Symmetry adapted harmonic oscillator functions

Quantum mechanical excited energy states of two- or three-dimensional harmonic oscillators are highly degenerate. A general method is presented to calculate the symmetry adapted quantum mechanical oscillator functions with respect to the symmetry group of the oscillator. In the special case of the tetrahedral symmetry group the method is demonstrated by detailed formulas for the two- and three-dimensional case. A computer program is also available. The method can easily be modified for other symmetry groups.     

Multipart wave functions

Some wave functions separate into two or more distinct regions in phase space. Each region is characterized by a trajectory and a spread about that trajectory. The trajectory is the quantum mechanical current. We show that these regions correspond to parts of the wave function and that these parts are generally nonorthogonal.My interest in physics was inspired, as an undergraduate, by the remarkable books of Henry Margenau, particularlyFoundations of Physics andThe Mathematics of Physics and Chemistry. I never imagined then that I would have the good forture of being his student. It is a pleasure and an honor to dedicate this article to him as a small expression of my deep appreciation.     

Symmetry groups and spiral wave solution of a wave propagation equation

We study a third-order nonlinear evolution equation, which can be transformed to the modified KdV equation, using the Lie symmetry method. The Lie point symmetries and the one-dimensional optimal system of the symmetry algebras are determined. Those symmetries are some types of nonlocal symmetries or hidden symmetries of the modified KdV equation. The group-invariant solutions, particularly the travelling wave and spiral wave solutions, are discussed in detail, and a type of spiral wave solution which is smooth in the origin is obtained.     

Polarized deep inelastic scattering at high energies and parity violating structure functions

Zeitschrift für Physik C Particles and Fields - A comprehensive analysis of deep inelastic scattering of polarized charged leptons on polarized nucleons is presented; weak interaction...     

Hierarchical wave functions revisited

We study the hierarchical wave functions on a sphere and on a torus. We simplify some wave functions on a sphere or a torus using the analytic properties of wave functions. The open question, the construction of the wave function for quasielectron excitations on a torus, is also solved in this paper.     

Generalized Bethe-Faddeev wave functions

Bethe-Faddeev wave functions are generalized toN-body clusters. A comparison made between the Bethe approximation and the Day approximation in the three body cluster indicates that the two approximations should be competitive for the higher body clusters.     

Cusped gaussian wave functions

An orbital-fitting procedure has been used to fit the s- and p-orbitals expressed in terms of a cusped gaussian basis onto Clementi's accurate SCF orbitals. The results suggest that approximately double-zeta accuracy may be obtained for the s-orbitals with a basis of two 1s gaussians per orbital supplemented by a single 1s cusp function for the representation of the orbitals at and near the nucleus, and for the p-orbitals with a basis of three 2p gaussians per orbital plus the 2p cusp function. It is proposed that for the s-orbitals the cusped gaussian basis is superior to both the all-gaussian and Slater function bases for the purpose of molecular calculations. The results are less conclusive for the p-orbitals and, although the efficiency of the basis increases with increasing nuclear charge, further investigation is necessary before a definite conclusion may be reached as to the usefulness of the cusped gaussian basis for the representation of p-orbitals.     

Symmetry groups, density-matrix equations and covariant Wigner functions

A representation theory for Lie groups is developed taking the Hilbert space, say , of the w^*-algebra standard representation as the representation space. In this context the states describing physical systems are amplitude wave functions but closely connected with the notion of the density matrix. Then, based on symmetry properties, a general physical interpretation for the dual variables of thermal theories, in particular the thermofield dynamics (TFD) formalism, is introduced. The kinematic symmetries, Galilei and Poincaré, are studied and (density) amplitude matrix equations are derived for both of these cases. In the same context of group theory, the notion of phase space in quantum theory is analysed. Thus, in the non-relativistic situation, the concept of density amplitude is introduced, and as an example, a spin-half system is algebraically studied; Wigner function representations for the amplitude density matrices are derived and the connection of TFD and the usual Wigner-function methods are analysed. For the Poincaré symmetries the relativistic density matrix equations are studied for the scalar and spinorial fields. The relativistic phase space is built following the lines of the non-relativistic case. So, for the scalar field, the kinetic theory is introduced via the Klein–Gordon density-matrix equation, and a derivation of the Jüttiner distribution is presented as an example, thus making it possible to compare with the standard approaches. The analysis of the phase space for the Dirac field is carried out in connection with the dual spinor structure induced by the Dirac-field density-matrix equation, with the physical content relying on the symmetry groups. Gauge invariance is considered and, as a basic result, it is shown that the Heinz density operator (which has been used to develope a gauge covariant kinetic theory) is a particular solution for the (Klein–Gordon and Dirac) density-matrix equation.     

Completeness of pair wave functions

A rather general proof is given of orthonormality and completeness of pair wave functions. Because of the close relationship between RPA and the pair-theory method, the pattern of the proof also applies to RPA wave functions.     

Quantum mechanics without wave functions

The phase space formulation of quantum mechanics is based on the use of quasidistribution functions. This technique was pioneered by Wigner, whose distribution function is perhaps the most commonly used of the large variety that we find discussed in the literature. Here we address the question of how one can obtain distribution functions and hence do quantum mechanics without the use of wave functions.     

Probability space of wave functions

It is well known that in quantum mechanics, when the mean value of an observable is given, entropy maximization (von Neumann, Born, Jaynes) can be successfully applied for constructing a probability distribution on the set of possible values of that observable. In this paper, the entropy maximization technique is extended to the complex domain in order to construct an unbiased probability measure on the set of all wave functions. In particular, a justification and a generalization of the Wiener-Siegel probability distribution of Gaussian type in the differential space of wave functions are given.     

Quantum-mechanical modeling without wave functions

It has been shown that the variational principle can be used as the practical way to find the electron density and the total energy in terms of the density functional theory without solving the Kohn-Sham equations (the so-called orbital-free approach). The equilibrium interatomic distances and binding energies found using examples of diatomic systems Si₂, Al₂, and P₂ are in good agreement with the published data. The results obtained for Si-Al, Si-P, and Al-P dimers are also close to the results obtained by the Kohn-Sham method.     

Photoionisation using Kohn-Sham wave functions

The determination of photoionisation cross-sections using the Kohn-Sham wave functions from accurate density functional theory (DFT) calculations is considered. The continuum electrons wave function is specified by its momentum measurable in the detector rather than fixed angular momentum and energy. The method is applied to the test case of a water molecule, where the DFT calculations show excellent agreement to experiment in the ionisation energies if a vertical transition is considered. The continuum electron is described by an analytical wave function and the matrix elements are calculated in both in length and velocity form of the dipole operator. The cross-sections agree well with the experiment in particular for the velocity form.     

Supersymmetric ground state wave functions

The construction of explicit supersymmetric ground states is considered in a variety of quantum mechanical systems. For broad classes of supersymmetric hamiltonians it is not difficult to find closed-form zero-energy ground-state wave functions.