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Transactions of Beijing Institute of Technology

Time-Dependent Perturbation Theory in Quantum Phase Space

Qian‐Shu Li

Published Jan 1, 2009• 0 citations
OpenAlex

Abstract

Time-dependent perturbation theory is discussed within the quantum phase space representation as formulated by Torres-Vega and Frederick(T-F) and the quantum transition probability is studied.Time-dependent perturbation formula in T-F quantum phase space is established,in accordance with the calculation theory in T-F phase space.As an example,the one dimensional harmonic oscillator system,affected by an external electric field,is used to calculate in T-F phase space,the result indicate that the principle of the method is correct.Quantum transition probability of the system's first excited state in T-F phase space is discussed.

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Time-dependent perturbation theory is discussed within the quantum phase space representation as formulated by Torres-Vega and Frederick(T-F) and the quantum transition probability is studied.Time-dependent perturbation formula in T-F quantum phase space is established,in accordance with the calculation theory in T-F phase space.As an example,the one dimensional harmonic oscillator system,affected by an external electric field,is used to calculate in T-F phase space,the result indicate that the principle of the method is correct.Quantum transition probability of the system's first excited state in T-F phase space is discussed.
The perturbation quantum system was discussed within the framework of quantum phase space representation formulated by Torres-Vega and Frederick(TF) and studied the nondegenerate perturbation theory.The energy and the wave functions of the system in quantum phase space representation has been obtained;the one dimensional anharmonic oscillator system was used for an example to analyses the behaviour of quantum system in quantum phase space representation.The actual physical system in quantum phase space representation by using the approximate methods has been discussed.
The non-degeneracy perturbation theory was investigated in the Torres-Vega and Frederick(T-F) phase space.On the basis of one perturbation,the second and third perturbation formulas of energies and function were deduced.Contrast the probability density of coordinate and momentum with phase space by harmonic oscillator,show that the quantum phase space can get more information than coordinate representation's or momentum representation's when study the perturbation of system in quantum phase space.
In this paper, we solve the rigorous solutions of the stationary Schrödinger equations for the harmonic oscillator within the framework of the quantum phase-space representation established by Torres-Vega and Frederick. We obtain the phase-space eigenfunctions of the three-dimensional isotropic harmonic oscillator in rectangular and spherical coordinate systems, and discuss the relationship between the eigenfunctions of the two coordinate systems. We also extend the "Fourier-like" projection transformations of the phase-space wave function to three-dimensional systems, and obtain the eigenfunctions in position and momentum representation through the transformations of the eigenfunction in the phase-space representation.
This chapter presents approximation methods in quantum mechanics, focusing on both time-independent and time-dependent perturbation theory. The time-independent formulation is developed up to second-order corrections for both nondegenerate and degenerate systems. Applications include the analysis of the Zeeman and Stark effects, where energy level shifts due to magnetic and electric fields are calculated. The time-dependent perturbation theory is introduced, with harmonic perturbations used to derive the first-order transition probabilities and to establish Fermi's golden rule. As an application, the ionization probability of a hydrogen atom under a linearly polarized electromagnetic wave is computed in detail. The treatment is extended to include the case of an unpolarized electromagnetic field. Metadata/Keywords: perturbation theory, Zeeman effect, Stark effect, time-dependent perturbation, harmonic perturbation, Fermi's golden rule, hydrogen ionization
If the perturbation is turned on for a brief period of time and then turned off, the particle ends up in the original potential. But it may have been induced to transition from one eigenstate to another during the perturbation. The calculation of the probability of that transition is the topic of time-dependent perturbation theory (TDPT). Again, formula are presented rather than derived, and two examples are provided: a linear perturbation, and an oscillating perturbation. These examples, respectively, show how selection rules arise, and how light of a given frequency excites energy transitions of the same frequency. This latter phenomenon is called resonance .