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Quantum Mechanics: Time-Dependent Perturbation Theory

By Dragica Vasileska1, Gerhard Klimeck2

1. Arizona State University 2. Purdue University

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Abstract

Time-dependent perturbation theory, developed by Paul Dirac, studies the effect of a time-dependent perturbation V(t) applied to a time-independent Hamiltonian H0. Since the perturbed Hamiltonian is time-dependent, so are its energy levels and eigenstates. Therefore, the goals of time-dependent perturbation theory are slightly different from time-independent perturbation theory. We are interested in the following quantities: (1) The time-dependent expected value of some observable A, for a given initial state. (2)The time-dependent amplitudes of those quantum states that are energy eigenkets (eigenvectors) in the unperturbed system.

The first quantity is important because it gives rise to the classical result of an A measurement performed on a macroscopic number of copies of the perturbed system. For example, we could take A to be the displacement in the x-direction of the electron in a hydrogen atom, in which case the expected value, when multiplied by an appropriate coefficient, gives the time-dependent electrical polarization of a hydrogen gas. With an appropriate choice of perturbation (i.e. an oscillating electric potential), this allows us to calculate the AC permittivity of the gas.

The second quantity looks at the time-dependent probability of occupation for each eigenstate. This is particularly useful in laser physics, where one is interested in the populations of different atomic states in a gas when a time-dependent electric field is applied. These probabilities are also useful for calculating the "quantum broadening" of spectral lines (see line broadening).

To better understand time-dependent perturbation theory and Fermi's Golden rule, below we provide reading material, slides and homework assignments.

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Researchers should cite this work as follows:

  • www.eas.asu.edu/~vasilesk
  • Dragica Vasileska; Gerhard Klimeck (2008), "Quantum Mechanics: Time-Dependent Perturbation Theory," http://nanohub.org/resources/5021.

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In This Series

  1. Bulk Monte Carlo Lab

    27 Apr 2008 | Tools | Contributor(s): Dragica Vasileska, Mark Lundstrom, Stephen M. Goodnick, Gerhard Klimeck

    This tool calculates the bulk values of the carrier drift velocity and average electron energy in any material in which the conduction band is represented by a three valley model. Examples include Si, Ge and GaAs.

  2. Slides: Time-Dependent Perturbation Theory

    10 Jul 2008 | Teaching Materials | Contributor(s): Dragica Vasileska, David K. Ferry

    www.eas.asu.edu/~vasileskNSF

  3. Reading Material: Time-Dependent Perturbation Theory

    10 Jul 2008 | Teaching Materials | Contributor(s): Dragica Vasileska

    www.eas.asu.edu/~vasileskNSF

  4. Time-Dependent Perturbation Theory: an Exercise

    10 Jul 2008 | Teaching Materials | Contributor(s): Dragica Vasileska, Gerhard Klimeck

    www.eas.asu.edu/~vasilesk NSF

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