B. Bagchi, C. Quesne, M. Znojil
The continuity equation relating the change in time of the position probability density to the gradient of the probability current density is generalized to PT-symmetric quantum mechanics. The normalization condition of eigenfunctions is modified in accordance with this new conservation law and illustrated with some detailed examples.
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Classes of the nonlinear Schrodinger-type equations compatible with the Galilei relativity principle are described. Solutions of these equations satisfy the continuity equation.
The continuity equation and its applications in phase space quantum mechanics
该研究推导含内禀自由度系统的连续性方程的量子相空间形式，给出一维Dirac方程的相空间对应，讨论不同情形下运动与散射的相空间表示并分析非束缚态Wigner函数性质。
Recent years have seen increased interest in complexified Bohmian mechanical trajectory calculations for quantum systems, both as a pedagogical and computational tool. In the latter context, it is essential that trajectories satisfy probability conservation, to ensure they are always guided to where they are most needed. In this paper, probability conservation for complexified Bohmian trajectories is considered. The analysis relies on time-reversal symmetry considerations, leading to a generalized expression for the conjugation of wavefunctions of complexified variables. This in turn enables m…
The local conservation of a physical quantity whose distribution changes with time is mathematically described by the continuity equation. The corresponding time parameter, however, is defined with respect to an idealized classical clock. We consider what happens when this classical time is replaced by a non-relativistic quantum-mechanical description of the clock. From the clock-dependent Schrödinger equation (as analogue of the time-dependent Schrödinger equation) we derive a continuity equation, where, instead of a time-derivative, an operator occurs that depends on the flux (probability cu…
Remarkably we find that for a ring with linear boundary conditions such that the eigenvector and its derivative are continuous, there does not seem to be a way for the well-known de Broglie relation to be gauge invariant. Certain nonlinear boundary conditions assure gauge invariance, and lead to eigenfunctions with a discontinuous but differentiable phase and a continuous spectrum. A discrete subset of this spectrum forms a Hilbert space, while another subset is excluded by the nonlinear boundaries. We conclude that discontinuous momentum eigenfunctions are tenable, and that it is possible tha…
Signal-state quantum mechanics is used to discuss quantum mechanical particle decay probabilities and the quantum Zeno effect. This approach avoids the assumption of continuous time, conserves total probability and requires neither non-Hermitian Hamiltonians nor the ad-hoc introduction of complex energies. The formalism is applied to single channel decays, the ammonium molecule, and neutral Kaon decay processes.
A variational scheme for the derivation of generalized, symmetry-induced continuity equations for Hermitian and non-Hermitian quantum mechanical systems is developed. We introduce a Lagrangian which involves two complex wave fields and whose global invariance under dilation and phase variations leads to a mixed continuity equation for the two fields. In combination with discrete spatial symmetries of the underlying Hamiltonian, the mixed continuity equation is shown to produce bilocal conservation laws for a single field. This leads to generalized conserved charges for vanishing boundary curre…
In 1989, Deutsch gave a basic physical explanation of why quantum-mechanical probabilities are squares of amplitudes. Essentially, a general state vector is transformed into a highly symmetric equal-amplitude superposition. The argument was recently elaborated and publicised by DeWitt. It has remained incomplete, however, inasmuch as both authors anticipate the usual normalization (sum of amplitudes squared) of state vectors. In the present paper, a thought experiment is devised in which Deutsch's idea is demonstrated independently of the normalization, exploiting further symmetries instead.
We present a variational approach which shows that the wave functions belonging to quantum systems in different potential landscapes, are pairwise linked to each other through a generalized continuity equation. This equation contains a source term proportional to the potential difference. In case the potential landscapes are related by a linear symmetry transformation in a finite domain of the embedding space, the derived continuity equation leads to generalized currents which are divergence free within this spatial domain. In a single spatial dimension these generalized currents are invariant…
Quantum statistical mechanics from a Bohmian perspective
本研究基于多粒子位置空间的概率流构建量子统计力学一般表述，提出对应的玻姆轨迹诠释，在与标准量子理论预测一致的前提下，为宏观量子统计现象提供直观微观解释，并探讨了熵、混态及热力学的玻姆视角理解。
Trajectory-based approaches to quantum mechanics include the de Broglie-Bohm interpretation and Nelson's stochastic interpretation. It is shown that the usual route to establishing the validity of such interpretations, via a decomposition of the Schroedinger equation into a continuity equation and a modified Hamilton-Jacobi equation, fails for some quantum states. A very simple example is provided by a quantum particle in a box, described by a wavefunction initially uniform over the interior of the box. For this example there is no corresponding continuity or modified Hamilton-Jacobi equation,…
A few modified textbook Rayleigh-Schrödinger perturbative representations of bound states are reviewed. They were all inspired by an adaptive re-split of the Hamiltonian, using nonstandard bases and the flexibility of normalization of the wave functions.
We present the coherent states of the harmonic oscillator in the framework of the generalized (gravitational) uncertainty principle (GUP). This form of GUP is consistent with various theories of quantum gravity such as string theory, loop quantum gravity, and black-hole physics and implies a minimal measurable length. Using a recently proposed formally self-adjoint representation, we find the GUP-corrected Hamiltonian as a generator of the generalized Heisenberg algebra. Then following Klauder's approach, we construct exact coherent states and obtain the corresponding normalization coefficient…
Among ${\cal P}$-pseudo-Hermitian Hamiltonians $H ={\cal P}^{-1} H^\dagger \cal P}$ with real spectra, the ''weakly pseudo-Hermitian" ones (i.e., those employing non-self-adjoint ${\cal P} \neq {\cal P}^\dagger$) form a remarkable subfamily. We list some reasons why it deserves a special attention. In particular we show that whenever ${\cal P} \neq {\cal P}^\dagger$, the current involutive operator of charge ${\cal C}$ gets complemented by a nonequivalent alternative involutive quasiparity operator ${\cal Q}$. We show how, in this language, the standard quantum mechanics is restored via the tw…
Macroscopic realism, as introduced by Leggett and Garg, is the world view in which properties of macroscopic systems exist independent of and are not influenced by measurement. Motivated by classical physical laws such as Newtonian mechanics or Maxwell's electrodynamics, in this work we add the restrictive postulate that the observables of macroscopic objects are evolved continuously through space and time. Quantum theory violates both macroscopic realism and the continuity assumption. While decoherence or collapse models (e.g. due to a universal noise background or gravitational self energy) …
In this paper, we introduce the Schrodinger equation with a general kinetic energy operator. The conservation law is proved and the probability continuity equation is deducted in a general sense. Examples with a Hermitian kinetic energy operator include the standard Schrodinger equation, the relativistic Schrodinger equation, the fractional Schrodinger equation, the Dirac equation, and the deformed Schrodinger equation. We reveal that the Klein-Gordon equation has a hidden non-Hermitian kinetic energy operator. The probability continuity equation with sources indicates that there exists a diff…
Space of states of PT symmetrical quantum mechanics is examined. Requirement that eigenstates with different eigenvalues must be orthogonal leads to the conclusion that eigenfunctions belong to the space with an indefinite metric. The self consistent expressions for the probability amplitude and average value of operator are suggested. Further specification of space of state vectors yield the superselection rule, redefining notion of the superposition principle. The expression for the probability current density, satisfying equation of continuity and vanishing for the bound state, is proposed.
Superposition states are at the origin of many paradoxes in quantum mechanics. By unraveling the von Neumann equation for density matrices, we develop a superposition-free formulation of quantum mechanics. Stochastic quantum jumps are a key feature of this approach, in blatant contrast with the continuity of the deterministic Schrödinger equation. We explain how quantum entanglement arises. Our superposition-free formulation results offers a new perspective on quantum mechanics.
The ubiquitous ether coming from the ancient times up to middle of the twenty century is replaced by a superfluid quantum space. It represents by itself a Bose-Einstein condensate consisting of enormous amount of virtual particle-antiparticle pairs emerging and disappearing in an infinitely ongoing dance. Flowing of this medium in the non-relativistic limit is described by the modified Navier-Stokes equation along with the continuity equation. The first equation admits the splitting on to two coupled equations. They are the quantum Hamilton-Jacobi equation and the equation for vorticity. The q…
We consider a Finslerian type geometrization of the non-relativistic quantum mechanics in its hydrodynamical (Madelung) formulation, by also taking into account the effects of the presence of the electromagnetic fields on the particle motion. In the Madelung representation the Schrödinger equation can be reformulated as the classical continuity and Euler equations of classical fluid mechanics in the presence of a quantum potential, representing the quantum hydrodynamical evolution equations. The equation of particle motion can then be obtained from a Lagrangian similar to its classical counter…
It has been shown that local quantum measurements and no-signaling imply quantum correlations (Barnum et al. Phys. Rev. Lett., 104, 140401 (2010)). This entails that superquantum correlations will be superluminaly signaling if we admit the validity of local quantum mechanics. We present a specific, simple instance of this situation. Our result also has the interpretation that replacing the local (trivial) dynamics in a generalized non-signaling theory (GNST) by one that admits local unitaries, results in a probability distribution that may be positive and normalization-preserving but violates …
A fluid analog of the information flux in the phase-space associated to purity and von Neumann entropy are identified in the Weyl-Wigner formalism of quantum mechanics. Once constrained by symmetry and positiveness, the encountered continuity equations provide novel quantifiers for non-classicality (non-Liouvillian fluidity) given in terms of quantum decoherence, purity and von Neumann entropy fluxes. Through definitions in the Weyl-Wigner formalism, one can identify the quantum fluctuations that distort the classical-quantum coincidence regime, and the corresponding quantum information profil…
Expectation value dynamics within real Hilbert space quantum mechanics
本文基于实Hilbert空间方法研究了复波函数与四元数波函数的物理期望值动力学方程，通过连续性方程等验证了形式体系一致性，并引入广义位置与角动量算符为后续研究提供新方向。
Quantum information theorists have created axiomatic reconstructions of quantum mechanics (QM) that are very successful at identifying precisely what distinguishes quantum probability theory from classical and more general probability theories in terms of information-theoretic principles. Herein, we show how one such principle, Information Invariance & Continuity, at the foundation of those axiomatic reconstructions maps to "no preferred reference frame" (NPRF, aka "the relativity principle") as it pertains to the invariant measurement of Planck's constant h for Stern-Gerlach (SG) spin measure…
Operator formulation of Classical mechanics: Levi-Civita map and equivalence of central forces in 2-dimensions
本文在经典力学算符表述下求解二维开普勒势与谐振子势的类薛定谔方程及经典波函数，并利用 Levi-Civita 映射和 Sundman 时间重参数化证明了两类中心力模型的等价性。
Quantum mechanics is often developed in the position representation, but this is not necessary, and one can perform calculations in a representation-independent fashion, even for wavefunctions. In this work, we illustrate how one can determine wavefunctions, aside from normalization, using only operators and how those operators act on state vectors. To do this in plane polar and spherical coordinates requires one to convert the translation operator into those coordinates. As examples of this approach, we illustrate the solution of the Coulomb problem in two and three dimensions without needing…
Solution of the Schrödinger's equation in the zero order WKB approximation is analyzed. We observe and investigate several remarkable features of the WKB$_0$ method. Solution in the whole region is built with the help of simple connection formulas we derive from basic requirements of continuity and finiteness for the wave function in quantum mechanics. We show that, for conservative quantum systems, not only total energy, but also momentum is the constant of motion. We derive the quantization conditions for two and more turning point problems. Exact energy eigenvalues for solvable and some ``i…
A generalized Euler equation of fluid dynamics is derived for describing many-body states of quantum mechanics. The Eulerian Eq. can be viewed as representing the interaction of two substates, where each substate has its own velocity and pressure fields. These field quantities are given by maps of the wavefunction. For one-body systems, the Eulerian Eq. can model either a fluid or particle description of quantum states. The generalized Euler Eq. is shown to be the gradient of an equation representing the total-energy of the two substates, having two energy fields. This total-energy Eq. is a ge…
A Clarifying Note on the Position-Momentum Correspondence: Pontryagin Duality, Fourier Transport, and Physical Normalization
该研究在范畴量子力学框架内，基于装备希尔伯特空间，提出波粒二象性与德布罗意关系的表述，利用傅里叶-庞特里亚金对偶关联位置与动量的可观测量结构，确定了二象性的结构形式。
Quantum Mechanics and Neural Networks
本文证明欧几里得时间量子力学可由神经网络表示，并提出通过参数空间分裂或马尔可夫性质实现反射正性，从而构建深度神经网络量子系统，数值实验成功恢复海森堡不确定性、非平凡对易子和能谱等经典结果。
Completeness of Energy Eigenfunctions for the Reflectionless Potential in Quantum Mechanics
本文针对量子力学中无反射势，利用产生湮灭算符类比和宇称本征态方法，证明了束缚态与散射态构成完备集，并展示了由连续本征态求束缚态波函数的途径。
The Completeness of Eigenstates in Quantum Mechanics
本文研究量子力学中本征态的完备性，通过八种情况的理论证明和数值模拟，定义了通用初始态和谱函数，揭示了测量概率振幅与展开函数之间的坐标-动量变换关系。
Given a positive energy solution of the Klein-Gordon equation, the motion of the free, spinless, relativistic particle is described in a fixed Lorentz frame by a Markov diffusion process with non-constant diffusion coefficient. Proper time is an increasing stochastic process and we derive a probabilistic generalization of the equation $(dτ)^2=-\frac{1}{c^2}dX_νdX_ν$. A random time-change transformation provides the bridge between the $t$ and the $τ$ domain. In the $τ$ domain, we obtain an $\M^4$-valued Markov process with singular and constant diffusion coefficient. The square modulus of the K…
Photon quantum mechanics with a position observable
本文提出了一种基于第二量子化的光子量子力学理论，通过扩展庞加莱算符和引入位置参数，建立了描述光子在光学电路中行为的协变模型，并探讨了其与传统量子力学的关系。
On the energy density in quantum mechanics
本文提出通过改变量子粒子势阱尺寸来探测系统，证明移动阱壁的平均功与特定能量密度定义相关，且该关系可推广至二维和三维系统。
The formalism for histories-based generalized quantum mechanics developed in two earlier papers is applied to the treatment of histories (of particles or fields or more general objects) in curved spacetimes (which need not admit foliation in spacelike hypersurfaces). The construction of the space of temporal supports (a partial semigroup generalizing the space of finite time sequences employed in traditional temporal description of histories) employs spacelike subsets of spacetime having dimensionality less than or equal to three. Definition of symmetry is sharpened by the requirement of conti…
A generalization of the Hamilton-Jacobi theory to arbitrary dynamical systems, including non-Hamiltonian ones, is considered. The generalized Hamilton-Jacobi theory is constructed as a theory of ensemble of identical systems moving in the configuration space and described by the continual equation of motion and the continuity equation. For Hamiltonian systems, the usual Hamilton-Jacobi equations naturally follow from this theory. The proposed formulation of the Hamilton-Jacobi theory, as the theory of ensemble, allows interpreting in a natural way the transition from quantum mechanics in the S…
We extend classical Maxwell field theory to a first quantized theory of the photon by deriving a conserved Lorentz four-current whose zero component is a positive definite number density. Fields are real and their positive (negative) frequency parts are interpreted as absorption (emission) of a positive energy photon. With invariant plane wave normalization, the photon position operator is Hermitian with instantaneously localized eigenvectors that transform as Lorentz four-vectors. Reality of the fields and wave function ensure causal propagation and zero net absorption of energy in the absenc…
The Born rule is derived from operational assumptions, independent of the normalization of the state. Unlike Gleason's theorem, the argument applies even if probabilities are defined for only a single resolution of the identity, so it applies to all the major foundational approaches to quantum mechanics. There are important points of contact with Deutsch's program for deducing the probabilistic interpretation of quantum mechanics from decision thoery, as recently completed by Wallace. Decision theory can be used to supplement the present derivation, in application to the Everett interpretation…
Quantum mechanics in configuration space in context
本文探讨构型空间量子力学，该理论通过将经典位置-速度态提升为可区分的量子态，避免了正则量子化中动量的概念不一致性，增强了量子与经典力学的连续性。
David Bohm shown that the Schr{ö}dinger equation, that is a "visiting card" of quantum mechanics, can be decomposed onto two equations for real functions - action and probability density. The first equation is the Hamilton-Jacobi (HJ) equation, a "visiting card" of classical mechanics, to be modified by the Bohmian quantum potential. And the second is the continuity equation. The latter can be transformed to the entropy balance equation. The Bohmian quantum potential is transformed to two Bohmian quantum correctors. The first corrector modifies kinetic energy term of the HJ equation, and the s…
Time Dilation and center of Mass Normalization in the Page-Wootters Formalism
该研究在Page-Wootters量子力学框架下，揭示复合量子钟的相对论效应，发现内部能量对质心的反作用会引发时间非局域的薛定谔方程，还得到干涉可见度的二次方能量修正项，可通过差分测量分离。
Mémoire sur la mécanique quantique et l'approche ondulatoire
针对哥本哈根诠释违背因果性与实在性原则的问题，本文延续德布罗意思想，研究波动力学作为传统量子力学的替代方案，梳理了导波理论、德布罗意-玻姆理论等相关理论与类比。
The eigenvalue problem in quantum mechanics is reduced to quantization of the classical action of the physical system. State function of the system, $ψ_0(ϕ)$, is written in the form of superposition of two plane waves in the phase space. Quantization condition is derived from the basic requirements of continuity and finiteness for $ψ_0(ϕ)$ in the whole region.
We address two questions regarding square-well potentials from a didactic perspective. The first question concerns whether or not the justification of the standard a priori omission of the potential's vertical segments in the analysis of the eigenvalue problem is licit. The detour we follow to find out the answer considers a trapezoidal potential, includes the solution, analytical and numerical, of the corresponding eigenvalue problem and then analyzes the behavior of that solution in the limit when the slope of the trapezoidal potential's ramps becomes vertical. The second question, obviously…
Least constraint approach to non-relativistic quantum mechanics
本文提出基于最小约束原理的变分原理，通过定义量子约束泛函来描述实际运动与外力作用下无约束运动的偏差，推导出量子欧拉方程并等价于薛定谔方程，为几何约束和耗散力提供统一的处理方法。
Reversible lattice dynamics embody basic features of physics that govern the time evolution of classical information. They have finite resolution in space and time, don't allow information to be erased, and easily accommodate other structural properties of microscopic physics, such as finite distinct state and locality of interaction. In an ideal quantum realization of a reversible lattice dynamics, finite classical rates of state-change at lattice sites determine average energies and momenta. This is very different than traditional continuous models of classical dynamics, where the number of …
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 this comment, we point out some shortcomings in two papers "Fractional quantum mechanics" [Phys. Rev. E 62, 3135 (2000)] and "Fractional Schroedinger equation" [Phys. Rev. E 66, 056108 (2002)]. We prove that the fractional uncertainty relation does not hold generally. The probability continuity equation in fractional quantum mechanics has a missing source term, which leads to particle teleportation, i.e., a particle can teleport from one place to another. Since the relativistic kinetic energy can be viewed as an approximate realization of the fractional kinetic energy, the particle teleport…
Bohm developed the Bohmian mechanics (BM), in which the Schrödinger equation is transformed into two differential equations: A continuity equation and an equation of motion similar to the Newtonian equation of motion. This transformation can be executed both for single-particle systems and for many-particle systems. Later, Kuzmenkov and Maksimov used basic quantum mechanics for the derivation of many-particle quantum hydrodynamics (MPQHD) including one differential equation for the mass balance and two differential equations for the momentum balance, and we extended their analysis in a prework…
Many quantization schemes rely on analogs of classical mechanics where the connections with classical mechanics are indirect. In this work I propose a new and direct connection between classical mechanics and quantum mechanics where the quantum mechanical propagator is derived from a variational principle. I identify this variational principle as a generalized form of Hamilton's principle. This proposed variational principle is unusual because the physical system is allowed to have imperfect information, i.e., there is incomplete knowledge of the physical state. Two distribution functionals ov…
Ermakov-Lewis Invariants in Stationary Bohm-Madelung Quantum Mechanics
本文研究了静态量子力学中波姆-马德卢形式下，当薛定谔方程的哈密顿量对角化且可分离时，厄马科夫-皮尼方程及其不变量的自然出现，揭示了非线性振幅方程的隐藏结构。
Probability Density in Relativistic Quantum Mechanics
针对相对论量子力学中Dirac与Foldy-Wouthuysen密度哪个是自旋1/2有质量粒子正确概率密度的问题，通过Newton-Wigner方法分析定域态与洛伦兹变换性质，证明只有Dirac密度满足全部物理准则，给出该概率密度的清晰一致诠释。
The classical limit of wave quantum mechanics is analyzed. It is shown that the general requirements of continuity and finiteness to the solution $ψ(x)=Ae^{iϕ(x)}+ Be^{-iϕ(x)}$, where $ϕ(x)=\frac 1\hbar W(x)$ and $W(x)$ is the reduced classical action of the physical system, result in the asymptote of the exact solution and general quantization condition for $W(x)$, which yields the exact eigenvalues of the system.
We propose a history state formalism for a Dirac particle. By introducing a reference quantum clock system it is first shown that Dirac's equation can be derived by enforcing a timeless Wheeler-DeWitt-like equation for a global state. The Hilbert space of the whole system constitutes a unitary representation of the Lorentz group with respect to a properly defined invariant product, and the proper normalization of global states directly ensures standard Dirac's norm. Moreover, by introducing a second quantum clock, the previous invariant product emerges naturally from a generalized continuity e…
Gibbs paradox in the context of statistical mechanics addresses the issue of additivity of entropy of mixing gases. The usual discussion attributes the paradoxical situation to classical distinguishability of identical particles and credits quantum theory for enabling indistinguishability of identical particles to solve the problem. We argue that indistinguishability of identical particles is already a feature in classical mechanics and this is clearly brought out when the problem is treated in the language of information and associated entropy. We pinpoint the physical criteria for indistingu…
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