What is it about?
(Uploaded: 2018-04-18). The energy levels for a vibrating-rotating molecule are found by expand-ing the Hamiltonian H in the normal coordinates so that it gets the form Ho+V where Ho consists of independent harmonic oscillators and V is a sum of cubic, quartic, ... anharmonicites and rotational terms. The various contributions to the energies can then be extracted from the perturbation taken into account through second order of V, considered as a perturbation. The familiar sum over intermediate states appearing in second order can - with some effort - be worked out analytically because of a finite number of terms in the sum . About 1939 the so-called contact transformation (CT) was invented to avoid this sum. Here one first writes V as v+w where w collects the terms which contributes to second order. Thereafter the Hamiltonian is subjected to a unitary transformation exp(F) where F is an antihermitian operator (found by some "intelligent guesses") so that [F, Ho] = w. Then the perturbation in the transformed Hamiltonian H'=Ho+V' gives no contribution in second order of V'. So, ignoring the subsequent orders, CT works without the appearance of intermediate states . With Van Vleck Transformation (VVT) as considered in the foregoing Part I and Part II, in-creased accuracy can be obtained by use of successive transformations ... exp(F') exp(F). But the sum over intermediate states is not avoided. However: There is a certain arbitrariness of the anti-hermitian F, F',.. . We show how to fix this so CT appears as a special case of VVT.
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Why is it important?
The original theory for CT looks as something invented independently of the traditional perturba-tion theory with its sum over immediate states. And it furthermore looks rather complicate because it has been introduced in direct connection to the complicate vibration-rotation problems it is meant for. We have considered the theory free from any special application and found the connection between CT and an appropriately adjusted version of VVT by writing the normal coordinates and their conjugate momenta in terms creation and annihilation operators. Thereby the perturbation can be written as a sum of terms, each of which changes a given energy level by a certain fixed amount. Hence the term can only couple a given level with ONE other. This eliminates the need for a sum over intermediate states. We think that this understanding of the structure of the problem is of importance by itself. It is always interesting when seemingly different approaches can be revealed as being too sides of the same thing.
Perspectives
A systematic description of VVT and CT through higher orders is probably too specialized a subject for textbooks on quantum mechanics in general. Second order problems, however, are so important that they are described in practical all such books. And the fundamental idea described above seems both simple and useful: Write the perturbation V as v+w and see if an anti-hermitian solution to [Ho, F] = w can be found without using the intermediate states. When that is so, this sum can be avoided in the second order corrections to the energy. Actually Dalgarno and Lewis have described the little "trick" long ago - and Schiff has included it in his classic text book Quantum Mechanics ( 3rd. ed, McGraw-Hill, 1968, p. 266) . However, the example chosen there is about the effect of an electric field on the ground state of hydrogen. Harris has shown how to use the trick on a vibration-rotation problem known from nuclear physics - but none of these authors mention the connection to CT and VVT. I have described it in "Note on the Harris cranking model calculation of the moment of inertia using the method of Dal-garno and Lewis" Am. J. Phys. 1979, p. 193 (DOI: 10.1119/1.11872) . I feel that this brief simple article contains the essence of the methods in question.
Dr Flemming Jørgensen
Nygårdsvej 43, 4700 Næstved http://www.naestved-gym.dk/
Read the Original
This page is a summary of: A projector formulation for the Van Vleck transformation, Molecular Physics, November 1975, Taylor & Francis,
DOI: 10.1080/00268977500102911.
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