Nonrelativistic Quantum Mechanics Research Articles

Specific heat at constant pressure, enthalpy and Gibbs free energy of boron nitride (BN) using [formula omitted]-deformed exponential-type potential

In this paper, we have obtained the thermodynamic properties of boron nitride (BN) using q-deformed exponential-type potential within the framework of the statistical mechanics. To this end, the energy spectra of the potential model are calculated using benefit approximation and appropriate coordinate transformation within the non-relativistic quantum mechanics framework. The partition function of BN has been obtained analytically and was used to find its thermodynamic functions such as specific heat at constant pressure, enthalpy and Gibbs free energy by the Poisson summation formula. The average absolute deviations of the predicted results from the experimental data are 3.9716%, 2.3567% and 8.5868% for specific heat at constant pressure, Gibbs free energy and enthalpy, respectively.

Yet again, quantum indeterminacy is not worldly indecision

It has been argued that non-relativistic quantum mechanics is the best hunting ground for genuine examples of metaphysical indeterminacy. Approaches to metaphysical indeterminacy can be divided into two families: meta-level and object-level accounts. It has been shown (Darby in Australasian Journal of Philosophy 88(2):27–245, 2010. https://doi.org/10.1080/00048400903097786; Skow in Philosophical Quarterly 60(241):851–858, 2010) that the most popular version of the meta-level accounts, namely the metaphysical supervaluationism proposed by Barnes and Williams (Oxford Studies in Metaphysics, Oxford University Press, Oxford, pp 103–148, 2011), fails to deal with quantum indeterminacy. Such a fact has been taken by many as a challenge to adapt supervaluationism to quantum cases. In this paper, I will focus on the very last of these attempts, i.e. the situation semantics account proposed by Darby and Pickup (Synthese 1–26, 2019). After having shown where quantum indeterminacy arises and having surveyed the assumptions endorsed by the participants of the debate, I turn to Darby and Pickup’s proposal. I argue that, despite the machinery introduced, their account still fails to account for quantum indeterminacy. After considering some possible counterarguments, I suggest in the conclusion that one can plausibly extend the argument to those meta-level approaches that treat quantum indeterminacy as worldly indecision.

Unitary, continuum, stationary perturbation theory for the radial Schrödinger equation

In nonrelativistic quantum mechanics, we require that the free and interacting Hamiltonians be related by a unitary transformation, as has been done by other authors. We then derive a stationary perturbation theory for the radial Schrödinger equation for scattering from a spherically symmetric potential. A resulting advantage over the more commonly used Green function method is that the expression for the interacting state vector is normalized to each order in the coupling constant, unlike, in general, the result of the Green function method. Other authors have applied the unitary transformation concept to time-dependent perturbation theory to give unitarity of the time evolution operator to each order in perturbation theory, with results that show improvement over the standard perturbation theory. In this paper, general formulas are obtained for the phase shifts at the first and second order in the coupling constant. We test the method on a simple system with a known exact solution and find complete agreement between our first- and second-order contributions to the s-wave phase shifts and the corresponding expansion to the second order of the exact solution.

On the Alleged Extra-Structures of Quantum Mechanics

I argue that a particle ontology naturally emerges from the basic dynamical equations of non-relativistic quantum mechanics, when the quantum continuity equation is realistically interpreted. This was recognized by J.J. Sakurai in his famous textbook “Modern Quantum Mechanics”, and then dismissed on the basis of the Heisenberg position–momentum uncertainty principle. In this paper, I show that the reasons of this rejection are based on a misunderstanding of the physical import of the uncertainty principle. As a consequence, a particle ontology can be derived from the quantum formalism without the need of additional ad hoc assumptions, and therefore it cannot be regarded as “extra-structure”.

A justification of the invariance of the speed of light by quantum theoretical considerations

phenomena can often be analysed in a visually clear way by thought experiments. In the present work a new thought experiment is analysed, which describes the emission of a photon by a special mechanism. The analysis leads to the finding, that there is a natural connection between the energy-time uncertainty and the Heisenberg position-momentum uncertainty principle. A deeper analysis of this thought experiment, especially with regard to a justification of the Doppler effect from the uncertainty principle, leads to the conclusion, that the speed of light is invariant. Therefore it seems reasonable to assume, that the origin of relativistic laws lies in non-relativistic quantum mechanics.

The equation for the wavefunction in nonrelativistic quantum mechanics*

The equation for the wavefunction in nonrelativistic quantum mechanics*

Energy spectra and magnetic properties of diatomic molecules in the presence of magnetic and AB fields with the inversely quadratic Yukawa potential

Within the framework of non-relativistic quantum mechanics, the inversely quadratic Yukawa potential is investigated in the presence of external magnetic and Aharonov–Bohm (AB) fields. The Schrodinger equation is solved via the Nikiforov–Uvarov functional analysis (NUFA) method, and the obtained energy eigenvalues are discussed. It is shown that the presence of the magnetic and AB fields removes the degeneracy. The partition function of the system and the thermodynamic properties of the system such as Helmholtz free energy, entropy, internal energy, specific heat and the magnetization and magnetic susceptibility of the system are calculated in detail. The magnetic properties of the QYP in the presence of the magnetic and AB fields are also discussed. The variation of the magnetic field as a function of temperature plays an importance role in understanding of many quantum systems in physics and chemistry

Thermodynamic Properties and Bound State Solutions of Schrodinger Equation with Inversely Quadratic Yukawa/Attractive Coulomb Potential Plus a Modified Kratzer Potential

In this paper, the energy spectra of the Inversely Quadratic Yukawa/ Attractive Coulomb Potential plus a Modified Kratzer Potential (IQYCFMP) are obtained using the formula method within the framework of non-relativistic quantum mechanics (Schrodinger Wave Equation). The energy spectra obtained was found to correspond with that obtained using WKB (Werner-Kramer-Brillouin). With the energy spectrum obtained the vibrational partition function is calculated in a closed form and is used to obtain an expression for other thermodynamic functions such as vibrational mean energy U, vibrational free energy F, vibrational entropy S, and vibrational specific capacity C for some diatomic molecules. The study reveals that the Inversely Quadratic Yukawa/ Attractive Coulomb Potential plus a Modified Kratzer Potential (IQYCFMP) can be used to predict the thermodynamic properties of diatomic molecules.

Direct Relativistic Extension of the Madelung-de-Broglie-Bohm Reformulations of Quantum Mechanics and Quantum Hydrodynamics

The basic equations of the non-relativistic quantum mechanics with trajectories and quantum hydrodynamics are extended to the relativistic domain. This is achieved by using a Schr?dinger-like equation, which describes a particle with mass and spin-0 and with the correct relativistic relation between its linear momentum and kinetic energy. Some simple but instructive free particle examples are discussed.

Spacetime Paths as a Whole

The mathematical similarities between non-relativistic wavefunction propagation in quantum mechanics and image propagation in scalar diffraction theory are used to develop a novel understanding of time and paths through spacetime as a whole. It is well known that Feynman’s original derivation of the path integral formulation of non-relativistic quantum mechanics uses time-slicing to calculate amplitudes as sums over all possible paths through space, but along a definite curve through time. Here, a 3+1D spacetime wave distribution and its 4-momentum dual are formally developed which have no external time parameter and therefore cannot change or evolve in the usual sense. Time is thus seen “from the outside”. A given 3+1D momentum representation of a system encodes complete dynamical information, describing the system’s spacetime behavior as a whole. A comparison is made to the mathematics of holograms, and properties of motion for simple systems are derived.

TWO-DIMENSIONAL NONLOCAL NONLINEAR SCHRODINGER EQUATION BASED ON THE ABLOWITZ-MUSSLIMANI SYMMETRY CONDITION

The nonlinear Schrodinger equation is a nonlinear partial differential equation and integrable equation that play an essential role in many branches of physics as nonrelativistic quantum mechanics, acoustics, and optics. In this work, motivated by the ideas of Ablowitz and Musslimani, we successfully obtain a two-dimensional nonlocal nonlinear Schrodinger equation where the nonlocality consists of reverse time fields as factors in the nonlinear terms. The nonlocal nonlinear Schrodinger equation admits a great number of good properties that the classical nonlinear Schrodinger equation possesses, e.g. PT-symmetric, admitting Lax-pair, and infinitely many conservation laws. We apply the Darboux transformation method to the two-dimensional nonlinear Schrodinger equation. The idea of this method is having a Lax representation, one can obtain various kinds of solutions of the Nth order with a spectral parameter. The exact solutions and graphical representation of obtained solutions are derived.

Normalized solutions to the Chern-Simons-Schrödinger system

In this paper, we study normalized solutions to the Chern-Simons-Schrödinger system, which is a gauge-covariant nonlinear Schrödinger system with a long-range electromagnetic field, arising in nonrelativistic quantum mechanics theory. The solutions correspond to critical points of the underlying energy functional subject to the L2-norm constraint. Our research covers several aspects. Firstly, in the mass subcritical case, we establish the compactness of any minimizing sequence to the associated global minimization problem. As a by-product of the compactness of any minimizing sequence, orbital stability of the set of minimizers to the minimization problem is achieved. In addition, we discuss the radial symmetry and uniqueness of minimizer to the minimization problem. Secondly, in the mass critical case, we investigate the existence and nonexistence of normalized solutions. Finally, in the mass supercritical case, we prove the existence of ground state and infinitely many radially symmetric solutions. Moreover, orbital instability of ground states is explored.

A New Approach to the Approximate Analytic Solution of the Three-Dimensional Schrӧdinger Equation for Hydrogenic and Neutral Atoms in the Generalized Hellmann Potential Model

Within the framework of nonrelativistic noncommutative quantum mechanics using the improved approximation scheme to the centrifugal term for any l-states via the generalized Bopp’s shift method and standard perturbation theory, we have obtained the energy eigenvalues of a newly proposed generalized Hellmann potential model (the GHP model) for the hydrogenic atoms and neutral atoms. The potential is a superposition of the attractive Coulomb potential plus Yukawa one, and new central terms appear as a result of the effects of noncommutativity properties of space and phase in the Hellmann potential model. The obtained energy eigen-values appear as a function of the generalized gamma function, the discrete atomic quantum numbers (j, n, l, s and m), infinitesimal parameters (a, b, б) which are induced by the position-position and phase-phase noncommutativity, and, the dimensional parameters (Θ, 0) of the GHP model, in the nonrelativistic noncommutative three-dimensional real space phase (NC: 3D-RSP). Furthermore, we have shown that the corresponding Hamiltonian operator with (NC: 3D-RSP) symmetries is the sum of the Hamiltonian operator of the Hellmann potential model and two operators, the first one is the modified spin-orbit interaction, while the second is the modified Zeeman operator for the hydrogenic and neutral atoms.

Probing the Planck scale: the modification of the time evolution operator due to the quantum structure of spacetime

The propagator which evolves the wave-function in non-relativistic quantum mechanics, can be expressed as a matrix element of a time evolution operator: i.e. GNR(x) = 〈x2|UNR(t)|x1〉 in terms of the orthonormal eigenkets |x〉 of the position operator. In quantum field theory, it is not possible to define a conceptually useful single-particle position operator or its eigenkets. It is also not possible to interpret the relativistic (Feynman) propagator GR(x) as evolving any kind of single-particle wave-functions. In spite of all these, it is indeed possible to express the propagator of a free spinless particle, in quantum field theory, as a matrix element 〈x2|UR(t)|x1〉 for a suitably defined time evolution operator and (non-orthonormal) kets |x〉 labeled by spatial coordinates. At mesoscopic scales, which are close but not too close to Planck scale, one can incorporate quantum gravitational corrections to the propagator by introducing a zero-point-length. It turns out that even this quantum-gravity-corrected propagator can be expressed as a matrix element 〈x2|UQG(t)|x1〉. I describe these results and explore several consequences. It turns out that the evolution operator UQG(t) becomes non-unitary for sub-Planckian time intervals while remaining unitary for time interval is larger than Planck time. The results can be generalized to any ultrastatic curved spacetime.

The Heisenberg Indeterminacy Principle in the Context of Covariant Quantum Gravity

The subject of this paper deals with the mathematical formulation of the Heisenberg Indeterminacy Principle in the framework of Quantum Gravity. The starting point is the establishment of the so-called time-conjugate momentum inequalities holding for non-relativistic and relativistic Quantum Mechanics. The validity of analogous Heisenberg inequalities in quantum gravity, which must be based on strictly physically observable quantities (i.e., necessarily either 4-scalar or 4-vector in nature), is shown to require the adoption of a manifestly covariant and unitary quantum theory of the gravitational field. Based on the prescription of a suitable notion of Hilbert space scalar product, the relevant Heisenberg inequalities are established. Besides the coordinate-conjugate momentum inequalities, these include a novel proper-time-conjugate extended momentum inequality. Physical implications and the connection with the deterministic limit recovering General Relativity are investigated.

Localizability and elementary particles

The well-definedness of particles of any kind depends on the limits, approximations, or other conditions that may or may not be involved, for example, whether there are interactions and whether ostensibly related energy is localizable. In particular, their theoretical status differs between its non-relativistic and relativistic versions: One can properly define interacting elementary particles in single-system non-relativistic quantum mechanics, at least in the case of non-zero mass systems; by contrast, one is severely challenged to define even these properly in the relativistic quantum field theories that now underlie the study of particle physics. Here, the impact of localizability on this status is reviewed in relation to the work of Paul Busch on positive operator valued measures that significantly probes the relevance of quantum unsharpness to it.

Answering Mermin\u2019s challenge with conservation per no preferred reference frame

In 1981, Mermin published a now famous paper titled, “Bringing home the atomic world: Quantum mysteries for anybody” that Feynman called, “One of the most beautiful papers in physics that I know.” Therein, he presented the “Mermin device” that illustrates the conundrum of quantum entanglement per the Bell spin states for the “general reader.” He then challenged the “physicist reader” to explain the way the device works “in terms meaningful to a general reader struggling with the dilemma raised by the device.” Herein, we show how “conservation per no preferred reference frame (NPRF)” answers that challenge. In short, the explicit conservation that obtains for Alice and Bob’s Stern-Gerlach spin measurement outcomes in the same reference frame holds only on average in different reference frames, not on a trial-by-trial basis. This conservation is SO(3) invariant in the relevant symmetry plane in real space per the SU(2) invariance of its corresponding Bell spin state in Hilbert space. Since NPRF is also responsible for the postulates of special relativity, and therefore its counterintuitive aspects of time dilation and length contraction, we see that the symmetry group relating non-relativistic quantum mechanics and special relativity via their “mysteries” is the restricted Lorentz group.

Energy spectra and thermal properties of diatomic molecules in the presence of magnetic and AB fields with improved Kratzer potential

In the present study, the improved screened Kratzer potential (ISKP) is investigated in the presence of external magnetic and Aharanov-Bohm (AB) fields within the framework of non-relativistic quantum mechanics. The Schrodinger equation is solved via the Nikiforov-Uvarov Functional Analysis (NUFA) method and the energy spectra and the corresponding wave function for the ISKP in the presence of external magnetic fields are obtained in a closed form. The obtained energy spectra are used to study some selected diatomic molecules. It is observed that the present of the magnetic and AB fields removes the degeneracy for different values of the control parameter. The thermodynamic and magnetic properties of the ISKP in the present of the magnetic and AB fields are also evaluated. The effects of the control potential parameter on the thermodynamic and magnetic properties of the selected diatomic molecules are discussed.

Locality Is Dead! Long Live Locality!

Several decades of theory and experiment into EPR correlations have led to the widely held belief that reality is non-local, in spite of the fact that this violates special relativity. To date, no experiment has shown a violation of special relativity, and EPR experiments do not demonstrate the existence of superluminal information exchange, merely correlations which violate certain inequalities. Every “loophole” in these hidden variable theories has been thought plugged. However, there is much confusion in the literature due to conflation of the terms ‘locality”, “realism”, “hidden variables”, “non-contextuality”. The presence of local hidden variables is thought to necessarily lead to a Kolmogorov probability structure (hence non-contextuality), but this is an assumption, one which is not true in general once context effects are taken into account. Treated as an observational theory, several authors have shown no incompatibility between quantum mechanics and locality, and that the Bell scenario is actually about whether reality is contextual. This paper proposes a descriptive theory by assuming a generated reality (following Whitehead’s Process Theory) which can violate the principle of continuity and possess non-Kolmogorov probability structure, and reproduce the results of non-relativistic quantum mechanics, while allowing only causally local information exchange, without hidden variables. A generated reality is thus compatible with both quantum mechanics and special relativity, reproducing all of the results expected from quantum mechanics while still maintaining causally local realism. This process model thus appears to be an ideal candidate for developing theories for the unification of quantum mechanics and general relativity.

Particles, fields, and the measurement of electron spin

This article compares treatments of the Stern-Gerlach experiment across different physical theories, building up to a novel analysis of electron spin measurement in the context of classical Dirac field theory. Modeling the electron as a classical rigid body or point particle, we can explain why the entire electron is always found at just one location on the detector (uniqueness) but we cannot explain why there are only two locations where the electron is ever found (discreteness). Using non-relativistic or relativistic quantum mechanics, we can explain both uniqueness and discreteness. Moving to more fundamental physics, both features can be explained within a quantum theory of the Dirac field. In a classical theory of the Dirac field, the rotating charge of the electron can split into two pieces that each hit the detector at a different location. In this classical context, we can explain a feature of electron spin that is often described as distinctively quantum (discreteness) but we cannot explain another feature that could be explained within any of the other theories (uniqueness).