Orthogonality and Boundary Conditions in Quantum Mechanics

One-dimensional particle states are constructed according to orthogonality conditions, without requiring boundary conditions. Free particle states are constructed using Dirac's delta function orthogonality conditions. The states (doublets) depend on two quantum numbers: energy and parity (+ or —). With the aid of projection operators the particles are confined to a constrained region, in a way similar to the action of an infinite well potential. From the resulting overcomplete basis, only the mutually orthogonal states are selected. Four solutions are found, corresponding to different non-commuting Hamiltonians. Their energy eigenstates are labeled with the main quantum number n and parity + or —. The energy eigenvalues are functions of n only. The four cases correspond to different boundary conditions: (I) The wave function vanishes on the boundary (energy levels: l⁺,2^–,3⁺,4^–,...), (II) the derivative of the wavefunction vanishes on the boundary (energy levels 0⁺,l^–,2⁺,3^–,...), (III) periodic boundary conditions (energy levels: 0⁺,2⁺,2^–,4⁺,4^–6⁺,6^–,...), (IV) periodic boundary conditions (energy levels: l⁺,1^–,3⁺,3^–,5⁺,5^–,...). Among the four cases, only solution (III) forms a complete basis in the sense that any function in the constrained region, can be expanded with it. By extending the boundaries of the constrained region to infinity, only solution (III) converges uniformly to the free particle states. Orthogonality seems to be a more basic requirement than boundary conditions. By using projection operators, confinement of the particle to a definite region can be achieved in a conceptually simple and unambiguous way, and physical operators can be written so that they act only in the confined region.     

Information,Quantum Mechanics,and Gravity

This is a basically expository article, with some new observations, tracing connections of the quantum potential to Fisher information, to Kähler geometry of the projective Hilbert space of a quantum system, and to the Weyl-Ricci scalar curvature of a Riemannian flat spacetime with quantum matter.Á Denise     

Expectation Values of Observables in Time-Dependent Quantum Mechanics

Let U(t) be the evolution operator of the Schrödinger equation generated by a Hamiltonian of the form H ₀(t) + W(t), where H ₀(t) commutes for all twith a complete set of time-independent projectors . Consider the observable A=_j P _jjwhere _j j , >0, for jlarge. Assuming that the matrix elements of W(t) behave as for p>0 large enough, we prove estimates on the expectation value for large times of the type where >0 depends on pand . Typical applications concern the energy expectation H₀(t) in case H ₀(t) H ₀or the expectation of the position operator x²(t) on the lattice where W(t) is the discrete Laplacian or a variant of it and H ₀(t) is a time-dependent multiplicative potential.     

From Classical Hamiltonian Flow to Quantum Theory: Derivation of the Schrödinger Equation

It is shown how the essentials of quantum theory, i.e., the Schrödinger equation and the Heisenberg uncertainty relations, can be derived from classical physics. Next to the empirically grounded quantisation of energy and momentum, the only input is given by the assumption of fluctuations in energy and momentum to be added to the classical motion. Extending into the relativistic regime for spinless particles, this procedure leads also to a derivation of the Klein-Gordon equation. Comparing classical Hamiltonian flow with quantum theory, then, the essential difference is given by a vanishing divergence of the velocity of the probability current in the former, whereas the latter results from a much less stringent requirement, i.e., that only the average over fluctuations and positions on the average divergence be identical to zero.     

Non-Existence of Liouvillian Solutions for a Free Quantum Particle on a Torus Surface,Part 1: Polar States

We consider a quantum particle constrained to the surface of a torus that we parametrize by its azimuthal and polar angle. We show that the corresponding Schrödinger equation does not have closed-form solutions (in the sense of Liouvillian functions) that depend on the polar angle only. It follows that if there are any wavefunctions in closed form, they must contain nondegenerate, special functions.     

Optical solitons in (1 + 1) and (2 + 1) dimensions

This paper addresses the nonlinear Schrödinger's equation that serves as the model to study the propagation of optical solitons through nonlinear optical fibers. The main focus of this paper is the aspect of integrability. There are a couple of integration tools that are employed to obtain the exact solutions to the model. Fan's F-expansion approach is applied to extract several forms of solutions to the model. This integration mechanism displays cnoidal waves, snoidal waves and several other solutions; needless to mention that these solutions, in the limiting case, leads to bright, dark and singular soliton solutions. The study then rolls over to the (2 + 1)-dimensions where, in addition, the semi-inverse variational principle is applied to extract a bright soliton solution, along with the necessary constraint conditions. There is also a display of several numerical simulations.     

A Tentative Expression of the Károlyházy Uncertainty of the Space-Time Structure Through Vacuum Spreads in Quantum Gravity

In the existing expositions of the Károlyházy model, quantum mechanical uncertainties are mimicked by classical spreads. It is shown how to express those uncertainties through entities of the future unified theory of general relativity and quantum theory.     

Quantum mechanics as applied mathematical statistics

Basic mathematical apparatus of quantum mechanics like the wave function, probability density, probability density current, coordinate and momentum operators, corresponding commutation relation, Schrödinger equation, kinetic energy, uncertainty relations and continuity equation is discussed from the point of view of mathematical statistics. It is shown that the basic structure of quantum mechanics can be understood as generalization of classical mechanics in which the statistical character of results of measurement of the coordinate and momentum is taken into account and the most important general properties of statistical theories are correctly respected.     

Gauge fields, point interactions and few-body problems in one dimension

Point interactions for the second derivative operator in one dimension are studied. Every operator from this family is described by the boundary conditions which include a 2x2 real matrix with the unit determinant and a phase. The role of the phase parameter leading to unitarily equivalent operators is discussed in the present paper. In particular, it is shown that the phase parameter is not redundant (contrary to previous studies) if nonstationary problems are concerned. It is proven that the phase parameter can be interpreted as the amplitude of a singular gauge field. Considering the few-body problem we extend the range of parameters for which the exact solution can be found using the Bethe Ansatz.     

Quantization as Selection Problem

Quantum systems exhibit a smaller number of energetic states than classical systems (A. Einstein, 1907, Die Plancksche Theorie der Strahlung und die Theorie der spezifischen Wärme, Ann. Phys. 22, 180ff). We take up the selection criterion for this in two parts. (1) The selection problem between classical and nonclassical mechanical systems is formulated in terms of possible and impossible configurations (among others, this overcomes the difficulties occurring when discussing the behavior of quantum particles in terms of paths). (2) The (nonclassical) selection of the quantum states is formulated, using recurrence relations and the energy law. The reformulation of “quantization as eigenvalue problem” in terms of “quantization as selection problem” allows one to derive Schrödinger’s stationary equation from classical mechanics through a straightforward and unique procedure; the nonstationary and multibody equations are subsequently acquired within the same frame. In contrast to the (classical) eigenvalue problem, the (nonclassical) selection problem can be formulated and solved without any reference to additional a priori assumptions on the nature of the quantum system, such as the wave-corpuscle dualism or an underlying wave equation or the existence of Planck’s finite action parameter. The existence of such an additional parameter—as the only additional one—is inherent in the procedure. Within our axiomatic-deductive approach, we modify classical mechanics only where it itself indicates an inherent limitation.     

Exact Solvability and Quantum Integrability of a Derivative Nonlinear Schrödinger Model

Using a variant of quantum inverse scattering method (QISM) which is directly applicable to field theoretical systems, we derive all possible commutation relations among the operator valued elements of the monodromy matrix associated with an integrable derivative nonlinear Schrödinger (DNLS) model. From these commutation relations we obtain the exact Bethe eigenstates for the quantum conserved quantities of DNLS model. We also explicitly construct the first few quantum conserved quantities including the Hamiltonian in terms of the basic field operators of this model. It turns out that this quantum Hamiltonian has a new kind of coupling constant which is quite different from the classical one. This fact allows us to apply QISM to generate the spectrum of quantum DNLS Hamiltonian for the full range of its coupling constant.     

Quantum fluctuations as deviation from classical dynamics of ensemble of trajectories parameterized by unbiased hidden random variable

A quantization method based on replacement of c-number by c-number parameterized by an unbiased hidden random variable is developed. In contrast to canonical quantization, the replacement has a straightforward physical interpretation as a statistical modification of classical dynamics of an ensemble of trajectories, and implies a unique operator ordering. We then apply the method to develop quantum measurement without wave function collapse and external observer a la pilot-wave theory.     

PT-Symmetric Calogero-Type Model

We examine angular (Pöschl-Teller) Schrödinger equation. The domain is deformed into the complex plane. We derive its solutions that are subject to Dirichlet boundary conditions.     

应用于量子含时波包方法的高精度透明边界条件

本文提出了一种适用于量子含时波包方法的透明边界条件. 该量子含时波包方法结合分离变量表象方法使用的时候,本质上是具有谱收敛性质的. 相对于以往的复数吸收势,该透明边界条件的突出优点是其对于波函数的能谱分布并不敏感. 采用具有共振态的一维势垒散射模型,本文对该方法的特点做了阐述.     

Fresnel Type Path Integral for the Stochastic Schrödinger Equation

A path integral representation is obtained for the stochastic partial differential equation of Schrödinger type arising in the theory of open quantum systems subject to continuous nondemolition measurement and filtering, known as the a posteriori or Belavkin equation. The result is established by means of Fresnel-type integrals over paths in configuration space. This is achieved by modifying the classical action functional in the expression for the amplitude along each path by means of a stochastic Itô integral. This modification can be regarded as an extension of Menski's path integral formula for a quantum system subject to continuous measurement to the case of the stochastic Schrödinger equation.     

Envelope self-similar solutions for the nonautonomous and inhomogeneous nonlinear Schrödinger equation

By means of the similarity transformation connecting with the solvable stationary equation, the self-similar combined Jacobian elliptic function solutions and fractional form solutions of the generalized nonlinear Schrödinger equation (NLSE) are obtained when the dispersion, nonlinearity, and gain or absorption are varied. The propagation dynamics in a periodic distributed amplification system is investigated. Self-similar cnoidal waves and corresponding localized waves including bright and dark similaritons (or solitons) for NLSE and arch and kink similaritons (or solitons) for cubic-quintic NLSE are analyzed. The results show that the intensity and the width of chirped cnoidal waves (or similaritons) change more distinctly than that of chirp-free counterparts (or solitons).     

An Example of Embedded Singular Continuous Spectrum for One-Dimensional Schrödinger Operators

We present a new example of a potential such that the corresponding Schrödinger operator in the halfaxis has singular continuous spectrum embedded in the absolutely continuous spectrum. The singular part is supported in an essentialy dense set. This generalizes a result of C. Remling [3].Mathematics Subject Classification: 34L40, 81Q10.     

Foundations of the Quaternion Quantum Mechanics

We show that quaternion quantum mechanics has well-founded mathematical roots and can be derived from the model of the elastic continuum by French mathematician Augustin Cauchy, i.e., it can be regarded as representing the physical reality of elastic continuum. Starting from the Cauchy theory (classical balance equations for isotropic Cauchy-elastic material) and using the Hamilton quaternion algebra, we present a rigorous derivation of the quaternion form of the non- and relativistic wave equations. The family of the wave equations and the Poisson equation are a straightforward consequence of the quaternion representation of the Cauchy model of the elastic continuum. This is the most general kind of quantum mechanics possessing the same kind of calculus of assertions as conventional quantum mechanics. The problem of the Schrödinger equation, where imaginary ‘i’ should emerge, is solved. This interpretation is a serious attempt to describe the ontology of quantum mechanics, and demonstrates that, besides Bohmian mechanics, the complete ontological interpretations of quantum theory exists. The model can be generalized and falsified. To ensure this theory to be true, we specified problems, allowing exposing its falsity.