The Van Vleck Transformation: Principle vectors and geometric visualization

What is it about?

(Uploaded 2018-04-18). The canonical Van Vleck Transformation is discussed as the unitary factor in the Polar Decomposition of a very simple operator S which obeys previously unrecognized commutation relations. These simplify algebraic manipulation substantially and give direct connections between Van Vleck's exponential form of the transformation to that found by Bloch and Soliverez who pioneered the geometric approach. The article builds on a relatively recent paper by S. Kvaal (Phys. Rev. C. 2008, 78, 044330) who has shown how the canonical VVT can be considered as a product of commuting rotations in 2-dimensional planes, each rotation transferring a principal vector from the zeroth order space to its partner which is the nearest unit vector in the subspace of the perturbed states. Visualizing these two spaces of perturbed and unperturbed states as 2-dimensional planes in a space of dimension 3, the transformation is a rotation around the line of intersection - as well as a reflection in the plane that intersect the angle between them. In the general case this angle is just replaced with an operator. Various other operators are expressed in terms of the trigonometric functions of this angle-operator.

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Why is it important?

Among the various possibilities for transformation, the canonical VVT seems to be the one with all the attractive algebraic and geometric properties. The algebraic include that the exponential form lends itself to use of the Baker-Campbell-Haussdorf formula. And Kvaal has shown how use of a standard program for SVD can give the matrices for the most important canonical operators once we know a matrix which diagonalizes the perturbed hamiltonian within an (appropriately truncated) basis. The geometric properties include some striking visualizations. In certain review-like sections I compare the new formulations with some previous.

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