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International Journal of Quantum Chemistry

On formulas in closed form for Van Vleck expansions

Edwin K. Gora

Published Jul 1, 1972• 8 citations

Abstract

Abstract Canonical transformations have been widely used to simplify Hamiltonians and other operators. In molecular and in solid state theory, the so‐called Van Vieck expansion is usually employed for this purpose while in theories of particles interacting with fields a combination of canonical transformations in closed form with Van Vleck type expansions has been found effective. For some of the transformations used in applications formulas in closed form are well known. It will be shown here that such formulas can be derived whenever the transformation function is bilinear in the canonical variables, and further that the use of matrix operators makes it possible to simplify these derivations substantially. The Cayley‐Hamilton theorem is then used to express the expansions for the matrix operators in closed form. The number of separate operator terms appearing in the formulas thus obtained is the same as the rank of the matrices used. To calculate the coefficients of these operator terms a new type of special functions is introduced. The resulting linear canonical transformations include generalized rotations in both ordinary and phase‐space. Explicit results have been obtained for several two‐ to four‐dimensional problems.

DOI: 10.1002/qua.560060412

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Abstract Canonical transformations have been widely used to simplify Hamiltonians and other operators. In molecular and in solid state theory, the so‐called Van Vieck expansion is usually employed for this purpose while in theories of particles interacting with fields a combination of canonical transformations in closed form with Van Vleck type expansions has been found effective. For some of the transformations used in applications formulas in closed form are well known. It will be shown here that such formulas can be derived whenever the transformation function is bilinear in the canonical variables, and further that the use of matrix operators makes it possible to simplify these derivations substantially. The Cayley‐Hamilton theorem is then used to express the expansions for the matrix operators in closed form. The number of separate operator terms appearing in the formulas thus obtained is the same as the rank of the matrices used. To calculate the coefficients of these operator terms a new type of special functions is introduced. The resulting linear canonical transformations include generalized rotations in both ordinary and phase‐space. Explicit results have been obtained for several two‐ to four‐dimensional problems.

A general N-dimensional linear transformation is written, wherein each original variable xi, pi (i=1, ···, N) is expressed in closed form as a linear function of the 2N transformed variables xi′, pi′ A set of 2N2-N relationships between the linear coefficients is then derived for the transformation to be canonical. This procedure provides one with the most general form of N-dimensional linear canonical quantum mechanical transformation, while giving the operators in closed form. The procedure is compared with the alternate method of using unitary generators, and its advantages over that method are discussed.

The transformations of the type which convert an exponential into a Gaussian and vice-versa and their applications in various areas of many-body physics are discussed. A new and general method of obtaining such transformations is given using the method of moments. It is compared with other methods which could be employed to obtain such transformations. In atomic physics, we have shown how such transformations can be used to obtain electron interaction energy for the ground state of Helium and Wigner transform for the ground state of H atom. It is shown how to bring angular momentum operators to linear form so that one can use the usual property of rotation operator to calculate their matrix elements. A new way of calculating the approximate eigenvalues of a Hamiltonian is given which combines the variational principles with the principle of maximum entropy. The anharmonic oscillator Hamiltonian is used to illustrate this new method. An interesting aspect of these transformations is that one could combine them with other transformations like Grassmann integration to calculate quantities of physical interest in closed form. A general matrix element of the harmonic oscillator is given which can be used to calculate usual quantities like the trace and density matrix. Some future applications are also discussed.

Expansion formulas for density matrices are derived with the use of the calculus of ordered exponentials. Using the formulas the expressions for virial coefficients are explicitly obtained. The expression of the third virial coefficient is calculated to obtain its expansion formula in terms of

\hbar^{2}\beta/M

. The expansion formulas for density matrices can be used to derive a new formula of multiple scattering, which involves Luttinger and Kohn's formula as a special case. The expansion formula for the normalized density matrix is also given. Expansion formulas which are to the applied to irreversible processes and relaxation phenomena are also given.

Abstract We explore the relation of Van Vleck-Primas perturbation theory of quantum mechanics with the Lie-series-based perturbation theory of Hamiltonian systems in classical mechanics. In contrast to previous works on the relation of quantum and classical perturbation theories, our approach is not based on the conceptual similarities between the two methods. Instead, we show that for quantum systems with a finite-dimensional Hilbert space, the Van Vleck-Primas procedure can be recast exactly into a classical perturbation problem. As a non-obvious consequence, this approach gives a new way of calculating the geometric phase of quantum systems using tools from the theory of classical canonical transformations.

The form of the complex Hamiltonian evolution equations (linear and nonlinear) is deduced here from that of the classical Hamiltonian systems. Their properties and those of the associated continuous Poisson bracket are studied. It is shown that if any space-localized solution of the above type equations is expanded in any Hilbert space basis the expansion coefficients are canonically conjugate variables and obey Hamiltonian systems of complex equations. The last is proved also for sets of “canonically conjugate” functionals of general type. As an application, commutation relations which are equivalent in form to Dirac’s quantization rules (the constant in place of ℏ is undetermined) are obtained. The question of how a nonlinear field theory may contain quantum mechanics as a special case is discussed in view of this paper’s results.

A size-extensive multidimensional many-body theory is developed from an order-expanded van Vleck transformation. This provides an effective Hamiltonian in a model space consisting of a set of determinants whose zeroth-order energies may be nondegenerate. Expressions for the effective Hamiltonian in terms of the perturbation and a set of resolvents generalized from the Rayleigh–Schrödinger form are given. Perturbative evaluation of the resultant formulas via diagrammatic expansion is illustrated and discussed. The diagrams required through second order for a model space consisting of a Hartree–Fock solution plus selected singly and doubly excited determinants are presented, and their relation to those employed in the method of Hose and Kaldor is discussed.