12: Time-Dependent Perturbation Theory
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Consider a system whose Hamiltonian can be written H(t)=H0+H1(t). Here, H0 is again a simple time-independent Hamiltonian whose eigenvalues and eigenstates are known exactly. However, H1 now represents a small time-dependent external perturbation. Let the eigenstates of H0 take the form H0ψm=Emψm. 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 ψn of the unperturbed Hamiltonian then it is found in some other eigenstate at a subsequent time (because ψn 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)
