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Communications in Theoretical Physics

Wave Functions for Time-Dependent Dirac Equation under GUP*

Meng-Yao Zhang, Chao‐Yun Long, Zheng‐Wen Long

Published Apr 1, 2018• 1 citation
Publisher

Abstract

Abstract In this work, the time-dependent Dirac equation is investigated under generalized uncertainty principle (GUP) framework. It is possible to construct the exact solutions of Dirac equation when the time-dependent potentials satisfied the proper conditions. In (1+1) dimensions, the analytical wave functions of the Dirac equation under GUP have been obtained for the two kinds time-dependent potentials.

DOI
10.1088/0253-6102/69/4/383

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Abstract In this work, the time-dependent Dirac equation is investigated under generalized uncertainty principle (GUP) framework. It is possible to construct the exact solutions of Dirac equation when the time-dependent potentials satisfied the proper conditions. In (1+1) dimensions, the analytical wave functions of the Dirac equation under GUP have been obtained for the two kinds time-dependent potentials.
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In this work we construct two classes of exact solutions for the most general time-dependent Dirac Hamiltonian in 1+11+11+1 dimensions. Some problems regarding some formal solutions in the literature are discussed. Finally the possibility of existence of a generalized Lewis-Riesenfeld invariant connected with such solutions is discussed.
Solutions of the Schrödinger equation with an exact time dependence are derived as eigenfunctions of dynamical invariants which are constructed from time-independent operators using time-dependent unitary transformations. Exact solutions and a closed form expression for the corresponding time evolution operator are found for a wide range of time-dependent Hamiltonians in d dimensions, including non-Hermitean -symmetric Hamiltonians. Hamiltonians are constructed using time-dependent unitary spatial transformations comprising dilatations, translations and rotations and solutions are found in several forms: as eigenfunctions of a quadratic invariant, as coherent state eigenfunctions of boson operators, as plane wave solutions from which the general solution is obtained as an integral transform by means of the Fourier transform, and as distributional solutions for which the initial wavefunction is the Dirac δ-function. For the isotropic harmonic oscillator in d dimensions radial solutions are found which extend known results for d = 1, including Barut–Girardello and Perelomov coherent states (i.e., vector coherent states), which are shown to be related to eigenfunctions of the quadratic invariant by the ζ-transformation. This transformation, which leaves the Ermakov equation invariant, implements SU(1, 1) transformations on linear dynamical invariants. coherent states are derived also for the time-dependent linear potential. Exact solutions are found for Hamiltonians with electromagnetic interactions in which the time-dependent magnetic and electric fields are not necessarily spatially uniform. As an example, it is shown how to find exact solutions of the time-dependent Schrödinger equation for the Dirac magnetic monopole in the presence of time-dependent magnetic and electric fields of a specified form.
Abstract The generalized uncertainty principle (GUP) is a natural consequence of many proposed quantum gravity theories which further indicates the modification of the quantum mechanics and classical mechanics. A unified classical-quantum framework also leads to the interpretation of a Winger phase-space distribution of a state as a KvN wave function. In this article, we utilize the classical-quantum unified framework to calculate the expression for the GUP modified Wigner function of a classical harmonic oscillator in the configuration space instead of the conventional approach of utilising the momentum space. Further, we plot the modified Wigner functions and show that they are similar to what has been obtained before in the past literature.