John R. Klauder

On the Completeness of the coherent states

We study subsets of coherent states based on square lattices in the complex plane, namely, {z.sfnc;Zm,n〉} where Zm,n=γ(m+in) for m,n=0, ±1, ±2, … Analyticity arguments suffice to establish completeness if 0 π. The completeness of the case γ=π, stated without proof by von Neumann, is established by invoking square integrability along with analyticity.

The modification of electron energy levels by impurity atoms

Abstract The influence of localized impurities in solids may be represented either by (i) the effect of randomly distributed classical scattering centers for which only averaged quantities are meaningful or, as proved herein, (ii) the effect of an equivalent quantum description which incorporates both the random character and the process of averaging. Adopting the latter viewpoint, we readily make contact with diagrammatic perturbation methods familiar in field theory and many-body studies. As an application of this formalism, various approximations, all of infinite order in the perturbation, are derived for the average density of states in an idealized solid for which the effects of the impurities prevail and the periodic lattice may be neglected. The results of these approximations are compared with the exact results obtained by Frisch and Lloyd for a special model in which the impurity potential may be represented as an attractive one-dimensional delta-function. One of the approximations discussed is a direct analog of the Brueckner model, and for the delta-function potential example this approximation can be solved exactly. In the case of a high density of impurities, our solutions demonstrate the well-known value of propagator modification, while, for a low density of impurities, several solutions exhibit an impurity band that is isolated from the conduction band. Finally, we discuss the average wave function, a quantity that is open to some arbitrariness because a wide latitude exists in the choice of a rule by which a wave function is paired with a set of impurity sites. Two distinct “pairingrules” are analyzed in detail: one rule restricts the wave and its derivative to have specified initial values but demands neither normalizability nor singlevaluedness; the other rule requires that every acceptable wave be an eigenvector whose phase, relative to the other waves, is selected in a definite manner. Differential equations are presented which are obeyed by the two differently defined average wave functions in the case of a one-dimensional delta-function potential.

Numerical Integration of Multiplicative-Noise Stochastic Differential Equations

A two-stage, Runge–Kutta algorithm for vector Ito (and, by transform, also Stratonovich) stochastic differential equations with multiplicative noise has been developed. The method is second order accurate; but, for vanishing drift the algorithm yields a martingale independent of step size. Several examples are included to illustrate our method. A discussion of errors shows that sample size can be as important as truncation.

Coherent State Quantization of Constraint Systems

We study the quantization of systems with general first- and second-class constraints from the point of view of coherent state phase-space path integration, and show that all such cases may be treated, within the original classical phase space, by using suitable path-integral measures for the Lagrange multipliers which ensure that the quantum system satisfies the appropriate quantum constraint conditions. Unlike conventional methods, our procedures involve no {delta}-functionals of the classical constraints, no need for dynamical gauge fixing of first-class constraints nor any average thereover, no need to eliminate second-class constraints, no potentially ambiguous determinants, as well as no need to add auxiliary dynamical variables expanding the phase space beyond its original classical formulation, including no ghosts. Additionally, our procedures have the virtue of resolving differences between suitable canonical and path-integral approaches, and thus agree with previous results obtained by other methods for such cases. Several examples are considered in detail. {copyright} 1997 Academic Press, Inc.

Continuous‐Representation Theory. III. On Functional Quantization of Classical Systems

The form of Schrodingers equation in a continuous representation is indicated for general systems and analyzed in detail for elementary Bose and Fermi systems for which illustrative solutions are given. For any system, a natural continuous representation exists in which state vectors are expressed as continuous, bounded functions of the corresponding classical variables. The natural continuous representation is generated by a suitable set S of unit vectors labeled by classical variables for which, for the system in question, the quantum action functional restricted to the domain S is equivalent to the classical action. When a classical action is viewed in this manner it contains considerable information about the quantum system. Augmenting the classical action with some physical significance of its variables, we prove that the classical theory virtually determines the quantum theory for the Bose system, while it uniquely determines the quantum theory for the Fermi system.

Vestigial effects of singular potentials in diffusion theory and quantum mechanics

Repulsive singular potentials of the form λV (x) =λ‖x−c‖−α, λ≳0, in the Feynman−Kac integral are studied as a function of α. For α≳2 such potentials completely suppress the contribution to the integral from paths that reach the singularity, and thus, unavoidably, certain vestiges of the potential remain even after the coefficient λ↓0. For 2⩾α⩾1 careful definition by means of suitable counterterms at the point of singularity (similar in spirit to renormalization counter terms in field theory) can lead to complete elimination of effects of the potential as λ↓0. For α<1 no residual effects of the potential exist as λ↓0. In order to prove these results we rely on the theory of stochastic processes using, in particular, local time and stochastic differential equations. These results established for the Feynman−Kac integral conform with those known in the theory of differential equations. In fact, a variety of vestigial effects can arise from suitable choices of counter terms, and these correspond in a natural ...

Quantum‐mechanical path integrals with Wiener measure for all polynomial Hamiltonians. II

The coherent‐state representation of quantum‐mechanical propagators as well‐defined phase‐space path integrals involving Wiener measure on continuous phase‐space paths in the limit that the diffusion constant diverges is formulated and proved. This construction covers a wide class of self‐adjoint Hamiltonians, including all those which are polynomials in the Heisenberg operators; in fact, this method also applies to maximal symmetric Hamiltonians that do not possess a self‐adjoint extension. This construction also leads to a natural covariance of the path integral under canonical transformations. An entirely parallel discussion for spin variables leads to the representation of the propagator for an arbitrary spin‐operator Hamiltonian as well‐defined path integrals involving Wiener measure on the unit sphere, again in the limit that the diffusion constant diverges.

Continuous Representation Theory Using the Affine Group

We present a continuous representation theory based on the affine group. This theory is applicable to a mechanical system which has one or more of its classical canonical coordinates restricted to a smaller range than − ∞ to ∞. Such systems are especially troublesome in the usual quantization approach since, as is well known from von Neumanns work, the relation [P, Q] = −iI implies that P and Q must have a spectrum from − ∞ to ∞ if they are to be self‐adjoint. Consequently, if the spectrum of either P or Q is restricted, at least one of the operators, say Q, is not self‐adjoint and does not have a spectral resolution. Thus Q cannot generate a coordinate representation. This leads us to consider a different pair of operators, P and B, both of which are self‐adjoint and which obey [P, B] = −iP. The Lie group corresponding to this latter algebra is the affine group, which has two unitarily inequivalent, irreducible representations, one in which the spectrum of P is positive. Using the affine group as our ki...

Continuous‐Representation Theory. IV. Structure of a Class of Function Spaces Arising from Quantum Mechanics

A rigorous development of the continuous representation of Hilbert space by bounded, continuous, multidimensional phase‐space functions ψ(p, q) is presented. It is shown that these functions form a closed subspace of L2(p, q) whose elements are functions and not equivalence classes. Differential properties are investigated and it is pointed out that there are a multitude of definitions whereby ψ(p, q) possesses continuous derivatives of all orders. In one of these definitions, each ψ(p, q) is proportional to a multidimensional, entire function f(q − ip), establishing a connection between Bargmanns Hilbert space of entire functions and one example of a continuous representation. Attention is devoted to the purely functional characterization of the continuous representation by means of the reproducing kernel as a special case of Aronszajns general theory. Properties of various operators in a continuous representation are carefully defined.

Direct-Product Representations of the Canonical Commutation Relations

We consider direct-product representations of the canonical commutation relations. An irreducible representation is defined on each of the incomplete direct-product spaces (IDPS) of von Neumann. We prove that two such representations are unitarily equivalent if and only if the corresponding IDPS are weakly equivalent, for which simple analytic tests exist. The matrix elements of these representations, coupled with a Friedrichs-Shapiro type of integral, fulfill group orthogonality relations. This classification into unitary equivalence classes also applies to direct-product representations of the canonical anticommutation relations.