12: Time-Dependent Perturbation Theory

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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.

12.1: Preliminary Analysis
12.2: Two-State System
12.3: Spin Magnetic Resonance
12.4: Perturbation Expansion
12.5: Harmonic Perturbation
12.6: Electromagnetic Radiation
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.
12.9: Radiation from Harmonic Oscillator
12.10: Selection Rules (Hydrogen Atoms)
12.11: 2P-1S Transitions in Hydrogen
12.12: Intensity Rules
12.13: Forbidden Transitions
12.E: Time-Dependent Perturbation Theory (Exercises)

Contributors and Attributions

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

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