6 Perturbation Theory
N. Panchapakesan and J.D. Anand
- Learn what is electromagnetic interaction and its strength.
- Learn why perturbation theory can be applied for most of the processes – involving charged particles and radiation field.
- Learn how the quantum theory of radiation is applied to absorption and emission of light by electrons in atom using perturbation theory.
- Learn what is time dependent perturbation theory and the transition probability per unit time.
3.1 Schrodinger Equation for matter interacting with electromagnetic waves
where e is the charge of the electron. This gives us the following Schrodinger equation
We shall take as quantized field vector and scalar potential is taken in the electron Coulomb interaction term variables as it is not a quantized variable in Coulomb gauge. If N electrons of an atom participate in the interaction the total Hamiltonian H satisfies the equation
The first part of HO is the particle kinetic interaction of the electrons and the last term energy plus the instantaneous Coulomb is E.M. field energy.
3.2 Quantized Electromagnetic Field
We have already studied Quantization of the electromagnetic field using radiation or Coulomb gauge. As studied earlier the vector potential operator? (? , ?) in Heinsenberg picture is written as
Eq. 3.14 and 3.15 are the two most important results of quantized radiation theory.
4. Time dependent Perturbation Theory
As the total Hamiltonian H = H0+HI is time dependent, the energy of the system is not conserved and so there are no stationary states. In this case we shall apply the time dependent perturbation theory. We shall assume that the unperturbed Hamiltonian 0 is time independent and forms the major part of the Hamiltonian, and has stationary, orthogonal and complete set of states. Let 0 be the wave function of the Hamiltonian 0. The wave function carries the time dependence. This is like Schrodinger picture. We assume that the interaction Hamiltonian is small (a perturbation) compared to 0 and is time dependent and produces a change or mixing of the eigen wave functions of 0. This is Heisenberg picture for . This mixed picture for the whole Hamiltonian is called the Interaction picture. We shall seek the solution of the Schrodinger’s equation
4.1 Initial Condition and Expansion in powers of
Suppose now that perturbation H1 begins to act at time t=0, and at t = 0, when the perturbation has not acted yet, let only one unperturbed state ??0 be there.
and so on. Eq.(4.11) can be interpreted as transition in two stages. First from state i to state m and next from state m to state n . The state m is called the intermediate state.
Now in the final state we have a number of photon states lying in the range and + . So the transition probability to go to state n has to be summed over all the states available for transition. As the energy interval is small we assume that the matrix element is constant and multiply the transition probability eq. (4.17) by the number of states in the small interval of energy dE, that is by the density of states d = dE,
Thus the transition probability for emission of photons in the states ? and ? + ?? is given by
usefulness in the next module for absorption or decay of atom and for scattering processes..
- Summary
In this module we have studied the emission and absorption of photons by charged particles like electrons in the atoms. It is seen that by treating electrons non-relativistically and describing photons by quantized radiation field we obtain the transition probability per unit time by using the techniques of time dependent perturbation theory.
The application of the expressions of transition probabilities under for various processes like absorption , emission will be taken up in detail in the next few modules.
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Learn More
The following books are some among many.
- Advanced Quantum Mechanics by J.J. Sakurai (Pearson Education, Singapore 1998)
- Quantum Mechanics Vol. 3 by L.D. Landau and E.M. Lifshitz (Pergamon Press, Oxford, Reprinted 1981)
- Quantum Electrodynamics, Vol. 4 by V.B. Berestetskii, E.M. Lifshitz and L.P. Pitaevskii (Pergamon Press, Oxford, 1982)
Web Links
www.tcm.phy.cam.ac.uk/~bds10/aqp/lec18pdf
This is more about the application of time dependent perturbation theory to simple but important systems like two level systems.
Interesting Facts
Non relativistic perturbation theory in the form discussed here was started by P.A,M. Dirac in 1927 and was used by particle physicists for about 25 years. The relativistic theory with Feynman diagrams then took over . However the condensed matter theorists still use it in many cases.
The Golden Rule is taught in Quantum mechanics courses. Here we have given a short revision to enable us to apply it for the electromagnetic field. See the web link for other applications. The name Golden rule was given by Enrico Fermi in his course on quantum mechanics give at University of Chicago, USA.