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Physics LibreTexts

8: Time-Dependent Perturbation Theory

  • Page ID
    1129
  • Suppose that the Hamiltonian of the system under consideration can be written

    \[H = H_0 + H_1(t) \label{739}\]

    where \(H_0\) does not contain time explicitly, and \(H_1\) is a small time-dependent perturbation. It is assumed that we are able to calculate the eigenkets of the unperturbed Hamiltonian:

    \[H_0 \,\vert n\rangle = E_n \,\vert n\rangle. \label{740}\]

    We know that if the system is in one of the eigenstates of \(H_0\) then, in the absence of the external perturbation, it remains in this state for ever. However, the presence of a small time-dependent perturbation can, in principle, give rise to a finite probability that a system initially in some eigenstate \(\vert i\rangle\) of the unperturbed Hamiltonian is found in some other eigenstate at a subsequent time (because \( \vert i\rangle\) is no longer an exact eigenstate of the total Hamiltonian). In other words, a time-dependent perturbation allows the system to make transitions between its unperturbed energy eigenstates.

    Contributors

    • Richard Fitzpatrick (Professor of Physics, The University of Texas at Austin)

      \( \newcommand {\ltapp} {\stackrel {_{\normalsize<}}{_{\normalsize \sim}}}\) \(\newcommand {\gtapp} {\stackrel {_{\normalsize>}}{_{\normalsize \sim}}}\) \(\newcommand {\btau}{\mbox{\boldmath$\tau$}}\) \(\newcommand {\bmu}{\mbox{\boldmath$\mu$}}\) \(\newcommand {\bsigma}{\mbox{\boldmath$\sigma$}}\) \(\newcommand {\bOmega}{\mbox{\boldmath$\Omega$}}\) \(\newcommand {\bomega}{\mbox{\boldmath$\omega$}}\) \(\newcommand {\bepsilon}{\mbox{\boldmath$\epsilon$}}\)