Jens Peder Dahl

The Morse oscillator in position space, momentum space, and phase space

We present a unified description of the position‐space wave functions, the momentum‐space wave functions, and the phase‐space Wigner functions for the bound states of a Morse oscillator. By comparing with the functions for the harmonic oscillator the effects of anharmonicity are visualized. Analytical expressions for the wave functions and the phase space functions are given, and it is demonstrated how a numerical problem arising from the summation of an alternating series in evaluating Laguerre functions can be circumvented. The method is applicable also for other problems where Laguerre functions are to be calculated. The wave and phase space functions are displayed in a series of curves and contour diagrams. An Appendix discusses the calculation of the modified Bessel functions of real, positive argument and complex order, which is required for calculating the phase space functions for the Morse oscillator.

Wigner's phase space function and atomic structure

We have constructed the Wigner function for the ground state of the hydrogen atom and analysed its variation over phase space. By means of the Weyl correspondence between operators and phase space functions we have then studied the description of angular momentum and resolved a dilemma in the comparison with early quantum mechanics. Finally we have discussed the introduction of local energy densities in coordinate space and demonstrated the validity of a local virial theorem.

On the Group of Translations and Inversions of Phase Space and the Wigner Functions

Grossmann and Royer have recently shown that the Wigner functions are closely related to the set of all translations and inversions of phase space. This allows the phase space representation of quantum mechanics to be constructed directly on the group of phase space translations and inversions. Starting from this observation, we have derived analytical expressions for the matrix elements of the translation and inversion operators, in the harmonic oscillator representation, without introducing coordinate or momentum wavefunctions.

The Wigner function

The basic relations of the Weyl-Wigner representation of quantum mechanics are reviewed. It is stressed that this representation is unique and based on a phase-space operator which corresponds to an observable in Diracs sense.

A new interpretation of Hund's first rule

The diagonal elements of the first and second order spinless density matrices have been calculated for the lowest excited1P and3P terms of Be, B+ and C++ using wavefunctions at different levels of approximations published in the literature. The analysis of these functions has resulted in a new interpretation of Hunds first rule in terms of an anisotropic screening effect.

Introduction to the quantum world of atoms and molecules

The rise of atomic theory the birth of quantum mechanics wave mechanics particle in a box quantum-mechanical operators the free particle the harmonic oscillator the central-field problem the hydrogen atom the spinning electron the periodic table by electron counting the variational method diatomic molecules vibration and rotation of diatomic molecules atomic term symbols atomic terms, wavefunctions and energies electronic terms of diatomic molecules the Hartree-Fock method density-functional theory.

Physical origin of the Runge - Lenz vector

The dynamical symmetry of the non-relativistic Kepler problem has attracted much attention in the scientific literature. In the present paper, we show that the Runge - Lenz vector, which accounts for the presence of this symmetry, has its physical origin in the generator of Lorentz transformations for the relativistic two-body problem. We reach this conclusion by considering the relativistic two-body problem, for the electromagnetic as well as for the gravitational interaction, in the approximation.

Dynamical Equations for the Wigner Functions

The motion of a classical particle is conveniently described as taking place in phase space, i.e., the direct product of configuration space and momentum space. The wavefunction associated with a quantum mechanical particle is, on the other hand, a quantity in configuration space or momentum space, and phase space might only seem to be of interest in cases where the uncertainty relation between position and momentum may be ignored.

Erratum: On the dirac-kepler problem. The johnson-lippmann operator, supersymmetry, and normal-mode representations

The relativistic Kepler problem is discussed, with emphasis on the exact supersymmetry of the problem. It is shown that the supersymmetry is generated by the Johnson-Lippmann operator. Two related operators are found to generate new supersymmetries in an extended function space. Each of these supersymmetries may be disguised as radial supersymmetries. The radial supersymmetries are discussed and it is shown that each of them defines a normal-mode representation of the hydrogen-atom radial functions. Thus, one obtains two different, but equivalent, analytical expressions for these functions. The expressions are well known, but are rederived here in the light of the new understanding. Finally, the nonrelativistic image of the relativistic supersymmetry is constructed and its generators shown to be identical with those recently presented in the literature. 24 refs., 1 fig.

On exact and approximate exchange-energy densities

Based on correspondence rules between quantum-mechanical operators and classical functions in phase space we construct exchange-energy densities in position space. Whereas these are not unique but depend on the chosen correspondence rule, the exchange potential is unique. We calculate this exchange-energy density for 15 closed-shell atoms, and compare it with kinetic- and Coulomb-energy densities. It is found that it has a dominating local-density character, but electron-shell effects are recognizable. The approximate exchange-energy functionals that have been proposed so far are found to account only poorly for the observed behaviors. Instead we use our results in proposing an alternative functional that depends on both first- and second-order derivatives of the electron density.