SchrÖdinger Wave Functional in Quantum Yang–Mills Theory from Precanonical Quantization

We address the issue of the relation between the canonical functional Schr\"odinger representation in quantum field theory and the approach of precanonical field quantization proposed by the author, which requires neither a distinguished time variable nor infinite-dimensional spaces of field configurations. We argue that the standard functional derivative Schr\"odinger equation can be derived from the precanonical Dirac-like covariant generalization of the Schr\"odinger equation under the formal limiting transition $\gamma^0\varkappa\rightarrow\delta(\mathbf{0})$, where the constant $\varkappa$ naturally appears within the precanonical quantization as the inverse of a small "elementary volume" of space. We obtain a formal explicit expression of the Schr\"odinger wave functional as a continuous product of the Dirac algebra valued precanonical wave functions, which are defined on the finite-dimensional covariant configuration space of the field variables and space-time variables.

Yang-Mills fields in the temporal gauge are transformed into pure gauge variables and gauge-invariant first integrals of Gauss's-law differential equations. It is observed that the Schr\"odinger equation becomes separable in the new variable system.

A variational theory called free-complement (FC) ${s}_{ij}$ theory for solving the Schr\odinger equation of atoms and molecules using only one- and two-electron integrals over Slater or Gaussian functions is proposed. It is derived from the scaled Schr\odinger equation [Phys Rev. Lett. 93, 030403 (2004)] by replacing the two-electron part ${r}_{ij}$ of the scaling $g$ operator with ${s}_{ij}={r}_{ij}^{2}$, both of which solve the divergence difficulty inherent to the original Schr\odinger equation by avoiding the interelectron collisions. The ${s}_{ij}$ function can be rewritten with only one-electron functions so that when the initial wave function of the FC theory is composed of only one-electron functions, the FC wave function including ${s}_{ij}$ can be rewritten with only one-electron functions. Therefore, the variational calculations of the FC ${s}_{ij}$ theory can be performed with only one- and two-electron integrals. However, in comparison with the ${r}_{ij}$ function, the ${s}_{ij}$ function behaves less efficiently when two electrons come close to each other: the electron-electron cusp condition is not satisfied with the FC ${s}_{ij}$ theory, though the wave function has explicit ${r}_{ij}$ dependence. On the other hand, the electron-nuclear cusp condition is satisfied, even with the Gaussian functions, for the presence of the electron-nuclear function ${r}_{iA}$ in the $g$ operator. Test applications of the FC ${s}_{ij}$ theory were done to He, Li, and the $^{5}S^{\mathrm{o}} {\mathit{sp}}^{3}$ state of carbon atom and to hydrogen molecule using the local-molecular orbital (MO)- and valence bond (VB)-type initial functions. We examined both Slater and Gaussian functions. As the order of the FC theory increased, the wave functions were improved and the energies approached from a few kcal/mol to less than 1 kcal/mol accuracies relative to the known exact values. Thus, with the FC ${s}_{ij}$ theory, the Schr\odinger equation was solved to less than 1 kcal/mol accuracy for all the systems examined. The costs for the ${s}_{ij}$ calculations were much smaller than those of the ${r}_{ij}$ theory. However, for the $^{5}S^{\mathrm{o}} {\mathit{sp}}^{3}$ state of the carbon atom, we observed that the convergence rate became slow when the calculation came close to $\ensuremath{\sim}1$ kcal/mol accuracy. This suggests that for the FC ${s}_{ij}$ theory, the calculations would become easier if the required accuracy is within a few kcal/mol for the absolute energy. Actually, what we need in chemical studies are mostly related to the difference energy, whose accuracy could be better than that of the absolute energy with the robust variational method. Two lines of future studies were suggested.

The article contains a review and new results of some mathematical models relevant to the interpretation of quantum mechanics and emulating well-known quantum gauge theories, such as scalar electrodynamics (Klein–Gordon–Maxwell electrodynamics), spinor electrodynamics (Dirac–Maxwell electrodynamics), etc. In these models, evolution is typically described by modified Maxwell equations. In the case of scalar electrodynamics, the scalar complex wave function can be made real by a gauge transformation, the wave function can be algebraically eliminated from the equations of scalar electrodynamics, and the resulting modified Maxwell equations describe the independent evolution of the electromagnetic field. Similar results were obtained for spinor electrodynamics. Three out of four components of the Dirac spinor can be algebraically eliminated from the Dirac equation, and the remaining component can be made real by a gauge transformation. A similar result was obtained for the Dirac equation in the Yang–Mills field. As quantum gauge theories play a central role in modern physics, the approach of this article may be sufficiently general. One-particle wave functions can be modeled as plasma-like collections of a large number of particles and antiparticles. This seems to enable the simulation of quantum phase-space distribution functions, such as the Wigner distribution function, which are not necessarily non-negative.

Doubly excited ridge states of atoms are states in which two electrons reach high and comparable excitations. Our goal is to calculate sequences for some of these states by using a wave function that treats the pair of electrons as a single entity in solving the two-electron Schr\"odinger equation in hyperspherical coordinates: R, \ensuremath{\alpha}, ${\mathrm{\ensuremath{\theta}}}_{12}$. For double escape of slow electrons, the so-called Wannier theory predicts that the wave function will be concentrated in the region \ensuremath{\alpha}=\ensuremath{\pi}/4, ${\mathrm{\ensuremath{\theta}}}_{12}$=\ensuremath{\pi} (that is, ${\mathbf{r}}_{1}$=-${\mathbf{r}}_{2}$) and we expect something similar for the ridge states. By expanding the Schr\"odinger equation around these points and retaining the first nontrivial quadratic dependences in \ensuremath{\alpha} and ${\mathrm{\ensuremath{\theta}}}_{12}$, we seek a solution in which the form of the wave function in these two variables is analytically determined as in the Wannier theory. The R dependence of this wave function is then handled numerically and differs from the double-escape solution only in the exponential decay appropriate for bound states. For L=0 a polynomial in R is included, giving the proper number of nodes for the solution. Numerical eigenvalues for autoionizing ridge states are calculated and compared with available results.

The generators of the Poincar\'e symmetry of scalar electrodynamics are quantized in the functional Schr\"odinger representation. We show that the factor ordering which corresponds to (minimal) Dirac quantization preserves the Poincar\'e algebra, but (minimal) reduced quantization does not. In the latter, there is a van Hove anomaly in the boost-boost commmutator, which we evaluate explicitly to lowest order in a heat kernal expansion using \ensuremath{\zeta}-function regularization. We illuminate the crucial role played by the gauge orbit volume element in the analysis. Our results demonstrate that preservation of extra symmetries at the quantum level is sometimes a useful criterion to select between inequivalent, but nevertheless self-consistent, quantization schemes.

We study the third quantization of a closed Friedmann-Robertson-Walker universe with a cosmological constant by using the functional Schr\"odinger equation. It is shown that fluctuations of the third-quantized universe field rapidly decrease in the course of cosmic expansion.

We have theoretically studied the optical absorption coefficient of quantum wires in electric and strain fields with intermixing interfaces. The potential profiles are governed by the intermixing heterojunctions, internal strain due to the lattice mismatch, the external electric field and stress. The second Fick's law describes the wires' intermixing heterojunctions due to the alloy interdiffusion. We adopt the Green's function method to solve the Poisson equation for the displacement to determine the internal strain due to the lattice mismatch. The single-band Schr\"odinger equation in the effective mass approximation is used to describe the conduction subband structures while the four-band Kohn-Luttinger Hamiltonian is used to describe the valence subband structures. In solving the Schr\"odinger equation and the Kohn-Luttinger Hamiltonian, we expand the wave functions for the electrons and holes by using linear combinations of the two-dimensional harmonic oscillator wave functions to yield two matrix equations. These matrix equations are numerically solved for their eigen-energies and their corresponding eigenfunctions. We investigate some physical properties of the unstrained quantum wire $(\mathrm{Ga}\mathrm{As}∕\mathrm{Al}\mathrm{As})$ and the strained wire $(\mathrm{Cd}\mathrm{Se}∕\mathrm{Zn}\mathrm{Se})$ as examples. When the interdiffusion becomes stronger, on one hand, the internal strain is relaxed leading to the band gap shrinkage (if the initial strain ${\ensuremath{\epsilon}}_{0}<0$) and, on the other hand, the potential profiles are deformed and the effective band gap becomes wider. Variations of the transition energies with external stress show the anticrossing effect and variations of the transition energies with external electric field show the field emission effect. The hole effective masses can be enhanced or become electronlike by applying stress to the wire. The oscillator strengths of the dipole-allowed intersubband transitions for the $y$ and $z$ polarizations are calculated to infer possible optical transitions. The optical absorption coefficients together with the joint density of states are calculated.

We extend to k \ensuremath{\ne} 0 the work in a previous paper on the solution of the Schr\"odinger equation in a periodic potential. We begin with a perturbation method, and expand the wave function and energy in powers of $k$. The equations determining the wave function to any given order in $k$ are then just inhomogeneous linear equations, which are practical to solve. We give formulas for evaluating the energy from these wave functions: For bands which are not degenerate at k = 0 these involve straightforward summations over the reciprocal lattice; for bands which are degenerate at k = 0, we show how to derive the secular determinant which gives the splitting of these bands. The elements of this determinant can also be calculated as sums, which are practical to evaluate, over the reciprocal lattice. We present a simple numerical example to show how some of the above results work out in practice. Next, we discuss tight-binding formulas in the present formalism, and the practical problem of evaluating these formulas. We give simple expressions, involving the radial derivatives of the atomic momentum space wave functions, for evaluating the energy in powers of $k$; these formulas do not make near-neighbor approximations. Then, for those cases for which neither the tight-binding approximation nor an expansion in powers of $k$ is appropriate, we discuss the direct numerical solution of the equations and give a convenient partial wave expansion for this purpose. Finally, we briefly discuss the relation to each other of the various approximations presented in this paper.

Graphical representations of three-particle wave functions obtained by direct solution of the Schr\odinger equation with the correlation-function hyperspherical-harmonic (CFHH) method are obtained and analyzed for ground and excited states of the helium atom and for the ground state of the \ensuremath{\mu}dd mesomolecular ion. The inclusion of adequate singular and cluster-correlation behavior is shown to be of crucial importance for a proper description of the wave function. In the CFHH method the wave function is a product of a correlation function and of a smooth factor expanded into hyperspherical-harmonic (HH) functions. While the HH expansion by itself is not able to reproduce a correct form of the wave function, the inclusion of the correlations results in its proper description even for low values of the maximal global momenta ${\mathit{K}}_{\mathit{m}}$.

A variational principle for the time-dependent Schr\"odinger equation with an anisotropic Gaussian wave packet as a trial wave function is used to calculate the nonperturbative response of a hydrogen atom to a strong linearly polarized laser field. This method leads to equations of motion for the position expectation value and allows one to study a classical (\ensuremath{\Elzxh}=0) limit as well as the first quantum corrections. These corrections are shown to be responsible for a strong enhancement in the generation of the third and fifth harmonic by the bound-state part of the electronic wave function for field intensities of the order of ${10}^{14}$ W/${\mathrm{cm}}^{2}$ and a laser frequency corresponding to three-photon ionization. For a low-frequency field a strong nonperturbative enhancement of harmonics of orders 9 to 15 is observed. It is concluded that the plateau at even higher harmonics observed recently in experiment and in numerical calculations is caused by the ionizing parts of the wave function that are not represented in the present calculation.

Central, tensor, and spin-orbit potentials are calculated for the neutron-proton triplet state. They allow explicit solution of the Schr\"odinger equation and fit the low-energy data: effective range, scattering length, deuteron binding energy, quadrupole moment, and $D$-state probability. The deuteron wave function as well as scattering wave functions which result from these potentials are also computed. The potential and wave functions which fit the low-energy singlet state data are also obtained.

An effective single-band Hamiltonian representing a crystal electron in a uniform magnetic field is constructed from the tight-binding form of a Bloch band by replacing $\ensuremath{\hbar}\stackrel{\ensuremath{\rightarrow}}{\mathrm{k}}$ by the operator $\stackrel{\ensuremath{\rightarrow}}{\mathrm{p}}\ensuremath{-}\frac{e\stackrel{\ensuremath{\rightarrow}}{A}}{c}$. The resultant Schr\"odinger equation becomes a finite-difference equation whose eigenvalues can be computed by a matrix method. The magnetic flux which passes through a lattice cell, divided by a flux quantum, yields a dimensionless parameter whose rationality or irrationality highly influences the nature of the computed spectrum. The graph of the spectrum over a wide range of "rational" fields is plotted. A recursive structure is discovered in the graph, which enables a number of theorems to be proven, bearing particularly on the question of continuity. The recursive structure is not unlike that predicted by Azbel', using a continued fraction for the dimensionless parameter. An iterative algorithm for deriving the clustering pattern of the magnetic subbands is given, which follows from the recursive structure. From this algorithm, the nature of the spectrum at an "irrational" field can be deduced; it is seen to be an uncountable but measure-zero set of points (a Cantor set). Despite these-features, it is shown that the graph is continuous as the magnetic field varies. It is also shown how a spectrum with simplified properties can be derived from the rigorously derived spectrum, by introducing a spread in the field values. This spectrum satisfies all the intuitively desirable properties of a spectrum. The spectrum here presented is shown to agree with that predicted by A. Rauh in a completely different model for crystal electrons in a magnetic field. A new type of magnetic "superlattice" is introduced, constructed so that its unit cell intercepts precisely one quantum of flux. It is shown that this cell represents the periodicity of solutions of the difference equation. It is also shown how this superlattice allows the determination of the wave function at nonlattice sites. Evidence is offered that the wave functions belonging to irrational fields are everywhere defined and are continuous in this model, whereas those belonging to rational fields are only defined on a discrete set of points. A method for investigating these predictions experimentally is sketched.

A method is presented for the determination, with relatively small effort, of analytic one-electron wave functions for an electron in an approximately self-consistent screened Coulomb field. The method enables one to avoid direct numerical solution of the Schr\"odinger equation. Instead, for each state one has to adjust two parameters to make a certain expression for the screening function reasonably approximate that which follows from the Thomas-Fermi ion for the atom in question. When this is done, one has automatically a simple, analytic expression for the eigenfunction and the corresponding energy eigenvalue for that state. In particular, pairs of parameters have been obtained for states of several alkali atoms, including all of the normally filled states of sodium, the $6s$ state of potassium and most of the important excited states of cesium. The analytic wave functions thus obtained are simple enough so that the expectation values of many quantities can be easily calculated by analytic methods, a few of which are illustrated. Thus, the normalization constants are calculated for three of the states of cesium. Moreover, calculations for the fine structure and hyperfine structure energy splittings of cesium are carried out as a test of the goodness of the wave functions. Good agreement is obtained with experiment and with previous calculations.

An analytically solvable composite potential that can closely reproduce the combined potential of an $\ensuremath{\alpha}+\mathrm{nucleus}$ system consisting of attractive nuclear and repulsive electrostatic potentials is developed. The exact $s$-wave solution of the Schr\"odinger equation with this potential in the interior region and the outside Coulomb wave function are used to give a heuristic expression for the width or half-life of the quasibound state at the accurately determined resonance energy, called the $Q$ value of the decaying system. By using the fact that for a relatively low resonance energy, the quasibound state wave function is quite similar to the bound state wave function where the amplitude of the wave function in the interaction region is very large as compared to the amplitude outside, the resonance energy could easily be calculated from the variation of relative probability densities of inside and outside waves as a function of energy. By considering recent $\ensuremath{\alpha}$-decay systems, the applicability of the model is demonstrated with excellent explanations being found for the experimental data of $Q$ values and half-lives of a vast range of masses including superheavy nuclei and nuclei with very long lifetimes (of order ${10}^{22}$ s). Throughout the application, by simply varying the value of a single potential parameter describing the flatness of the barrier, we obtain successful results in cases with as many as 70 pairs of $\ensuremath{\alpha}+\mathrm{daughter}$ nucleus systems.