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), where
H is the quantum-mechanical Hamiltonian operator. It is found that the matrix element may be written as an ordinary
c-number integral of a
c-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, which is a function of the parameter
s and the
c-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 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 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). 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. 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.
In this paper, four distinct ideas are combined, which under a wide range of circumstances can give very rapidly converging series expansions for functional integrals. (1) Expansion of the functional being integrated in functional Taylor series. In the familiar case arising in quantum statistical mechanics, that of the Wiener integral of exp[−∫V(x(t))dt],V(x) being the perturbing potential, this is equivalent to expanding the characteristic functional of the probability functional of V(x(t)) in central moments of V(x(t)). The lowest-order term of the series is the approximation obtained by Feynman and Hibbs through a variational method. (2) Transfer of the harmonic term of the potential, when the functional integral is the quantum-statistical density matrix (Green's function of the Bloch equation), to the weighting function. This transforms the functional integral from a Wiener to an Uhlenbeck-Ornstein integral. The formal expressions for the terms of the expansion are somewhat more complicated, but they can be worked out, and the result is a great improvement in the speed of convergence of the series with decreasing temperature and/or decreasing relative magnitude of the anharmonic part of the potential. (3) ``Reservation of variables'' in the integration. This amounts to breaking the averaging process down into an average over subsets of the distributed random function (conditional average), followed by an averaging of these averages. Any step of this kind (it may be repeated within the subsets, etc.) gives an improvement of accuracy. (4) When the quantity being evaluated through functional integration is the partition function, the device introduced by Feynman and Hibbs, of interchanging the functional integration with the integration of the Green's function over the equated initial and final configuration-space points, may be combined with the above techniques. This eliminates one integration in the terms of the expansion and seems to improve accuracy at the same time. The general series obtained is correlated with more conventional operator techniques of quantum-mechanical perturbation theory, in order to answer the perennial question, does the path-integral method bring with it anything that could not be derived by other methods? It is in some sense a Feynman-Dyson expansion of the Green's function, but one that is further modified mathematically in a way characteristic only of the path-integral point of view, and which, moreover, improves its accuracy. It thus appears unlikely that the result is merely one of standard type disguised as a functional integral result. Sample numerical calculations are given to assay the accuracy of the methods, which are shown to compare very favorably with the traditional approximation of finite subdivision of the time interval.
It is shown by a rigorous development from first principles that a relatively simple form for the partition function of an imperfect Bose-Einstein gas can be obtained. This form is that of a
3N-dimensional integral (
N being the total number of particles) the integrand of which can be completely determined for any given interparticle interaction provided certain, in general small, many-body quantum mechanical effects are ignored. The final evaluation of the partition function in closed form requires that this integrand be approximated by functions sufficiently tractable that the
3N-dimensional integral can be performed.
A formulation is given whereby the grand partition function of a many-body system satisfying Bose-Einstein or Fermi-Dirac statistics is expressed in terms of certain
U functions defined for the same system with Boltzmann statistics. It is then shown that these
U functions can be evaluated in successive approximations in terms of a binary kernel
B which can be computed from a solution of the two-body problem. The approach to the limit of infinite volume is studied. The example of a hard sphere interaction is discussed in some detail.
In statistical mechanics, involving quantum objects, one sometimes uses sums if a simple series sum rule exists. For example, in obtaining the Planck distribution 1/ (1-exp(hf/T)) one uses the sum of a geometric series. Otherwise, it seems one uses integrals, arguing that energy levels are very close together. For the Planck distribution with 1/ (1-exp(fon/T)) where n is 1,2,3…, one usually uses an integral, when a sum really applies. In (1), it is argued that such an integral does not yield the same result as the sum for this Planck distribution example. In particular, (1) considers fo as a very small number and shows explicitly that the summation depends on fo, while the integral does not. Given that this applies to the partition function, it carries over to all statistical quantities calculated from the partition function, as noted in (1). Here, we consider other examples. For example, if one tries to replace the geometric series sum used to obtain the Planck distribution, in the first place, with an integral, one immediately has a degeneracy issue. A one dimensional integral over dp (p=magnitude of momentum) yields one answer, while a three-dimensional result Integral 4*3.14 pp dp yields another. The geometric sum, however, does not include the degeneracies of p so it is not clear why one would use a 3-dimensional integral. Even if one uses a one dimensional integral, it does not yield the same result as the geometric sum. In fact, the geometric sum depends on the value of f/T, while the integral does not - it depends on T. Another example is the partition function for a single quantum particle in a box with infinite potential walls (e.g. a Fermi gas). Again an integral is used, but it seems that using p=po a where a is a small number proportional to 1/L, the length of the box and summing should yield an “a” dependent result, which the integral does not give.