Kiyoung Kim
Using classical statistics, Schrodinger equation in quantum mechanics is derived from complex space model. Phase-space probability amplitude, that can be defined on classical point of view, has connections to probability amplitude in internal space and to wave function in quantum mechanics. In addition, the physical entity of wave function in quantum mechanics is confirmed once again.
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In the present article, we discuss one of the basic relations of Quantum Mechanics - the Uncertainty Relation (UR). In 1930, few years after Heisenberg, Erwin Schrodinger generalized the famous Uncertainty Relation in Quantum Mechanics, making it more precise than the original. The present study discusses recent generalizations of Schrodinger's work and explains why his paper remains almost forgotten in the last century.
We present the pedagogical method of Tridiagonal representation approach,an algebraic method for the solution of Schrodinger equation in nonrelativistic quantum mechanics for conventional potential functions. However, we solved a new three parameters potential function recently encountered using it as an example
Recently, Faria et al [Phys. Lett. A 305 (2002) 322] discussed an example in which the Heisenberg and the Schrodinger pictures of quantum mechanics gave different results. We identify the mistake in their reasoning and conclude that the example they discussed does not support the inequivalence of these two pictures.
In the paper the Schrodinger equation (SE) with gravity term is developed and discussed. It is shown that the modified SE is valid for particles with mass m<M_P, M_P is the Planck mass, and contains the part which, we argue describes the pilot wave. For m \to M_P the modified SE has the solution with oscillatory term, i.e. strings. Key words: Schrodinger-Newton equation; Planck time; pilot wave.
Mechanical Analogue for Schrodinger's Matter-Wave Equation
本文证明薛定谔一维四阶实值物质波方程存在精确的弯剪梁力学类比，但指出其二阶复值方程无法构造物理可实现的力学类比。
A weakly nonlocal extension of ideal fluid dynamics is derived from the Second Law of thermodynamics. It is proved that in the reversible limit the additional pressure term can be derived from a potential. The requirement of the additivity of the specific entropy function determines the quantum potential uniquely. The relation to other known derivations of Schrödinger equation (stochastic, Fisher information, exact uncertainty) is clarified.
We discuss the Bohmian mechanics by means of the deformed Schrödinger equation for position dependent mass, in the context of a $q$-algebra inspired by nonextensive statistics. A deduction of the Bohmian quantum formalism is performed by means of a deformed Fisher information functional, from which a deformed Cramér-Rao bound is derived. Lagrangian and Hamiltonian formulations, inherited by the $q$-algebra, are also developed. Then, we illustrate the results with a particle confined in an infinite square potential well. The preservation of the deformed Cramér-Rao bound for the stationary state…
We investigate two methods of constructing a solution of the Schrödinger equation from the canonical transformation in classical mechanics. One method shows that we can formulate the solution of the Schrödinger equation from linear canonical transformations, the other focuses on the generating function which satisfies the Hamilton-Jacobi equation in classical mechanics. We also show that these two methods lead to the same solution of the Schrödinger equation.
Beginning with ordinary quantum mechanics for spinless particles, together with the hypothesis that all experimental measurements consist of positional measurements at different times, we characterize directly a class of nonlinear quantum theories physically equivalent to linear quantum mechanics through nonlinear gauge transformations. We show that under two physically-motivated assumptions, these transformations are uniquely determined: they are exactly the group of time-dependent, nonlinear gauge transformations introduced previously for a family of nonlinear Schrödinger equations. The gene…
Propagation effects are analyzed for electromagnetic (EM) waves which satisfy the one-soliton non-linear Schrodinger (NLS) equation in a dispersive wave guide. The coupling between momentum and frequencies due to dispersion relation is treated by a coupled Hamiltonian-Momentum operator with equal-space commutation relations(CR). Kerr interactions in the soliton are treated. The classical solution of the one-soliton with possible quantum effects are analyzed. The integrability of nonlinear equations is related to compatibility-condition for scattering matrices and such relation is applied to th…
Unitary equivalence of Schrödinger and Heisenberg pictures survives in nonlinear quantum mechanics
本文证明在非线性量子力学中，当哈密顿量依赖于瞬时期望值时，薛定谔绘景与海森堡绘景的幺正等价性仍成立，尽管交织算子依赖初始状态，但物理预言一致。
A Real-Valued Description of Quantum Mechanics with Schrodinger's 4th-order Matter-Wave Equation
本文通过变分法证明薛定谔四阶实值物质波方程可精确复现二阶复值方程的本征值，并揭示其遗漏的负排斥能级，从而建立非相对论量子力学的实值描述。
The derivation of the time dependent Schrödinger equation with transversal and longitudinal relaxation, as the quantum mechanical analog of the classical Landau-Lifshitz-Bloch equation, has been described. Starting from the classical Landau-Lifshitz-Bloch equation the transition to quantum mechanics has been performed and the corresponding von-Neumann equation deduced. In a second step the time Schrödinger equation has been derived. Analytical proofs and computer simulations show the correctness and applicability of the derived Schrödinger equation.
The representation of a Schrodinger equations as a classic Hamiltonian system allows to construct a unified perturbation theory both in classic, and in a quantum mechanics grounded on the theory of canonical transformations, and also to receive asymptotic estimations of affinity of the precisian approximated solutions of Schrodinger equations
A new non-perturbative approach is proposed to solve time-independent Schrödinger equations in quantum mechanics and chromodynamics (QCD). It is based on the homotopy analysis method (HAM), which was developed by the author for highly nonlinear equations since 1992 and has been widely applied in many fields. Unlike perturbative methods, this HAM-based approach has nothing to do with small/large physical parameters. Besides, convergent series solution can be obtained even if the disturbance is far from the known status. A nonlinear harmonic oscillator is used as an example to illustrate the val…
We introduce the set of bicomplex numbers $\mathbb{T}$ which is a commutative ring with zero divisors defined by $\mathbb{T}=\{w_0+w_1 \bold{i_1}+w_2\bold{i_2}+w_3 \bold{j}| w_0,w_1,w_2,w_3 \in \mathbb{R}\}$ where $\bold{i^{\text 2}_1}=-1, \bold{i^{\text 2}_2}=-1, \bold{j}^2=1,\ \bold{i_1}\bold{i_2}=\bold{j}=\bold{i_2}\bold{i_1}$. We present the conjugates and the moduli associated with the bicomplex numbers. Then we study the bicomplex Schrödinger equation and found the continuity equations. The discrete symmetries of the system of equations describing the bicomplex Schrödinger equation are o…
In the Schrödinger-cat gedanken experiment a cat is in a quantum superposition of two macroscopically distinct states, alive and dead.The paradoxical interpretation of quantum mechanics is that the cat is not in one state or the other, alive or dead, immediately prior to its measurement. Because of this apparent defiance of macroscopic reality, quantum superpositions of states macroscopically distinct have generated much interest. Here we address the issue of proving a contradiction with macroscopic reality objectively, through the testable predictions of quantum mechanics. We consider the pre…
We show that QM can be represented as a natural projection of a classical statistical model on the phase space $Ω= H\times H,$ where $H$ is the real Hilbert space. Statistical states are given by Gaussian measures on $Ω$ having zero mean value and dispersion of very small magnitude $α$ (which is considered as a small parameter of the model). Such statistical states can be interpreted as fluctuations of the background field, cf. with SED and Nelson's mechanics. Physical variables (e.g., energy) are given by maps $f: Ω\to {\bf R}$ (functions of classical fields). The conventional quantum represe…
The wave equation in quantum mechanics and its general solution in the phase space are obtained.
Analysis of quantum mechanics with real-valued Schrödinger equation,single-event quantum-path dynamics, Mauprtuis path in parameter space, and branching paths beyond semiclassics
本文从实场重建薛定谔方程，提出单事件量子路径动力学，建立量子莫培督-哈密顿原理，并揭示半经典轨迹在超越半经典区域后分支为相干片段，从而深入分析量子力学本质。
In this article we present a detailed description of an electron in a uniform magnetic field evolving under the Schrödinger equation using ladder operators. Based on this analysis, we describe the same physical system using the Dirac equation, known from relativistic quantum mechanics. The main differences between these two quantum mechanical approaches are discussed and we observe specifically how the relativistic phenomena modify the description of this particular quantum system.
Although quantum mechanics is a mature theory, fundamental problems discussed during its time of foundation have remained with us to this day. These problems are centered on the problematic relation between the quantum and classical worlds. The most famous element is the measurement problem, i.e., the measurement of a quantum system by a classical apparatus, and the concomitant phenomena of wave packet reduction, the appearance of probability, and the problems related to Schrödinger cat states. A fundamental question in this context is whether quantum mechanics can bootstrap itself to the clas…
The question whether quantum mechanics is complete and the nature of the transition between quantum mechanics and classical mechanics have intrigued physicists for decades. There have been many experimental breakthroughs in creating larger and larger quantum superposition and entangled states since Erwin Schrödinger proposed his famous thought experiment of putting a cat in a superposition of both alive and dead states in 1935. Remarkably, recent developments in quantum optomechanics and electromechanics may lead to the realization of quantum superposition of living microbes soon. Recent evide…
The Eherenfest theorem states that Schrodinger representation of quantum mechanics (wave mechanics) reproduces Newton laws of motion in terms of expectation values. Remarkably, the contrary is considered elusive and, indeed, many authors have tried to obtain wave mechanics starting from other alternative frameworks of classical mechanics (for instance, Hamilton-Jacobi theory). Despite this common opinion, we present here a simple method to make Newtonian dynamics develop naturally into Schrodinger representation. The proof is based on the assumption of matter waves and is laid out in three fun…
We derive some important features of the standard quantum mechanics from a certain classical-like model -- prequantum classical statistical field theory, PCSFT. In this approach correspondence between classical and quantum quantities is established through asymptotic expansions. PCSFT induces not only linear Schrödinger's equation, but also its {\it nonlinear generalizations.} This coupling with ``nonlinear wave mechanics'' is used to evaluate the small parameter of PCSFT.
A duality between an electrostatic problem in a three dimensional world and a quantum mechanical problem in a one dimensional world which allows one to obtain the ground state solution of the Schrödinger equation by using electrostatic results is generalized to three dimensions. Here, it is demonstrated that the same transformation technique is also applicable to the s-wave Schrödinger equation in three dimensions for central potentials. This approach leads to a general relationship between the electrostatic potential and the s-wave function and the electric energy density to the quantum mecha…
Very first time Elzaki transform is used in non relativistic quantum mechanics to solve $N$-dimensional Schrödinger equation in a closed form for different solvable potential models. A universal transformation scheme is introduced and a formula based approach is developed which shows how to apply Elzaki transform to differential equation with non-constant coefficients that generally appear in solving quantum mechanical initial value problems specially for multidimensional Schrödinger equation.
A Bohmian analysis of the so-called Schrödinger-Langevin or Kostin nonlinear differential equation is provided to study how thermal fluctuations of the environment affects the dynamics of the wave packet from a quantum hydrodynamical point of view. In this way, after obtaining the Schrödinger-Langevin-Bohm equation from the Kostin equation its application to simple but physically insightful systems such as the Brownian-Bohmian motion, motion in a gravity field and transmission through a parabolic repeller is studied. % If a time-dependent Gaussian ansatz for the probability density is assumed,…
This paper follows on from a previous one in which it was shown that it is possible, within a de Broglie-Bohm style ontology for quantum mechanics, to incorporate action and reaction between the particle and its guiding field while remaining consistent with the usual experimental predictions. Whereas the previous paper focussed on the Dirac equation, the present work addresses the Schrodinger case and demonstrates that the same two-way interaction can be achieved. The aim in each case is to increase the physical plausibility of such models. The transition to include the reaction of the particl…
We study, in the semi-classical approximation, the convergence of the quantum density and the quantum action, solutions to the Madelung equations, when the Planck constant h tends to 0. We find two different solutions which depend to the initial density . In the first case where the initial quantum density is a classical density rho_0(x), the quantum density and the quantum action converge to a classical action and a classical density which satisfy the statistical Hamilton-Jacobi equations. These are the equations of a set of classical particles whose initial positions are known only by the de…
One of the advantages of a reconstruction of quantum mechanics based on transparent physical axioms is that it may offer insight to naturally generalize quantum mechanics by relaxing the axioms. Here, we discuss possible extensions of quantum mechanics within a general epistemic framework based on an operational scheme of estimation of momentum given positions under epistemic restriction. The epistemic restriction is parameterized by a global-nonseparable random variable on the order of Planck constant, an ontic extension to the separable classical phase space variables. Within the estimation …
In this work, we follow the idea of the De Broglie's matter waves and the analogous method that Schrödinger founded wave equation, but we apply the more essential Hamilton principle instead of the minimum action principle of Jacobi which was used in setting up Schrödinger wave equation. Thus, we obtain a novel non-relativistic wave equation which is different from the Schrödinger equation, and relativistic wave equation including free and non-free particle. In addition, we get the spin 1/2 particle wave equation in potential field.
Fractal Schrödinger Equation: Implications for Fractal Sets
本文引入分形薛定谔方程，研究其在分形集合中的量子力学后果，包括氢原子概率密度、能级及谐振子分形概率密度函数，揭示量子力学与分形几何的深层关系。
The stochastic dissipative Schrodinger equation is derived for an open quantum system consisting of a sub-system able to exchange energy with a thermal reservoir. The resultant evolution of the wave function also gives the evolution of the density matrix, which is an explicit, stochastic form of the Lindblad master equation. A quantum fluctuation-dissipation theorem is also derived. The time correlation function is discussed.
The aim of this short Note is to show that the Schrödinger and Heisenberg pictures of quantum mechanics are not equivalent unless one uses a quantization rule clearly stated by Born and Jordan in their famous 1925 paper. This rule is sufficient and necessary to ensure energy conservation in Heisenberg's matrix mechanics. It follows, in particular, that Schrödinger and Heisenberg mechanics are not equivalent if one quantizes observables using the Weyl prescription.
Schrodinger's Wave Mechanics: Remarkable Dates and Place One Hundred Years Ago
回顾薛定谔1925年圣诞节在瑞士阿罗萨度假期间发现波方程的历史，强调波动力学的诞生背景。
This paper is concerned with the connection between density matrix method, supersymmetric quantum mechanics and Lewis-Riesenfeld invariant theory. It is shown that these three formulations share the common mathematical structure: specifically, all of them have the invariant operators which satisfies the Liouville-Von Neumann equation and the solutions to the time-dependent Schrödinger equation and/or Schrödinger eigenvalue equation can be constructed in terms of the eigenstates of the invariants.
Quantum simulation of the Liouville equation in classical mechanics with discontinuous potential via Schrödingerization
提出结合哈密顿保持格式与薛定谔化方法的量子算法，模拟具有不连续势的经典刘维尔方程，避免了经典方法对相空间维度的指数依赖，并验证了势垒处的透射/反射行为。
We show that a nonlinear Schrödinger wave equation can reproduce all the features of linear quantum mechanics. This nonlinear wave equation is obtained by exploring, in a uniform language, the transition from fully classical theory governed by a nonlinear classical wave equation to quantum theory. The classical wave equation includes a nonlinear classicality enforcing potential which when eliminated transforms the wave equation into the linear Schrödinger equation. We show that it is not necessary to completely cancel this nonlinearity to recover the linear behavior of quantum mechanics. Scali…
Recently, Olavo has proposed several derivations of the Schrodinger equation from different sets of hypothesis ("axiomatizations") [Phys. Rev. A 61, 052109 (2000)]. One of them is based on the infinitesimal inverse Weyl transform of a classically evolved phase space density. We show however that the Schrodinger equation can only be obtained in that manner for linear or quadratic potential functions.
Stochastic extensions of the Schrodinger equation have attracted attention recently as plausible models for state reduction in quantum mechanics. Here we formulate a general approach to stochastic Schrodinger dynamics in the case of a nonlinear state space of the type proposed by Kibble. We derive a number of new identities for observables in the nonlinear theory, and establish general criteria on the curvature of the state space sufficient to ensure collapse of the wave function.
The main aim of this note is to show that the formalism of multi-time equations in quantum mechanics meant to represent a manifestly Lorentz-invariant theory suffers from at least three imperfections: (1) It does not cover those physical systems for which the Schrödinger picture of quantum mechanics does not exist. (2) The integrabiliy condition which has been proposed in the formalism does not imply as necessary that for every partial Hamiltonian there exists a Schrödinger equation. (3) The formalism does not generally allow for interactions even in a non-relativistic multi-particle quantum s…
The probability operator for a generic non-equilibrium quantum system is derived. The corresponding stochastic, dissipative Schrödinger equation is also given. The dissipative and stochastic propagators are linked by the fluctuation-dissipation theorem that is derived from the unitary condition on the time propagator. The dissipative propagator is derived from thermodynamic force and entropy fluctuation operators that are in general non-linear.
Derivation of Fokker-Planck equation from Schrodinger dynamics
本文基于可逆薛定谔动力学，通过纯量子力学方法推导Fokker-Planck方程，并指出该情况下热力学熵对应于量子熵的粗粒化。
Enhancement of the effects due to the Schrödinger-Newton equation
本文提出通过周期性调制悬浮机械振子的囚禁频率，将薛定谔-牛顿方程引起的偏差放大最高六个数量级，从而为在现有磁悬浮技术上实验检验该方程提供可行方案。
In this paper, the classical Schrödinger equation, which allows the study of classical dynamics in terms of wave functions, is analyzed theoretically and numerically. First, departing from classical (Newtonian) mechanics, and assuming an additional single-valued condition for the Hamilton's principal function, the classical Schrödinger equation is obtained. This additional assumption implies inherent non-classical features on the description of the dynamics obtained from the classical Schrödinger equation: the trajectories do not cross in the configuration space. Second, departing from Bohmian…
Using the simplest but fundamental example, the problem of the infinite potential well, this paper makes an ideological attempt (supported by rigorous mathematical proofs) to approach the issue of «understanding» the mechanism of quantum mechanics processes, despite the well-known examples of the EPR paradox type. The new exact solution of the Schrödinger equation is analyzed from the perspective of quantum mechanics in the phase space. It is the phase space, which has been extensively used recently in quantum computing, quantum informatics and communications, that is the bridge towards classi…
According to classical non-relativistic Schrödinger equation, any local perturbation of wave function instantaneously affects all infinite region, because this equation is of parabolic type, and its solutions demonstrate infinite speed of perturbations propagation. From physical point of view, this feature of Schrödinger equation solutions is questionable. According to relativistic quantum mechanics, the perturbations propagate with speed of light. However when appropriate mathematical procedures are applied to Dirac relativistic quantum equation with finite speed of the wave function perturba…
The energy level structure of the modified Schrodinger equation can be consistent with Lamb shift
本文提出修正薛定谔方程，通过改进作用势中的修正函数，使类氢和类氦体系的能级结构与兰姆位移一致，避免了复杂的费曼图计算。
Using a new state-dependent, $λ$-deformable, linear functional operator, ${\cal Q}_ψ^λ$, which presents a natural $C^{\infty}$ deformation of quantization, we obtain a uniquely selected non--linear, integro--differential Generalized Schrödinger equation. The case ${\cal Q}_ψ^{1}$ reproduces linear quantum mechanics, whereas ${\cal Q}_ψ^{0}$ admits an exact dynamic, energetic and measurement theoretic {\em reproduction} of classical mechanics. All solutions to the resulting classical wave equation are given and we show that functionally chaotic dynamics exists.
Quantum mechanics describes successfully numerous quantum phenomena both microscopic and macroscopic, such as superconductivity. But the controversies about quantum mechanics, in the old days and present day, reveal fundamental obscurity in quantum mechanics. In this work reader attention is drawn first of all to the fact that the orthodox description of superconductivity and some other quantum phenomena uses the realistic interpretation of the wave function proposed by Schrodinger whereas the controversies take place until the present about the positivistic interpretation proposed by Born. Ca…
In 1926, Erwin Schrodinger wrote a series of papers that invented wave mechanics and set the foundation for much of the single-particle quantum mechanics that we teach today. In his first paper, he solved the Schrodinger equation using the Laplace method, which is a technique that is quite powerful, but rarely taught. This is unfortunate, because it opens the door to examining quantum mechanics from a complex-analysis perspective. Gaining this experience with complex analysis is a useful notion to consider when teaching quantum mechanics, as these techniques can be widely used outside of quant…
Probing quantum mechanics using nanoparticle Schrödinger cats
本文构建钠纳米粒子物质波干涉平台，在含逾7000原子、质量逾170000 Da的体系中实现μ=15.5的薛定谔猫态，刷新宏观量子叠加检验。
A procedure allowing to construct rigorously discrete as well as continuum deterministic evolution equations from stochastic evolution equations is developed using a Dirac's bra and ket notation. This procedure is an extension of an approach previously used by the authors coined Discrete Stochastic Evolution Equations. Definitions and examples of discretes as well as continuum one-dimensional lattices are developed in detail in order to show the basic tools that allows to construct Schrödinger-like equations. Extension to d-dimensional lattices are studied in order to provide a wider expositio…
In this paper, we demonstrate how the interpretation of quantum mechanics due to Landé resolves the Schrödinger cat paradox and disposes of the problem of wave function collapse.
Quantum Uncertainty and Particle Trajectories: From Heisenberg and Schrödinger to Constructive Dynamical Models
本文区分量子不确定性关系与广义不确定性原理，通过费曼路径、纳尔逊随机力学、玻姆力学及卡茨动力学证明不确定性关系不排除底层轨迹描述，从而驳斥其否定微观轨迹的本体论解读。
The fractional operators together with exponential quantum in coordinate and momentum space corresponding to the power of observables are introduced. Based on an exponential relation between energy and momentum, the fractional Schrödinger equations for the free particle and the one in potential fields in heterogeneous complex media are found. The fractional equation of motion and the fractional virial theorem for anomalous quantum mechanics are then developed. Applying the fractional virial theorem, we derive an anomalous hydrogen atom whose transition energy values are much higher than that o…
We construct quantum circuits for solving one-dimensional Schrödinger equations. Simulations of three typical examples, i.e., harmonic oscillator, square-well and Coulomb potential, show that reasonable results can be obtained with eight qubits. Our simulations show that simple quantum circuits can solve the standard quantum mechanical problems.
Using a simple geometrical construction based upon the linear action of the Heisenberg--Weyl group we deduce a new nonlinear Schrödinger equation that provides an exact dynamic and energetic model of any classical system whatsoever, be it integrable, nonintegrable or chaotic. Within our model classical phase space points are represented by equivalence classes of wavefunctions that have identical position and momentum expectation values. Transport of these equivalence classes without dispersion leads to a system of wavefunction dynamics such that the expectation values track classical trajector…
Schrödinger cat states of a 16-microgram mechanical oscillator
本研究制备出16.2微克机械谐振子的薛定谔猫态，实现了叠加态大小与相位的调控并研究其退相干动力学，为探索量子经典边界及相关量子技术应用提供了基础。
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