Using Baker-Campbell-Hausdorff formulae for the exponentials of the generators of the SU 2 and SU(1,1) groups, the matrix elements of spherical and hyperbolic rotations are calculated in the angular momentum basis.
Abstract This chapter presents orbital angular momentum as the generator or rotations in space. The operators which carry out rotations in space form a group, which is a symmetry group of the Hamiltonian when the interaction is spherically symmetric. In such a case, there are certain useful statements which we can make about the role of invariant and irreducible subspaces of the Hilbert space under the group operations and how to calculate matrix elements of tensor operators within such spaces, that is, the Wigner-Eckart theorem.
Semiclassical angular momentum matrices are calculated using a new contour-integral formula for matrix-elements. WKB spherical harmonic functions are found to be exaclty orthonormal with the contour-integral inner-product. Matrix-elements obtained from these wave-functions are accurate to about 2 % and the matrices obey the commutation relations expected of quantum angular momentum operators. The semiclassical wave-functions are related to a superposition of allowed classical orbits and this illustrates the connection to the Feynman pathsummation formulation of quantum mechanics for electrons in a spherically symmetric potential.
It is shown that a SU(1,1) algebra may be used to provide a unified description of the simple hamonic oscillator and the angular momentum algebras and a class of other semi-infinite algebras. A normal ordered representation of a Unitary operator U constructed from the generators of a SU(1,1) algebra, which is a generalisation of the Baker- Campbell - Hausdorff relation for Lie algebras, is given. It is shown that the normal ordered representatiion of U may be used to calculate expectation values which are functions of the parameters used to construct the operator. The functions so constructed satisfy certain recurrence relations and the entire set of functions may be interpreted in terms of diagrams similar to the Pascal triangle for binomial coefficients. Coherent states, squeezed states and rotation matrices of the angular momentum algebra emerge as special cases.
The Baker--Campbell--Hausdorff (BCH) formula plays a critical role in many branches of mathematics and physics. It expresses the logarithm of the product of exponentials of non-commuting operators as an infinite series of nested commutators of the operators involved. The Zassenhaus formula is the dual of the BCH formula: the exponential of a sum of operators is written as an infinite product of exponentials involving the operators and their commutators. In practical computations, however, one typically has to truncate the expansions, and so understanding the error committed by the resulting approximations and eventually providing suitable bounds for this error is of paramount interest. In this work we present a general strategy to derive rigorous error bounds and explicit error constants for the BCH and Zassenhaus formulas when the operators involved are skew-adjoint, as is the case for quantum evolution problems.