I gave out a copy of two pages from Fermi's own handwritten 1954 lecture "Notes on Quantum Mechanics", which were published by the Unversity of Chicago Press in 1961 (see lecture 23, pages 99-100). There, he didn't call the final formula his golden rule #2, though he did elsewhere; see Liboff (L4 p717 or L3 p739 ). If you can find a copy of his lectures on EBay go for it - a collector's item, I'm sure. Any offers? I'm not selling, but I'd like to know what my 12 shillings and sixpence is worth now.
The books and supplemental web pages do a good job of explaining how one gets this result, usually in the form of a transition from state k to state m, and being more specific about the type of transition, e.g. an atomic transition which is dipole allowed. The most obvious example is 2p to 1s transition in atomic hydrogen, which we talked about in lectures. In that case, we can write the matrix element as
d(E) conserves energy as E = Ek0 - Em0 - hw (3.2) |
Various approximations to the A.p form of V(t) are given in the books, and in particular it is shown that the emission of a photon at angular frequency w can be expressed in terms of the polarization vector e and the wave vector k as
The "Dipole approximation" comes about by noting that the wavevector k is small, since the optical wavelength is much larger than the spatial extent of the atomic states. Hence we can eventually express the Dipole matrix element of V(t) as
The combination er is an electric dipole, and you can think about this matrix element as coupling states m and k, which have different parity, creating or absorbing a photon. Equation (3.4) above also contains the polarization vector e, so if you write z = r cos(q) and express z in terms of A and A+, you can incorporate all that you know about the simple harmonic oscillator into the same framework. As Mathematicians would say, Q.E.D.! (Don't forget the Einstein A and B coefficients and the (1 + n) factor that is outlined in the previous section on The role of radiation.
Since having tried (once) to get all this material across at the end of semester when everyone is busy with other things, I have offered this topic amongst others as projects. Time-dependent PT was the subject of a project in 2000-01 by Jason Harris, and is available on his website, complete with animations. A printed version is available as a handout.
One aspect of time-dependent PT that I particularly enjoy is the derivation (quite complicated) which shows that the amplitude of the upper state (e.g. the 2p state in hydrogen) decays exponentially in time, as exp(-t/t). The time constant t also a measure of the linewidth Dw of the spectral line, such that Dwt is of order 2p. Sound familiar? Of course, it is not only the Uncertainty Principle, but it also works out quantitatively. Details are spelt out towards the end of Jason's project, but the short form answer corresponds to the following argument. The Fourier transform of the decaying exponential (exp(-const|t|) for t > 0) is the Lorentzian lineshape 1/((Dw)2 + t -2), expressed as a function of either energy or frequency. Check this out on our FT pages. Full circle: a good way to end the Semester.