The Dance of the Sheared Eigenfunctions

Symmetry has long been recognized as a powerful formal and computational tool in quantum mechanics, beginning with the seminal work of Wigner Wigner & Fano (1960) and Weyl Weyl (1950). Indeed, it is well understood that so-called “accidental degeneracies” were, in fact, not accidents at all but rather the result of “hidden symmetry” (e.g. the 2l+1 degeneracy of the hydrogen atom energy states ). Because of this fundamental role in quantum mechanics, the discovery of new symmetries (and their possible “breaking” by interactions) is of enormous interest Griffiths (1987). In the latter half of the 20th century, a new hidden symmetry was discovered that led to much speculation in relativistic quantum field theory as applied to elementary particles. The essence of this so-called “supersymmetry” (SUSY) is that for every boson, there is also a fermion of the same mass (energy) and vice versa. Of course, this has not been observed in nature, leading to speculation that there exists some interaction in nature that “breaks” the symmetry. On the other hand, there is also substantial opinion held bymany physicists that SUSY has no connection to physical reality. This chapter is not aimed at addressing such issues. Rather, it was observed by many Junker (1996) that one did not have to deal with quantum field theory to encounter SUSY. Indeed, SUSY is also an intrinsic feature of ordinary, non-relativistic quantum mechanics (SUSY-QM). In this case, attention has not been focused on whether the symmetry exists in nature. Instead, it has been used primarily as a pedagogical tool. The reason for this lies in the intimate connection of SUSY-QM and the ladder operator approach to the harmonic oscillator, angular momentum, and the hydrogen atom Dirac (1958). The essence of SUSY-QM is the factorization of the Hamiltonian for a one dimensional system in analogy with the harmonic oscillator. For most bound-state quantum systems, it is possible to define operators analogous to the lowering (â) and raising (â†) operators that factor the harmonic oscillator Hamiltonian. However, in the general case, these operators do not possess all the properties of the harmonic oscillator â and â†, but rather they behave as so-called “charge operators”. As such, the SUSY charge operators not only allow factorization of the one dimensional Hamiltonian, they form a Lie algebraic structure. This structure results in the generation of isospectral “sector Hamiltonians”. Unfortunately, almost all previous research concentrated on exactly soluble, one dimensional model systems. We became interested in the possibility of taking computational advantage of SUSY. Our idea was that symmetry in QM has long been known to lead to significant computational simplifications and advantages. We 5

In non-relativistic quantum mechanics, singular potentials in problems with spherical symmetry lead to a Schrodinger equation for stationary states with non-Fuchsian singularities both as r tends to zero and as r tends to infinity. In the sixties, an analytic approach was developed for the investigation of scattering from such potentials, with emphasis on the polydromy of the wave function in the r variable. The present paper extends those early results to an arbitrary number of spatial dimensions. The Hill-type equation which leads, in principle, to the evaluation of the polydromy parameter, is obtained from the Hill equation for a two-dimensional problem by means of a simple change of variables. The asymptotic forms of the wave function as r tends to zero and as r tends to infinity are also derived. The Darboux technique of intertwining operators is then applied to obtain an algorithm that makes it possible to solve the Schrodinger equation with a singular potential containing many negative powers of r, if the exact solution with even just one term is already known.

In this paper, some concepts of nonrelativistic many-particle quantum mechanics (e.g., product states, density matrix) are generalized to the relativistic domain using a framework called relativistic Schrodinger theory (RST). By using a general ansatz, the RST framework is simplified considerably and some of its field equations are solved directly. The RST approach is then compared with nonrelativistic quantum mechanics (QM) for the case of a product state (conventional QM) and its RST analog. It is shown that relativistic wave equations can he derived from the RST formalism, so that they coincide in the nonrelativistic limit with the well-known Hartree equations.

The supersymmetry in non-relativistic quantum mechanics is applied as an algebraic method to obtain the solutions of the Dirac equation with spherical symmetry electromagnetic potentials. We show that some of the superpotentials related to ground state of the solvable potentials in non-relativistic quantum mechanics can be used for studying of the Dirac equation.

Nonperturbative Regularization and Renormalization: Simple Examples from Nonrelativistic Quantum Mechanics

We use a new, distinctly “geometrical” interpretation of non-relativistic quantum mechanics (NRQM) to argue for the fundamentality of the 4D blockworld ontology. We argue for a geometrical interpretation whose fundamental ontology is one of spacetime relations as opposed to constructive entities whose time-dependent behavior is governed by dynamical laws. Our view rests on two formal results: Kaiser (1981 & 1990), Bohr & Ulfbeck (1995) and Anandan, (2003) showed independently that the Heisenberg commutation relations of NRQM follow from the relativity of simultaneity (RoS) per the Poincare Lie algebra. And, Bohr, Ulfbeck & Mottelson (2004a & 2004b) showed that the density matrix for a particular NRQM experimental outcome may be obtained from the spacetime symmetry group of the experimental configuration. This shows how the blockworld view is not only consistent with NRQM, not only an implication of our geometrical interpretation of NRQM, but it is necessary in a non-trivial way for explaining quantum interference and “non-locality” from the spacetime perspective. Together the formal results imply that contrary to accepted wisdom, NRQM, the measurement problem and so-called quantum non-locality do not provide reasons to abandon the 4D blockworld implication of RoS. But rather, the deep non-commutative structure of the quantum and the deep structure of spacetime as given by the Minkowski interpretation of special relativity (STR) are deeply unified in a 4D spacetime regime that lies between Galilean spacetime (G4) and Minkowski spacetime (M4). Taken together the aforementioned formal results allow us to model NRQM phenomena such as interference without the need for realism about 3N Hilbert space, establishing that the world is really 4D and that configuration space is nothing more than a calculational device. Our new geometrical interpretation of NRQM provides a geometric account of quantum entanglement and so-called non-locality free of conflict with STR and free of interpretative mystery. In section 2 we discuss the various tensions between STR and NRQM with respect to the dimensionality of the world. Section 3 is devoted to an explication of the Kaiser et al. results and their philosophical implications. Likewise, the Bohr et al. results and their implications are the subject of section 4. In section 5, we present our geometric interpretation of quantum entanglement and “non-locality.”

The aim of this paper is to show the possible significance, and usefulness, of various non-self-adjoint operators for suitable Observables in nonrelativistic and relativistic quantum mechanics, and in quantum electrodynamics. More specifically, this work deals with: (i) the maximal Hermitian (but not self-adjoint) time operator in nonrelativistic quantum mechanics and in quantum electrodynamics; (ii) the problem of the four-position and four-momentum operators, each one with its Hermitian and anti-Hermitian parts, for relativistic spin-zero particles. Afterwards, other physically important applications of non-self-adjoint (and even non-Hermitian) operators are discussed: in particular, (iii) we reanalyze in detail the interesting possibility of associating quasi-Hermitian Hamiltonians with (decaying) unstable states in nuclear physics. Finally, we briefly mention the cases of quantum dissipation, as well as of the nuclear optical potential.

Possible theoretical frameworks for measurement of (arrival) time in nonrelativistic quantum mechanics are reviewed. It is argued that the ambiguity between indirect measurements by a suitably introduced time operator and direct measurements by a physical clock particle has a counterpart in the corresponding classical framework of measurement of the Newtonian time based on the Hamiltonian mechanics.

This work deals with the new, relativistic direction in quantum econophysics, within the bounds of which a change of the classical paradigms in mathematical modelling of socio-economic system is offered. Classical physics proceeds from the hypothesis that immediate values of all the physical quantities, characterizing system’s state, exist and can be accurately measured in principle. Non-relativistic quantum mechanics does not reject the existence of the immediate values of the classical physical quantities, nevertheless not each of them can be simultaneously measured (the uncertainty principle). Relativistic quantum mechanics rejects the existence of the immediate values of any physical quantity in principle, and consequently the notion of the system state, including the notion of the wave function, which becomes rigorously nondefinable. The task of this work consists in econophysical analysis of the conceptual fundamentals and mathematical apparatus of the classical physics, relativity theory, non-relativistic and relativistic quantum mechanics, subject to the historical, psychological and philosophical aspects and modern state of the socio-economic modeling problem. We have shown that actually and, virtually, a long time ago, new paradigms of modeling were accepted in the quantum theory, within the bounds of which the notion of the physical quantity operator becomes the primary fundamental conception(operator is a mathematical image of the procedure, the action), description of the system dynamics becomes discrete and approximate in its essence, prediction of the future, even in the rough, is actually impossible when setting aside the aftereffect i.e. the memory. In consideration of the analysis conducted in the work we suggest new paradigms of the economical-mathematical modeling.

Aim of this paper is trying to show the possible significance, and usefulness, of various non-selfadjoint operators for suitable Observables in non-relativistic and relativistic quantum mechanics, and in quantum electrodynamics: More specifically, this work starts dealing with: (i) the hermitian (but not selfadjoint) Time operator in non-relativistic quantum mechanics and in quantum electrodynamics; with (ii) idem, the introduction of Time and Space operators; and with (iii) the problem of the four-position and fourmomentum operators, each one with its hermitian and anti-hermitian parts, for relativistic spin-zero particles. Afterwards, other physical applications of non-selfadjoint (and even non-hermitian) operators are briefly discussed. We briefly mention how non-hermitian operators can indeed be used in physics [as it was done, elsewhere, for describing Unstable States]; and some considerations are added on the cases of the nuclear optical potential, of quantum dissipation, and in particular of an approach to the measurement problem in QM in terms of a chronon. This paper is largely based on work developed, along the years, in collaboration with V.S.Olkhovsky, and, in smaller parts, with P.Smrz, with R.H.A.Farias, and with S.P.Maydanyuk.

Given the classical dynamics of a non-relativistic particle in terms of a Hamiltonian or an action, it is relatively straightforward to obtain the non-relativistic quantum mechanics (NRQM) of the system. These standard procedures, based on either the Hamiltonian or the path integral, however, do not work in the case of a relativistic particle. As a result we do not have a single-particle description of relativistic quantum mechanics (RQM). Instead, the correct approach requires a transmutation of dynamical variables from the position and momentum of a single particle to a field and its canonical momentum. Particles, along with antiparticles, reappear in a very nontrivial manner as the excitations of the field. The fact that one needs to adopt completely different languages to describe a relativistic and non-relativistic free particle implies that obtaining the NRQM limit of QFT is conceptually nontrivial. I examine this limit in several approaches (like, for e.g., Hamiltonian dynamics, Lagrangian and Hamiltonian path integrals, field theoretic description etc.) and identify the precise issues which arise when one attempts to obtain the NRQM from QFT in each of these approaches. The dichotomy of NRQM and QFT does not originate just from the square root in the Hamiltonian or from the demand of Lorentz invariance, as is sometimes claimed. The real difficulty has its origin in the necessary existence of antiparticles to ensure a particular notion of relativistic causality. Because of these conceptual issues, it turns out that one cannot, in fact, obtain some of the popular descriptions of NRQM by any sensible limiting procedure applied to QFT. To obtain NRQM from QFT in a seamless manner, it is necessary to work with NRQM expressed in a language closer to that of QFT. This fact has several implications, especially for the operational notion of space coordinates in quantum theory. A close examination of these issues, which arise when quantum theory is combined with special relativity, could offer insights in the context of attempts to combine quantum theory with general relativity.

In this work, the ‐deformed Schrödinger equations defined in different form of the ‐Hamiltonian for ‐harmonic oscillator are considered with symmetric, asymmetric, and non‐polynomial potentials. The spectrum of the ‐Hamiltonian is obtained by using the Rayleigh‐Ritz variational method in which the discrete ‐Hermite I polynomials are taken as the basis. As applications, ‐harmonic, purely ‐quartic, and ‐quartic oscillators are examined in the class of symmetric polynomial potentials. Moreover, the ‐version of Gaussian potential for an example of a non‐polynomial symmetric potential and a specific example of ‐version of asymmetric double well potential are presented. Numerous results are given for these potentials for several values of . The limit relation as is discussed. The obtained results of ground‐ and excited‐state energies of the purely ‐quartic oscillator and the accuracy of the ground‐state energy levels are compared with the existing results. Also, the results are compared with the classical case appearing in the literature in the limiting case .

A boundSl is given for the number of bound statesni in thelth partial wave corresponding to a spherically symmetric potential in non-relativistic quantum mechanics. This bound is given by\(n_\iota = \mathop \smallint \limits_0^\infty |V_a (l,r)|^{1/2} dr/\pi + 1\) whereVa(l, r) is the attractive part of the effective potentialV(r)+l(l+1)/r2. Extensive comparative study ofSi and the Bargmann inequality is made.

An important aspect of the physical sciences is to make inferences about physical parameters from data. In general, the laws of physics provide the means for computing the data values given a model. This is called the “forward problem”, see figure 1. In the inverse problem, the aim is to reconstruct the model from a set of measurements. In the ideal case, an exact theory exists that prescribes how the data should be transformed in order to reproduce the model. For some selected examples such a theory exists assuming that the required infinite and noise-free data sets would be available. A quantum mechanical potential in one spatial dimension can be reconstructed when the reflection coefficient is known for all energies [Marchenko, 1955; Burridge, 1980]. This technique can be generalized for the reconstruction of a quantum mechanical potential in three dimensions [Newton, 1989], but in that case a redundant data set is required for reasons that are not well understood. The mass-density in a one-dimensional string can be constructed from the measurements of all eigenfrequencies of that string [Borg,1946], but due to the symmetry of this problem only the even part of the mass-density can be determined. If the seismic velocity in the earth depends only on depth, the velocity can be constructed exactly from the measurement of the arrival time as a function of distance of seismic waves using an Abel transform [Herglotz, 1907; Wiechert, 1907]. Mathematically this problem is identical to the construction of a spherically symmetric quantum mechanical potential in three dimensions [Keller et al., 1956]. However, the construction method of Herglotz-Wiechert only gives an unique result when the velocity increases monotonically with depth [Gerver and Markushevitch, 1966]. This situation is similar in quantum mechanics where a radially symmetric potential can only be constructed uniquely when the potential does not have local minima [Sabatier, 1973].KeywordsInverse ProblemRayleigh WaveLove WaveModel VectorSeismic TomographyThese keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

The effect of sampling the electrostatic potential around a molecule on the quality of electrostatic potential derived charges is investigated. In addition, the effect of the number of expansion sites in a Distributed Multipole Analysis (DMA) on the quality of charges fitted to the DMA derived electrostatic potential is investigated. Sampling on constant electron density surfaces gives a better fit between the quantum mechanical potential and the potential derived from the fitted charges, compared to sampling on a van der Waals surface composed of intersecting spheres. The fit between the electrostatic potential derived from point charges and the quantum mechanical potential becomes poorer with increasing quality of the employed basis set. The inclusion of bond-centers into the calculations improves the fit between the Quantum Mechanical (QM) electrostatic potential and the DMA derived potential. The number of expansion sites needed for an accurate approximation of the QM electrostatic potential increases with increasing quality of the used basis set.