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International Journal of Theoretical Physics

Classical limit of the probability density for a perturbed change in the Hamiltonian

Der-Ruenn Su

Published May 1, 1971• 0 citations
Publisher
DOI
10.1007/bf00673802

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The stationary perturbation series is derived from the time-dependent perturbation series up to any order by using a formula derived from the theory of divergent series. The probability density of the perturbed state turns out as the evolution of the density operator of the pure unperturbed initial state as h -+ 0. This result, for h -+ 0, indicates that the particle moves in the classical trajectory with the Hamiltonian H = Ho + V immediately after V is turned on, if it is in the trajectory of H0 initially. This confirms the classical dynamics. Further, the adiabatic theorem is introduced in order to get the same conclusion for arbitrary finite potential V.
I present a derivation of the many qualitative changes to basic scattering theory that result when a (partially) coherent matter beam inelastically interacts with a quantum target. The weak beam limit, where only one object in the beam interacts with the quantum target, is developed. Some of the basic changes to scattering theory are as follows. (1) The transition probability can increase as the square of the number of particles in the beam. (2) The transition probability is not proportional to the convolution of the total inelastic cross section with the momentum distribution of the beam. (3) The transition probability to different final states of the target does not depend on the population in the initial states but depends on the amplitudes for the initial states. (4) The total transition probability depends on the differential cross section integrated over all final angles and on the differential cross section only in the forward scattering direction. (5) The transition probability can be nonzero even when the beam misses the target. (6) In the perturbative limit, the transition probability can be related to classical, large scale properties of the beam (such as the average density).
We provide a brief overview of approaches for calculating the density of states of quantum systems and random matrix Hamiltonians using the tools of free probability theory. For a given Hamiltonian of a quantum system or a generic random matrix Hamiltonian, which can be written as a sum of two non-commutating operators, one can obtain an expression for the density of states of the Hamiltonian from the known density of states of the two component operators by assuming that these operators are mutually free and by using the free additive convolution. In many examples of interacting quantum systems and random matrix models, this procedure is known to provide a reasonably accurate approximation to the exact numerical density of states. We review some of the examples that are known in the literature where this procedure works very well, and also discuss some of the limitations of this method in situations where the free probability approximation fails to provide a sufficiently accurate description of the exact density of states. Subsequently, we describe a perturbation scheme that can be developed from the subordination formulas for the Cauchy transform of the density of states and use it to obtain approximate analytical expressions for the density of states in various models, such as the Rosenzweig-Porter random matrix ensemble and the Anderson model with on-site disorder.