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Physical Review

Statistical Mechanics of Coupled Particles in the Schrödinger Representation

Robert H. Kraichnan

Published Nov 15, 1958• 3 citations
Publisher

Abstract

A perturbation treatment is presented for the operator $u(t)$ giving the evolution of the state vector of a system of $N$ particles coupled by 2-body forces. The total Hamiltonian is divided into a diagonal part ${H}_{0}$ and off-diagonal part ${H}_{1}$ in the representation with $N$-particle plane-wave states as basis. An effective pair-interaction operator ${\overline{H}}_{1}$ is defined which includes "vertex" corrections of all orders, and the off-diagonal part of $u$ is expressed exactly by an expansion which involves only ${\overline{H}}_{1}$ and the exact diagonal part of $u$. A closed set of equations determining ${\overline{H}}_{1}$ and the diagonal part of $u$ are obtained by retaining only the leading terms in the expansion. These equations include all the corrections included in the Brueckner approximation and, in addition, they contain 3-body effects, iterated to all orders, which are omitted in that approximation. No one-to-one correspondence between the eigenstates of ${H}_{0}$ and those of the total Hamiltonian is appealed to, and the ground state plays no special role. The connection with statistical mechanics follows the fact that the partition function is $\mathrm{Tr}[u(\ensuremath{-}\frac{i}{\mathrm{kT}})]$. The theory simplifies greatly for very large $N$.

DOI
10.1103/physrev.112.1056

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Physical review. D. Particles, fields, gravitation, and cosmology/Physical review. D. Particles and fields•Feb 15, 1973•0 citations
A perturbation treatment is presented for the operator u(t)u(t)u(t) giving the evolution of the state vector of a system of NNN particles coupled by 2-body forces. The total Hamiltonian is divided into a diagonal part H0{H}_{0}H0​ and off-diagonal part H1{H}_{1}H1​ in the representation with NNN-particle plane-wave states as basis. An effective pair-interaction operator H‾1{\overline{H}}_{1}H1​ is defined which includes "vertex" corrections of all orders, and the off-diagonal part of uuu is expressed exactly by an expansion which involves only H‾1{\overline{H}}_{1}H1​ and the exact diagonal part of uuu. A closed set of equations determining H‾1{\overline{H}}_{1}H1​ and the diagonal part of uuu are obtained by retaining only the leading terms in the expansion. These equations include all the corrections included in the Brueckner approximation and, in addition, they contain 3-body effects, iterated to all orders, which are omitted in that approximation. No one-to-one correspondence between the eigenstates of H0{H}_{0}H0​ and those of the total Hamiltonian is appealed to, and the ground state plays no special role. The connection with statistical mechanics follows the fact that the partition function is Tr[u(\ensuremath−ikT)]\mathrm{Tr}[u(\ensuremath{-}\frac{i}{\mathrm{kT}})]Tr[u(\ensuremath−kTi​)]. The theory simplifies greatly for very large NNN.
A system of bosons interacting by 2-body forces is treated in momentum space in the Heisenberg representation, assuming a statistically homogeneous and stationary state. The principle of weak statistical dependence of different momentum modes, previously applied to turbulence dynamics, is used to obtain an exact expansion expressing quadrilinear (2-body) time-displaced correlation functions of the second-quantized field variables in terms of bilinear time-displaced correlation functions and average impulse-response functions. The last-named functions are defined in terms of the exact Green's operator, for infinitesimal sources, of the nonlinear Heisenberg equations of motion. A set of coupled integral equations are then obtained which determine both the bilinear correlation functions and the response functions. By retaining in these equations only the lowest terms in the cited expansion, a theory is obtained which includes particle "self-energy" interactions with the medium of all orders. A modification of the theory is briefly outlined which includes, at finite temperatures, all the processes described at absolute zero by the Brueckner approximation and, in addition, 3-body effects, iterated to all orders, which are omitted in that approximation.
A formulation of the energy eigenvalue problem for a many-particle system is presented in several alternative forms and, following Brueckner and his collaborators, the energy of the system is expressed in terms of solutions to the two-body problem. This work bears considerable resemblance to Brueckner's "linked cluster" expansion. The present expansion is derived with the aid of an ordering operator and involves a sequence of "nearest neighbor" interactions. The particles tend to form "chains," the sequence of approximations involving "two-particle chains," "three-particle chains," etc. The rapidity of convergence of the method depends on the particle density and "temperature" and on the nature of the force between particles. The method lends itself readily to use in statistical mechanics. Some applications to the calculation of the equation of state of a gas are included.
In the conventional perturbation treatment of the many-body problem of interacting particles, the zero-order Hamiltonian corresponds to independent particles moving in a static over-all potential. A discussion of the effects of particle interactions or `correlations' shows that if one starts instead with a Hamiltonian representing noninteracting particles in a velocity-dependent over-all potential, deeper for slow, and shallower for fast particles, then part of the correlation effect is included already in zero order. In addition, a velocity-dependent over-all potential may be called for by velocity dependent interparticle forces or by exchange forces. The degree of the improvement in the convergence of a perturbation expansion based on a Hamiltonian with a velocity-dependent over-all potential is discussed and illustrated by a simple example in which the velocity dependence of the potential gives rise to a reduced "effective mass" of the particles.The many-body problem of a large, uniform system of interacting particles (e.g., the case of a heavy nucleus without surface effects) is formulated in detail in perturbation theory, starting with a velocity-dependent potential constant in space. The work involved in such a calculation turns out to be essentially the same as with a velocity-independent potential, the effect of the velocity dependence being to reduce each term in the perturbation expansion by a constant factor raised to a power equal to the number of "energy denominators" ${E}_{m}\ensuremath{-}{E}_{n}$ in the term in question. A simple equation is deduced for the optimum degree of velocity dependence of the over-all potential which ensures the most rapid convergence.
The concept of an effective Hamiltonian in Fock space is introduced. It is based on the division of the entire one-particle space into subspaces of ‘‘active’’ and ‘‘inactive’’ orbitals. The effective Fock space Hamiltonian has—for active model states—the same eigenvalues as the full Hamiltonian. The theory outlined in this context differs from that of paper I mainly in a different definition of the ‘‘diagonal part’’ of an operator, and in the fact that the ‘‘quasidegenerate case’’ applies throughout. The separation theorem, and as a consequence the connected diagram theorem, is shown to hold, in a more limited sense though, even for those normalizations where it did not in the context of universal wave and energy operators. Unlike in the theory of the ‘‘universal’’ operators of paper I the Fock space and n-particle Hilbert space approaches with analogous normalizations are no longer equivalent. In particular, the Primas normalization with a fully Lie-algebraic structure does not lead to a connected diagram expansion if it is formulated in n-particle Hilbert space, only so in a Fock space formulation. In n-particle Hilbert space with the present definition of the diagonal part of an operator the normalizations b (‘‘canonical’’) and c (‘‘Primas’’) happen to agree. As an alternative to the construction of the wave and energy operator W and L by perturbation theory the nonperturbative approach is presented as a generalization of the coupled-cluster method, in detail both in the intermediate and in the unitary normalization. In the unitary variant only a linear system for σ (the logarithm of the wave operator) has to be solved in order to get L correct through fifth order in perturbation theory with important contributions of higher orders included. A generalization of the Hartree–Fock method to Fock space theory is outlined, which guarantees stationarity of all (active) eigenstates with respect to one-particle transformations. A generalized electron pair theory is also defined. An analysis of the necessary computational steps shows that the nonperturbative approaches do not require significantly more computational effort than perturbation theory to the corresponding order. As a numerical example the H2 molecule in a small basis is discussed.
A general system composed of many weakly interacting bosons and/or fermions is considered. The object is to develop a procedure for renormalizing the single-particle creation operators, so as to remove the interactions to successively higher orders of perturbation. The basic idea is that creation operators ${{\ensuremath{\theta}}_{i}}^{\ifmmode\dagger\else\textdagger\fi{}}$ must satisfy the Hamiltonian commutator equations $[\mathcal{H},{{\ensuremath{\theta}}_{i}}^{\ifmmode\dagger\else\textdagger\fi{}}]={\ensuremath{\omega}}_{i}{{\ensuremath{\theta}}_{i}}^{\ifmmode\dagger\else\textdagger\fi{}}$, where H\mathcal{H}H is the Hamiltonian and ${\ensuremath{\omega}}_{i}$ are the particle energies. For the many-weakly-interacting-particle system, the zeroth-order boson and fermion creation operators satisfy these equations to zeroth order. It is shown that if the creation operators are renormalized so as to satisfy the Hamiltonian commutator equations to order mmm, and also to satisfy the appropriate boson commutators and fermion anticommutators to order mmm, then the problem is solved to order mmm. In particular, the vectors formed by operating on the ground state with the renormalized creation operators, according to the usual boson and fermion occupation-number representation, are eigenvectors of H\mathcal{H}H and are orthonormal, all to order mmm. A procedure is otained for finding the (m+1m+1m+1)-order contributions to the particle creation energies, in terms of the mmm-order operators. Explicit first-order calculations of these general results are provided for a system of bosons and a system of fermions, and these first-order results are shown to include similar results of Rayleigh-Schr\"odinger perturbation theory, the random-phase approximation, and the method of thermodynamic Green's functions. The problem of anharmonic lattice dynamics is studied in detail, and a method of undetermined coefficients is used to renormalize the phonon creation operators to first order. The phonon energies are calculated to second order, and this calculation shows that the interactions between renormalized phonons cannot be removed in second order. Statistical averages of the phonon energies give the energy shifts and lifetimes which have been calculated previously by various propagator techniques. In addition, the renormalized energies are used to calculate the temperature-dependent part of the Helmholtz free energy correct to second order. As a final example, electron-phonon interactions in a normal metal are studied. The electron and phonon creation operators are renormalized to first order, the particle energies are calculated to second order, and statistical averages of the particle energies recover the usual thermodynamic Green's-function results for energy shifts and lifetimes. These examples show the simplicity by which the renormalization procedure obtains a great amount of detailed information about the single-particle nature of a many-particle system.
This is the first in a series of papers. It deals with the linked-cluster summation of the perturbation series in quantum mechanics and quantum statistical physics. In particular, a formalism is developed for calculating the matrix element of the operator exp(+sH)\mathrm{exp}(+sH)exp(+sH), where HHH is the quantum-mechanical Hamiltonian operator. It is found that the matrix element may be written as an ordinary ccc-number integral of a ccc-number function over the phase space. This function is found by doing a linked-cluster summation of the usual perturbation series. It is in the form of an exponential of an effective Hamiltonian, Heff{H}_{\mathrm{eff}}Heff​, which is a function of the parameter sss and the ccc-number momentum and position coordinates. We have assumed that the Hamiltonian has no spin-dependent terms. The matrix element is written in such a form that it may be used for calculating the transition matrix elements in time-dependent perturbation theory as well as for calculating the canonical partition function ZN{Z}_{N}ZN​ in quantum statistical mechanics. The former is obtained by substituting $s=\ensuremath{-}\frac{\mathrm{it}}{\ensuremath{\hbar}}$ and the latter by substituting $s=\ensuremath{-}\ensuremath{\beta}$. The series for Heff{H}_{\mathrm{eff}}Heff​ depends only on the potential and is independent of the statistics. The effects of the statistics (Bose-Einstein, Fermi-Dirac, or Maxwell-Boltzmann) are all included in a certain factor multiplying exp(sHeff)\mathrm{exp}(s{H}_{\mathrm{eff}})exp(sHeff​). Thus we have succeeded in separating the statistical and quantum corrections. These latter arise from the noncommutativity of the position and momentum operators. The effective Hamiltonian is an infinite series. A diagrammatic technique is given for evaluating this series. It is an expansion in powers of the coupling constant and can also be expanded in powers of Planck's constant. In general, it is not possible to give a closed-form expression for Heff{H}_{\mathrm{eff}}Heff​. But for most practical cases it is sufficient to keep the first few powers of Planck's constant $\ensuremath{\hbar}$, because it is very small. We therefore have an entirely new method of approximation. The statistical effect may always be included exactly. No numerical estimates are made in this paper.