Latex, 18 pages, 8 figures; for the proceedings of the conference Non-locality of Quantum Mechanics and Statistical Inference, September 8-9, 2003, Kyoto Sangyo University, Kyoto, Japan
Defects or junctions in materials serve as a source of interactions for particles, and in idealized limits they may be treated as singular points yielding contact interactions. In quantum mechanics, these singularities accommodate an unexpectedly rich structure and thereby provide a variety of physical phenomena, especially if their properties are controlled properly. Based on our recent studies, we present a brief review on the physical aspects of such quantum singularities in one dimension. Among the intriguing phenomena that the singularities admit, we mention strong vs weak duality, supersymmetry, quantum anholonomy (Berry phase), and a copying process by anomalous caustics. We also show that a partition wall as a singularity in a potential well can give rise to a quantum force which exhibits an interesting temperature behavior characteristic to the particle statistics.
The equivalence of the Rivier-Margenau-Hill and Born-Jordan-Shankara phase space formalisms to the conventional operator approach of quantum mechanics is demonstrated. It is shown that in spite of the presence of singular kernels the mappings relating phase space functions and operators back and forth are possible.
We discuss a possible approach to the absorption problem in Quantum Mechanics based on using of singular attractive potentials in the corresponding Schrödinger equations. Possible criteria for selection of exact solutions of these equations are considered and it is shown that different models of absorption can be realized by a special choice of exact solutions. As an example, the motion of charged particles in the Aharonov-Bohm (AB) and the scalar attractive $ρ^{-2}$ potentials is investigated in detail. Other attractive potentials are briefly considered.
A modification of the spiked harmonic oscillator is studied in the case for which the perturbation potential contains both an inverse power and a linear term. It is then possible to evaluate trial functions by solving an integral equation due to the occurrence of the linear term. The general form of such integral equation is obtained by using a Green-function method, and adding a modified Bessel function of second kind which solves an homogeneous problem with Dirichlet boundary condition at the origin.
A simple methodology is suggested for the efficient calculation of certain central potentials having singularities. The generalized pseudospectral method used in this work facilitates {\em nonuniform} and optimal spatial discretization. Applications have been made to calculate the energies, densities and expectation values for two singular potentials of physical interest, {\em viz.,} (i) the harmonic potential plus inverse quartic and sextic perturbation and (ii) the Coulomb potential with a linear and quadratic term for a broad range of parameters. The first 10 states belonging to a maximum o…
For a class of singular potentials, including the Coulomb potential (in three and less dimensions) and $V(x) = g/x^2$ with the coefficient $g$ in a certain range ($x$ being a space coordinate in one or more dimensions), the corresponding Schrödinger operator is not automatically self-adjoint on its natural domain. Such operators admit more than one self-adjoint domain, and the spectrum and all physical consequences depend seriously on the self-adjoint version chosen. The article discusses how the self-adjoint domains can be identified in terms of a boundary condition for the asymptotic behavio…
We develop an extension of Bohmian mechanics by defining Bohm-like trajectories for quantum particles in a curved background space-time containing a spacelike singularity. As an example of such a metric we use the Schwarzschild metric, which contains two spacelike singularities, one in the past and one in the future. Since the particle world lines are everywhere timelike or lightlike, particles can be annihilated but not created at a future spacelike singularity, and created but not annihilated at a past spacelike singularity. It is argued that in the presence of future (past) spacelike singul…
We apply the causal interpretation of quantum mechanics to homogeneous quantum cosmology and show that the quantum theory is independent of any time-gauge choice and there is no issue of time. We exemplify this result by studying a particular minisuperspace model where the quantum potential driven by a prescribed quantum state prevents the formation of the classical singularity, independently on the choice of the lapse function. This means that the fast-slow-time gauge conjecture is irrelevant within the framework of the causal interpretation of quantum cosmology.
We provide a systematic study on the possibility of supersymmetry (SUSY) for one dimensional quantum mechanical systems consisting of a pair of lines $\R$ or intervals [-l, l] each having a point singularity. We consider the most general singularities and walls (boundaries) at $x = \pm l$ admitted quantum mechanically, using a U(2) family of parameters to specify one singularity and similarly a U(1) family of parameters to specify one wall. With these parameter freedoms, we find that for a certain subfamily the line systems acquire an N = 1 SUSY which can be enhanced to N = 4 if the parameters…
It is shown that if a potential in a nonrelativistic system of Fermi particles has a sufficiently strong singularity, anomalies (nonzero values of quantities formally equal to zero) will probably appear. For different types of singularities (in paticular, for the Coulomb potential), anomalies associated with the energy and total number of particles in the system are calculated. These anomalies may be beneficial in deriving a semiclassical description of electron- nuclear systems.
The analytic structure of the S-matrix of singular quantum mechanics is examined within a multichannel framework, with primary focus on its dependence with respect to a parameter ($Ω$) that determines the boundary conditions. Specifically, a characterization is given in terms of salient mathematical and physical properties governing its behavior. These properties involve unitarity and associated current-conserving Wronskian relations, time-reversal invariance, and Blaschke factorization. The approach leads to an interpretation of effective nonunitary solutions in singular quantum mechanics and…
Borel summable semiclassical expansions in 1D quantum mechanics are considered. These are the Borel summable expansions of fundamental solutions and of quantities constructed with their help. An expansion, called topological,is constructed for the corresponding Borel functions. Its main property is to order the singularity structure of the Borel plane in a hierarchical way by an increasing complexity of this structure starting from the analytic one. This allows us to study the Borel plane singularity structure in a systematic way. Examples of such structures are considered for linear, harmonic…
We study the possibility of supersymmetry (SUSY) in quantum mechanics in one dimension under the presence of a point singularity. The system considered is the free particle on a line R or on the interval [-l, l] where the point singularity lies at x = 0. In one dimension, the singularity is known to admit a U(2) family of different connection conditions which include as a special case the familiar one that arises under the Dirac delta-potential. Similarly, each of the walls at x = l and x = -l admits a U(1) family of boundary conditions including the Dirichlet and the Neumann boundary conditio…
Recently Thienel [Ann. Phys. (N.Y.) 280 (2000), 140; quant-ph/9809047] investigated the Pauli equation for an electron moving in a plane under the influence of a perpendicular magnetic field which is the sum of a uniform field and a singular flux tube. Here we criticise his claim that one cannot properly solve this equation by treating the singular flux tube as the limiting case of a flux tube of finite size.
The three-dimensional potential equation, motivated by representations of quantum mechanics, is investigated in four different scenarios: (i) In the usual Euclidean space $\mathbb{E}_{3}$ where the potential is singular but invariant under the continuous inhomogeneous orthogonal group $\mathcal{I}O(3)$. The invariance under the translation subgroup is compared to the corresponding unitary transformation in the Schrödinger representation of quantum mechanics. This scenario is well known but serves as a reference point for the other scenarios. (ii) Next, the discrete potential equation as a part…
A path-integral approach for the computation of quantum-mechanical propagators and energy Green's functions is presented. Its effectiveness is demonstrated through its application to singular interactions, with particular emphasis on the inverse square potential--possibly combined with a delta-function interaction. The emergence of these singular potentials as low-energy nonrelativistic limits of quantum field theory is highlighted. Not surprisingly, the analogue of ultraviolet regularization is required for the interpretation of these singular problems.
A multichannel S-matrix framework for singular quantum mechanics (SQM) subsumes the renormalization and self-adjoint extension methods and resolves its boundary-condition ambiguities. In addition to the standard channel accessible to a distant ("asymptotic") observer, one supplementary channel opens up at each coordinate singularity, where local outgoing and ingoing singularity waves coexist. The channels are linked by a fully unitary S-matrix, which governs all possible scenarios, including cases with an apparent nonunitary behavior as viewed from asymptotic distances.
The spectral singularity have been extensively studied over the last one and half decade for different non-Hermitian potentials in non-Hermitian quantum mechanics. The nature of spectral singularities have not been studied for the case of quaternionic potential. In the present work we perform an analytical study for the scattering from a quaternionic point interaction represented by delta function. New features of spectral singularities are observed which are different than the case of complex (non-quaternionic) point interaction. Most notable difference is the occurrence of spectral singulari…
We study the problem of the attractive inverse square potential in quantum mechanics with a generalized uncertainty relation. Using the momentum representation, we show that this potential is regular in this framework. We solve analytically the s-wave bound states equation in terms of Heun's functions. We discuss in detail the bound states spectrum for a specific form of the generalized uncertainty relation. The minimal length may be interpreted as characterizing the dimension of the system.
Classical statistical average values are generally generalized to average values of quantum mechanics, it is discovered that quantum mechanics is direct generalization of classical statistical mechanics, and we generally deduce both a new general continuous eigenvalue equation and a general discrete eigenvalue equation in quantum mechanics, and discover that a eigenvalue of quantum mechanics is just an extreme value of an operator in possibility distribution, the eigenvalue f is just classical observable quantity. A general classical statistical uncertain relation is further given, the general…
We study perturbation theory in certain quantum mechanics problems in which the perturbing potential diverges at some points, even though the energy eigenvalues are smooth functions of the coefficient of the potential. We discuss some of the unusual techniques which are required to obtain perturbative expansions of the energies in such cases. These include a point-splitting prescription for expansions around the Dirichlet (fermionic) limit of the $δ$-function potential, and performing a similarity transformation to a non-Hermitian potential in the Calogero-Sutherland model. As an application o…
A recent proposal of Sjoqvist et.al. to extend Pancharatnam's criterion for phase difference between two different pure states to the case of mixed states in quantum mechanics is analyzed and the existence of phase singularities in the parameter space of an interference experiment with particles in mixed states pointed out. In the vicinity of such singular points the phase changes sharply and precisely at these points it becomes undefined. A closed circuit in the parameter space around such points results in a measurable phase shift equal to 2nπ, where n is an integer. Such effects have earlie…
The eigenvalue problem of the Hamiltonian of an electron confined to a plane and subjected to a perpendicular time-independent magnetic field which is the sum of a homogeneous field and an additional field contributed by a singular flux tube, i.e. of zero width, is investigated. Since both a direct approach based on distribution-valued operators and a limit process starting from a non-singular flux tube, i.e. of finite size, fail, an alternative method is applied leading to consistent results. An essential feature is quantum mechanical supersymmetry at g=2 which imposes, by proper representati…
In order to resolve the measurement problem of Quantum Mechanics, non-unitary time evolution has been derived from the unitarity of standard quantum formalism. New wave functions of free and non-free quantum systems follow from Schroedinger equation after inserting an ansatz. Quantum systems show up as probability waves before measurement. A pure entangled state of a composite system evolves non-unitarily, only to disentangle itself into a definite state after reduction at the measurement point. A classical space-time point is created momentarily in this event. Unitarity is restored at that po…
We reconsider briefly the relation between "physical quantity" and "physical reality in the light of recent interpretations of Quantum Mechanics. We argue, that these interpretations are conditioned from the epistemological relation between these two fundamental concepts. In detail, the choice as ontic level of the concept affect, the relative interpretation. We note, for instance, that the informational view of quantum mechanics (primacy of the subjectivity) is due mainly to the evidence of the "random" physical quantities as ontic element. We will analyze four positions: Einstein, Rovelli, d…
Unitary evolution in PT-symmetric quantum mechanics with a time-dependent metric is found to yield a new class of adiabatic processes. As an explicit example, a Berry-like phase associated with a PT-symmetric two-level system is derived and interpreted as the flux of a fictitious monopole with a tunable charge plus a singular string component with non-trivial phase contributions. The Hermitian analog of our results is also discussed.
Several locally deterministic interpretations of quantum mechanics are presented and reviewed. The fundamental differences between these interpretations are made transparent by explicitly showing what information is carried locally by each physical system in an idealized experimental test of Bell's theorem. This also shows how each of these models can be locally causal and deterministic. First, a model is presented which avoids Bell's arguments through the assumption that space-time inflated from an initial singularity, which encapsulates the entire past light cone of every event in the univer…
Light propagation in distributed feedback optical structures with gain/loss regions is shown to provide an accessible laboratory tool to visualize in optics the spectral properties of the one-dimensional Dirac equation with non-Hermitian interactions. Spectral singularities and PT symmetry breaking of the Dirac Hamiltonian are shown to correspond to simple observable physical quantities and related to well-known physical phenomena like resonance narrowing and laser oscillation.
In this paper, starting from vortices we are finally lead to a treatment of Fermions as Kerr-Newman type Black Holes wherein we identify the horizon at the particle's Compton wavelength periphery. A naked singularity is avoided and the singular processes inside the horizon of the Black Hole are identified with Quantum Mechanical effects within the Compton wavelength. Inertial mass, gravitation, electromagnetism and even QCD type interactions emerge from such a description including relative strengths and also other features like the anomalous gyromagnetic ratio, the discreteness of the charge,…
A general theory is presented of quantum mechanics of singular, non-autonomous, higher derivative systems. Within that general theory, $n$-th order and $m$-th order Lagrangians are shown to be quantum mechanically equivalent if their difference is a total derivative.
We revisited how Weinberg's ideas in Nuclear Physics influenced our own work and lead to a renormalization group invariant framework within the quantum mechanical few-body problem, and we also update the discussion on the relevant scales in the limit of short-range interactions. In this context, it is revised the formulation of the subtracted scattering equations and fixed-point Hamiltonians applied to few-body systems, in which the original interaction contains point-like singularities, such as Dirac-delta and/or its derivatives. The approach is being illustrated by considering two-nucleons d…
In the calculation of quantum-mechanical singular-potential scattering, one encounters divergence. We suggest three renormalization schemes, dimensional renormalization, analytic continuation approach, and minimal-subtraction scheme to remove the divergence.
Quantum-mechanical wave functions in singular potentials: linear and nonlinear states
Interdisciplinary Graduate School of Engineering Sciences, Kyushu University（九州大学理工学研究科）；Department of Physical Electronics, School of Electrical and Computer Engineering, Faculty of Engineering, Tel Aviv University（特拉维夫大学工程学院电子物理系）；Instituto de Alta Investigación, Universidad de Tarapacá（塔拉帕卡大学高级研究所）
New applications of Feynman disentangling method in quantum mechanics are studied and the time-dependent singular oscillator problem is solved in this approach. The important role of representation group theory is discussed in this context.
Diagonalizable pseudo-Hermitian Hamiltonians with real and discrete spectra, which are superpartners of Hermitian Hamiltonians, must be $η$-pseudo-Hermitian with Hermitian, positive-definite and non-singular $η$ operators. We show that despite the fact that an $η$ operator produced by a supersymmetric transformation, corresponding to the exact supersymmetry, is singular, it can be used to find the eigenfunctions of a Hermitian operator equivalent to the given pseudo-Hermitian Hamiltonian. Once the eigenfunctions of the Hermitian operator are found the operator may be reconstructed with the hel…
We investigate N-extended supersymmetry in one-dimensional quantum mechanics on a circle with point singularities. For any integer n, N=2n supercharges are explicitly constructed and a class of point singularities compatible with supersymmetry is clarified. Key ingredients in our construction are n sets of discrete transformations, each of which forms an su(2) algebra of spin 1/2. The degeneracy of the spectrum and spontaneous supersymmetry breaking are briefly discussed.
We investigate $N$-extended supersymmetry in one-dimensional quantum mechanics on a circle with point singularities. For any integer $n$, $N=2n+1$ supercharges are explicitly constructed in terms of discrete transformations, and a class of singularities compatible with supersymmetry is clarified. In our formulation, the supersymmetry can be reduced to $M$-extended supersymmetry for any integer $M<N$. The degeneracy of the spectrum and spontaneous supersymmetry breaking are also studied.
The physical constructs underlying the properties of quantum mechanics are explored. Arguments are given that the particle wave function as well as photon and phonon quanta must derive from a more fundamental physical construct that has not yet been identified. An approach to identifying the construct is discussed and a specific construct is proposed and explained.
We continue our study of the quantum mechanical motion in the $x^2y^2$ potentials for $n=2,3$, which arise in the spatially homogeneous limit of the Yang-Mills (YM) equations. In the present paper, we develop a new approach to the calculation of the partition function $Z(t)$ beyond the Thomas-Fermi (TF) approximation by adding a harmonic (Higgs) potential and taking the limit $v\to 0$, where $v$ is the vacuum expectation value of the Higgs field. Using the Wigner-Kirkwood method to calculate higher-order corrections in $\hbar$, we show that the limit $v\to 0$ leads to power-like singularities …
The confluent algorithm, a degenerate case of the second order supersymmetric quantum mechanics, is studied. It is shown that the transformation function must asymptotically vanish to induce non-singular final potentials. The technique can be used to create a single level above the initial ground state energy. The method is applied to the free particle, one-soliton well and harmonic oscillator.
The Energy Eigenvalue for the Singular Wave Function of the Three Dimensional Dirac Delta Schrodinger Potential via Distributionally Generalized Quantum Mechanics
Even though quantum mechanics has existed for almost 100 years, questions concerning the foundation and interpretation of the theory still remain. These issues have gathered more attention in recent years, but does this mean that physicists are more aware of foundational issues concerning quantum mechanics? A survey was sent out to 1234 physicists affiliated to 8 different universities. 149 responded to the questions, which both concerned foundational issues related to quantum mechanics, specifically, as well as questions concerning interpretations of physical theories in general. The answers …
