the approximate eigenvalues can be analytically determined and can be lower or upper bounds. The accuracy of the method is tested with several examples, and an application to a N-body system with a minimal length is studied. Finally, a semiclassical interpretation is given for the generic formula of the eigenvalues.
The solution of Schrödinger’s equation for systems composed of a large number of strongly interacting particles is an unsolved problem. Indeed, even the classical three-body problem cannot be solved in general form. Fortunately, the problem of obtaining the Ψ-function of such a system is not only insoluble but unnecessary; such a detailed description of the many-body problem is not required by any experiment that is possible in real life. Any real experimental set-up contains a comparatively small number of particle indicators, and hence, in cases where a large number of particles take part in a given process, can give only their averaged characteristics. Any attempt to determine the coordinates of every particle of such a system would lead (because of the uncertainty relation) to a complicated excited state and would indeed change the properties of the system. Thus, the description of a macroscopic system by means of a wave function is an inadequate approach * : we need 262 instead methods of incomplete description, which determine only the relations between averaged quantities. An example of such a description is hydrodynamics, the equations of which determine only the average velocity of the particles at each point (i. e. the velocity field). A second example, closer to our problem, is the kinetic equation, which allows us to find the distribution of particles with respect to velocity and position. A knowledge of the one-particle distribution function f ( r ˜ , p ˜ , t ) https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780429497940/5adca591-a306-49c7-affc-2d249cfef85c/content/eq1304.tif"/> , which depends on the coordinate r ˜ https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780429497940/5adca591-a306-49c7-affc-2d249cfef85c/content/eq1305.tif"/> and velocity p ˜ https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780429497940/5adca591-a306-49c7-affc-2d249cfef85c/content/eq1306.tif"/> of the particles, allows us to calculate the mean values of additive quantities such as the density n ( r ˜ ) = ∑ i δ ( r ˜ - r ˜ i ) https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780429497940/5adca591-a306-49c7-affc-2d249cfef85c/content/eq1307.tif"/> or the momentum per unit volume j ˜ ( r ˜ ) = ∑ i p ˜ i δ ( r ˜ - r ˜ i ) https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780429497940/5adca591-a306-49c7-affc-2d249cfef85c/content/eq1308.tif"/> . The two-particle distribution function f ( r ˜ 1 , r ˜ 2 ; p ˜ 1 , p ˜ 2 ; t ) https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780429497940/5adca591-a306-49c7-affc-2d249cfef85c/content/eq1309.tif"/> allows us to find the correlations between the coordinates and the velocities of two particles and to determine the averages of quantities which depend on the coordinates of two particles, for instance the mean value of the pairwise interaction energy V = ∑ ik V ( r ˜ i - r ˜ k ) https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780429497940/5adca591-a306-49c7-affc-2d249cfef85c/content/eq1310.tif"/> . The periodic or weakly damped solutions of the equation for the distribution function give the eigenfrequencies of the system.
Approximation of eigen values and eigenvectors Many realistic problems that arise in theoretical physics lead to equations which cannot be given exact general solutions. Therefore, many attempts have been made to produce satisfactory methods to provide approximate solutions. The most common ones search for solutions in terms of power series and in terms of asymptotic expansions. The main idea behind approximation methods is to try to find approximate solutions of a given system by means of exact solutions of an approximate system (a ‘nearby’ system). We start with the eigenvalue equation associated with the Hamiltonian H and consider an approximating one, say H 0 , whose equations of motion can be explicitly solved. We introduce an interpolating family of Hamiltonians H ε = H 0 + ε H 1 such that, for a given value of ε, say ε, H ε ≡ H . Both eigenvalues and eigenvectors for H ε may be expanded as a series in ε and solved order by order. The simplest example to illustrate what we have in mind is provided by a second-order algebraic equation, e.g. x 2 − 3.03 x + 2.02 = 0. This equation may be written by means of an interpolating family x 2 − 3(1 + ε) x + 2(1 + ε) = 0, which yields the equation we want to solve when ε = 10 −2 . One can consider the approximate equation to be given by ε = 0, i.e. x 2 − 3 x + 2 = 0.
The methods of quantum chemistry and solid state theory to solve the many-body problem are reviewed. We start with the definitions of reduced density matrices, their properties (contraction sum rules, spectral resolutions, cumulant expansion, NNN-representability), and their determining equations (contracted Schrödinger equations) and we summarize recent extensions and generalizations of the traditional quantum chemical methods, of the density functional theory, and of the quasi-particle theory: from finite to extended systems (incremental method), from density to density matrix (density matrix functional theory), from weak to strong correlation (dynamical mean field theory), from homogeneous (Kimball-Overhauser approach) to inhomogeneous and finite systems. Measures of the correlation strength are discussed. The cumulant two-body reduced density matrix proves to be a key quantity. Its spectral resolution contains geminals, being possibly the solutions of an approximate effective two-body equation, and the idea is sketched of how its contraction sum rule can be used for a variational treatment.