What is it about?
(Uploaded 2018-04-18). Dalgarno and Lewis has given a "trick" by which the familiar sum over intermediate states going with second order perturbation theory can be worked out analytically in certain special cases. The most simple is that where the perturbation V can be written as the commutator [F, Ho] where Ho is the zeroth order Hamiltonian and F a certain operator. In general it is sufficient that the part of V which contributes to second order, can be written that way. Harris refers this "trick" to Sciff's classic book "Quantum Mechanics" [3.ed. p.266. NOT 2. ed. as by a mistake asserted in the present paper]. An important case - not observed in Schiff's book- is at hand if the zeroth order Hamiltonian is a sum of harmonic oscillators and the perturbation is a polynomial in the coordinates and the conj. momenta. Besides using this observation for further simplification of Harris' calculation, I briefly discuss how the method is related to the so-called contact transformation which, since its introduction in 1939, has been a standard method within vibration-rotation spectroscopy for molecules: One can choose F to be antihermitian and use it to subject the Hamiltonian to the unitary transformation exp(F). Thereby the perturbation takes the form of an expansion where the term of first order gives no energy correction in second order.
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Why is it important?
When a certain theoretical method arises in different forms in different fields of research, it is important/satisfactory to have the relationship established. Dalgarno and Lewis introduced their method to eliminate the sum over intermediate states. With the contact transformation method this sum does not appear at all - but the two methods are none the less two sides of the same. The contact transformation method has been build on the combination of intelligent guesses for the operator F. However, Jørgensen, Pedersen and Chedin (Mol. Phys., 30, 1377 (1975)) have described how it is related to the so-called Van Vleck transformation method in which an operator F always can be determined in a systematic way. This general way, however, gives F in terms of a sum over intermediate states. In Harris' cranking model calculation one finds how the (Coriolis-) interaction between rotation and vibration affects the rotational constants going with a given vibrational state in a nucleus. Not surprisingly this is also a central application for the contact transformation within molecular vibration-rotation spectroscopy. Other applications within this last subject have to do with centrifugal distortion and the effect of anharmonicities in the potentials.
Perspectives
Very few textbook show how the sum over intermediate states in second order perturbation theory can be eliminated by Dalgarno and Lewis's method - and when they do, they use the same example as used originally by Dalgarno and Lewis: The influence on the ground state of the hydrogen atom of the electric field from a nearby proton. This is indeed an example where the method shows its strength. But it does not seem so well suited for the more elementary textbooks on quantum mechanics. However, since practically all such books consider the harmonic oscillator as an example, it would probably be very helpful to consider instead the influence of anharmonicities on the energy levels. Once the idea is understood, it is straightforward to pass on to the other applications within vibration-rotation spectroscopy.
Dr Flemming Jørgensen
Nygårdsvej 43, 4700 Næstved http://www.naestved-gym.dk/
Read the Original
This page is a summary of: Note on the Harris cranking model calculation of the moment of inertia using the method of Dalgarno and Lewis, American Journal of Physics, February 1979, American Association of Physics Teachers (AAPT),
DOI: 10.1119/1.11872.
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