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Physical Review A

Matrix elements in the basis of hydrogenic wave functions using an integral transformation

Nazakat Ullah

Published Mar 1, 1992• 2 citations
Publisher

Abstract

An integral transformation is given which when applied to hydrogenic-type wave functions converts them into Gaussian form. This is particularly needed in the evaluation of the two-particle matrix elements of the interaction operator. Further examples of the use of the transformation are given by calculating the Wigner transform for the ground state of the H atom and matrix elements of the powers of the z coordinate.

DOI
10.1103/physreva.45.2807

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An integral transformation is given which when applied to hydrogenic-type wave functions converts them into Gaussian form. This is particularly needed in the evaluation of the two-particle matrix elements of the interaction operator. Further examples of the use of the transformation are given by calculating the Wigner transform for the ground state of the H atom and matrix elements of the powers of the z coordinate.
It is shown that the representation matrices of the quantum mechanical group in the coherent state basis can be used as bases for the expansion of arbitrary square integrable functions on the group. Although this basis is nonorthogonal the expansion can be inverted in a manner similar to an orthogonal basis. Next a coherent state ‘‘wave function’’ is defined in analogy with the Schrödinger wave function and the above matrix elements are used to determine the action of the group in the space of the coherent state wave functions. The function space in this realization is Bargmann’s Hilbert space of analytic functions. It is shown that the mixed basis matrix element between the coordinate and coherent state bases is essentially the integral kernel of Bargmann. The transpose of this matrix element yields the kernel for the inversion of the transform. In this construction the Bargmann transform appears as the unitary transformation connecting the Schrödinger and coherent state wave functions. This considerably simplifies Bargmann’s original arguments and yields the integral transform as well as its inversion formula in a simple way.
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.
The paper compares the methods used to calculate matrix elements of the operator of radail electron coordinates in an arbitrary order in the Foldy-Wouthuysen representation and with the use of the Dirac equation for 1s-states of the hydrogen-like and helium-like ions of transuranic elements. The obtained analytical and numerical results for 1s-states of the hydrogen-like and helium-like ions proves that the wave function reduction requirement is met with transformation to the Foldy-Wouthuysen representation and confirms that matrix elements can be calculated using only one component (either upper, or lower) of the Dirac bispinor wave function.
A prescription is given for obtaining the Green's function for N free particles which can have different masses. The approach is systematic and straightforward. A coordinate transformation of the Fourier integral representation of the N-particle noninteracting Green's function facilitates the integration over 3N-1 angular variables of wavenumber space. A single radial integral can then be evaluated. The resulting Green's function representation may be of use in applying the integral form of Schrödinger's equation to calculate the ground and excited states of atoms.