Bin Kang Cheng

Green function approach for general quantum graphs

We present a recursive prescription to calculate the exact Green function for general quantum graphs. For the closed case, the expression for the poles of G—which gives the individual eigenstates—has the structure of a semiclassical formula, where the sum over the periodic orbit is already performed. As applications we discuss eigenstate localization for a three-arm closed star and filter-like mechanisms for transmission throughout an open trident graph.

Green functions for generalized point interactions in one dimension: A scattering approach

Recently, general point interactions in one dimension has been used to model a large number of different phenomena in quantum mechanics. Such potentials, however, require some sort of regularization to lead to meaningful results. The usual ways to do so rely on technicalities that may hide important physical aspects of the problem. In this work we present a method to calculate the exact Green functions for general point interactions in one dimension. Our approach differs from previous ones because it is based only on physical quantities, namely, the scattering coefficients R and T to construct G. Renormalization or particular mathematical prescriptions are not invoked. The simple formulation of the method makes it easy to extend to more general contexts, such as for lattices of N general point interactions, on a line, on a half-line, under periodic boundary conditions, and confined in a box.

A generalized semiclassical expression for the eigenvalues of multiple well potentials

From the poles of a generalized semiclassical Greens function we derive expressions for the eigenvalues of 1D multiple well potentials. In the case of asymmetric and symmetric double wells, we also obtain analytical formulae for, respectively, the shift and splitting of energies. Our results are better than some approximations in the literature because they take more properly into account the tunnelling through the barriers forming the multiple well and depend on energy-dependent Maslov indices. We illustrate the good numerical precision of the method by discussing some case tests on double wells.

Extended Feynman formula for time-dependent forced harmonic oscillator

Abstract Horvathys modification of Feynmans path integral formula is generalized to the time-dependent forced harmonic oscillator. The propagator at caustics is then obtained by using its modified semi-group property. Finally, with our new formula, the propagator for a charged particle in a time-dependent electromagnetic field is evaluated exactly beyond and at caustics.

Asymptotic Green functions: a generalized semiclassical approach for scattering by multiple barrier potentials

We show how quantum mechanical barrier reflection and transmission coefficients and can be obtained from asymptotic Green functions. We exemplify our results by calculating such coefficients for the Rosen-Morse (RM) potential. For multiple barrier potentials, V(x) = ∑jV (j)(x), where each V (j) goes to zero for x→±∞, we derive the asymptotic Green functions by a generalized semiclassical approximation, which is based on the usual sum over classical paths considered only in the classically allowed regions and includes local quantum effects through the individual (j) and (j). The approach is applied to double RM potentials and to Woods-Saxon barriers. We obtain analytical expressions for the transmission and reflection probabilities of these potentials which are very accurate when compared with exact numerical calculations, being much better than the usual WKB approximation. Finally we briefly discuss how to extend the present method to other kinds of potential.

Exact propagator of the harmonic oscillator with a time-dependent mass

Abstract By finding a space and time transformation, the exact evaluation of the propagator for the harmonic oscillator with a time-dependent mass by the path integral method becomes possible. We then derive the wavefunctions from the propagator obtained. Finally, the propagator beyond and at caustics is evaluated by using its modified semi-group property and is confirmed by investigating the classical paths with two fixed end-positions.

Exact propagators for moving hard-wall potentials

The semiclassical approximation has been used to evaluate the exact propagators for two one-dimensional quantum systems: (a) a free particle interacting with one hard-wall potential moving with constant velocity, and (b) a free particle inside a rigid box with one wall moving uniformly in time. The results derived from these propagators are in agreement with those of the corresponding Schrodinger equations.

Extended Feynman formula for damped harmonic oscillator with time-dependent perturbative force

Abstract Horvathys modification of Feynmans path integral formula is extended to the damped harmonic oscillator with time-dependent pertubative force. The propagator at caustics is then obtained by introducing its modified semi-group property. Finally, our new results are confirmed by investigating the classical paths joining two end-point positions.

Classifying the general family of 1D point interactions: a scattering approach

We propose a classification scheme for the complete family of 1D point interactions. To do so, we first review the solutions of the wavefunctions and Green functions of the problem. Second, we derive the exact time-dependent propagators in such a way that we can write the expressions for the Ks in a very compact form. As they should, such expressions do reduce to the known results in the literature according to the potential parameter values. Then, we analyse in general terms how the different point interactions scatter off arbitrary initially localized wave packets. Finally, we show that the physical features associated with the scattering process can be used to establish a classification procedure. Moreover, these physical characteristics are directly related to the potential parameters leading to the many formulae for the Ks. As an application, we present numerical calculations for Gaussian wave packets.

Propagator for a harmonically bound charged particle in a constant magnetic field and with the vector potential of a solenoid

Abstract Performing the constrained path integral in polar coordinates, the propagator for a harmonically bound charged particle subjected both to a constant magnetic field and to the vector potential of an infinitely thin solenoid containing magnetic flux is evaluated exactly. The result is then used to derive the energy eigenfunctions, the energy eigenvalues and the Bloch density matrix of the system considered.