# 12: Time-Dependent Perturbation Theory

- Page ID
- 15805

Consider a system whose Hamiltonian can be written \[H(t) = H_0 + H_1(t).\] Here, \(H_0\) is again a simple time-independent Hamiltonian whose eigenvalues and eigenstates are known exactly. However, \(H_1\) now represents a small time-dependent external perturbation. Let the eigenstates of \(H_0\) take the form \[H_0\,\psi_m = E_m\,\psi_m.\] We know (see Section [sstat]) that if the system is in one of these eigenstates then, in the absence of an 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 if the system is initially in some eigenstate \(\psi_n\) of the unperturbed Hamiltonian then it is found in some other eigenstate at a subsequent time (because \(\psi_n\) 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. Let us investigate such transitions.

- 12.7: Electric Dipole Approximation
- n general, the wavelength of the type of electromagnetic radiation that induces, or is emitted during, transitions between different atomic energy levels is much larger than the typical size of an atom.

- 12.8: Spontaneous Emission
- In the absence of any external radiation, we would not expect an atom in a given state to spontaneously jump into an state with a higher energy. On the other hand, it should be possible for such an atom to spontaneously jump into an state with a lower energy via the emission of a photon whose energy is equal to the difference between the energies of the initial and final states. This process is known as spontaneous emission.

## Contributors and Attributions

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

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