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Journal of Applied Physics

Analytic Wigner distribution function for a split potential well

Kevin L. Jensen, Donald A. Shiffler, Jeanne M. Riga, J. R. Harris, Joel L. Lebowitz, Marc M. Cahay, John J. Petillo

Published Oct 9, 2019• 6 citations
Publisher

Abstract

The analytic Wigner function for a single well with infinite walls is extended to the configuration where half of the well is raised and the weighting of the wave functions is in accordance with a thermal Fermi-Dirac distribution. This requires a method for determining the energy eigenstates particularly near the height of the higher half-well. The methods provide a means for constructing analytic and accurate trajectories that can be used to model tunneling times. The fidelity of the trajectory model is assessed with respect to absorption of quanta related to changes within the allowable eigenstates.

DOI
10.1063/1.5110406

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The analytic Wigner function for a single well with infinite walls is extended to the configuration where half of the well is raised and the weighting of the wave functions is in accordance with a thermal Fermi-Dirac distribution. This requires a method for determining the energy eigenstates particularly near the height of the higher half-well. The methods provide a means for constructing analytic and accurate trajectories that can be used to model tunneling times. The fidelity of the trajectory model is assessed with respect to absorption of quanta related to changes within the allowable eigenstates.
The Wigner function is assembled from analytic wave functions for a one-dimensional closed system (well with infinite barriers). A sudden change in the boundary potentials allows for the investigation of time-dependent effects in an analytically solvable model. A trajectory model is developed to account for tunneling when the barrier is finite. The behavior of the density (the zeroth moment of the Wigner function) after an abrupt change in potential shows net accumulation and depletion over time for a weighting of energy levels characteristic of the supply function in field emission. However, for a closed system, the methods have application to investigations of tunneling and transmission associated with field and photoemission at short time scales.
Abstract In the present paper a method of finding the dynamics of the Wigner function of a particle in an infinite quantum well is developed. Starting with the problem of a reflection from an impenetrable wall, the obtained solution is then generalized to the case of a particle confined in an infinite well in arbitrary dimensions. It is known, that boundary value problems in the phase space formulation of the quantum mechanics are surprisingly tricky. The complications arise from nonlocality of the expression involved in calculation of the Wigner function. Several ways of treating such problems were proposed. They are rather complicated and even exotic, involving, for example, corrections to the kinetic energy proportional to the derivatives of the Dirac delta–function. The presented in the manuscript approach is simpler both from analytical point of view and regarding numerical calculation. The solution is brought to a form of convolution of the free particle solution with some function, defined by the shape of the well. This procedure requires calculation of an integral, which can be done by developed analytical and numerical methods.
The analytic nature of the transmission coefficient for a δ-function barrier makes it a useful tool to examine a variety of technologically important applications, such as photoemission from semiconductors with an alkali coating, the examination of tunneling times for wave packets incident on a barrier, and for parameterizing tunneling through the narrow barrier of a normal-superconducting point contact. The analytic model of a δ-function barrier inside a confining well is extended to the finite height and width rectangular barrier (a delta-function sequence). Methods to exactly evaluate the eigenstates are given and their dependencies are examined. The time evolution of a superposition of the lowest eigenstates is considered for barriers having comparable Gamow tunneling factors so as to quantify the impact of barrier height and shape on time evolution in a simple and exact system and, therefore, serve as a proxy for tunneling time. Last, density profiles and associated quantum potentials are examined for coupled wells to show changes induced by weaker and wider barriers.
We applied the Wigner Distribution Function, a distribution function of time and frequency based on an initial function of either of those variables, to a series of time based correlation functions. These time based correlation functions were the result of a 1-dimensional free particle wave packet, the reactant wave function, which had propagated through a quantum potential well and then had components of the reactant wave function that exited the opposite side of the well auto-correlated in time with a stationary 1-dimensional free particle wave packet, the product wave function. This process was undertaking in order to generate a 3-dimensional depiction, in time and frequency, of the reactant wave functions interaction with the quantum potential well. Fortran 77 code was utilized to generated the time propagation of the reactant wave function by means of the Split Operator Method, which was given the following initial set of conditions; x sub 0 = -20 (Bohr radii), k sub 0 = 3 (atomic units), and = 1 (Bohr radius). A series of potential wells with variable depths were implemented into the code. The code then computed the correlation in time of the exiting reactant wave function with a stationary wave function before applying the Wigner Distribution Function. When Wigner Distribution Function was applied to the time correlation function many recognizable features on the potential well were observed from the 3-dimensional plot generated including transmission resonance energy levels. The classical time of arrival was also captured by the Wigner Distribution Function. As a useful tool the Wigner Distribution Function provides more insight into the quantum interactions of chemical reactions in terms of time and frequency than traditional spectrographic analysis.
Two methods have been widely employed for computing tunneling rates in a double-well potential, one based on a transmission function, the second on energy splitting. Semiclassical calculations (Brickmann and Zimmermann) show that the transmission method leads to lower rates than the splitting method. It is shown here without employing the semiclassical approximation that a more accurate relation between the energy splitting and the transmittion function may be obtained by using a decomposition of the stationary states into scattering states. This relation is then used to provide an analytical basis for the usual heuristic picture in which the particle oscillates with classical frequency in one well and has a probability of tunneling to the other well equal to the transmission function in each classical period of oscillation. It is concluded that the transmission method should give more meaningful results, particularly in situations where interactions of the system with a heat bath have significant effects in the tunneling times predicted by the splitting method. This is in accord with previous work in which both methods were used in an attempt to fit experimental data for hydrogen diffusion in niobium.