We discuss the particle method in quantum mechanics which provides an exact scheme to calculate the time-dependent wavefunction from a single-valued continuum of trajectories where two spacetime points are linked by at most a single orbit. A natural language for the theory is offered by the hydrodynamic analogy, in which wave mechanics corresponds to the Eulerian picture and the particle theory to the Lagrangian picture. The Lagrangian model for the quantum fluid may be developed from a variational principle. The Euler-Lagrange equations imply a fourth-order nonlinear partial differential equation to calculate the trajectories of the fluid particles as functions of their initial coordinates using as input the initial wavefunction. The admissible solutions are those consistent with quasi-potential flow. The effect of the superposition principle is represented via a nonclassical force on each particle. The wavefunction is computed via the standard map between the Lagrangian coordinates and the Eulerian fields, which provides the analogue in this model of Huygens principle in wave mechanics. The method is illustrated by calculating the time-dependence of a free Gaussian wavefunction. The Eulerian and Lagrangian pictures are complementary descriptions of a quantum process in that they have associated Hamiltonian formulations that are connected by a canonical transformation. The de Broglie-Bohm interpretation, which employs the same set of trajectories, should not be conflated with the Lagrangian version of the hydrodynamic interpretation. The theory implies that the mathematical results of the de Broglie-Bohm model may be regarded as statements about quantum mechanics itself rather than about its interpretation.
We present a self-contained formulation of spin-free nonrelativistic quantum mechanics that makes no use of wavefunctions or complex amplitudes of any kind. Quantum states are represented as ensembles of real-valued quantum trajectories, obtained by extremizing an action and satisfying energy conservation. The theory applies for arbitrary configuration spaces and system dimensionalities. Various beneficial ramifications - theoretical, computational, and interpretational - are discussed.
A simple real-space model for the electron wavefunction is suggested, based on a transverse wave with helicity, rotating at mc^2/h. The mapping of the real two-dimensional vector phasor to the complex plane permits this to satisfy the standard time-dependent Schroedinger equation. This model is extended to provide an intuitive physical picture of electron spin. Implications of this model are discussed.
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…
It is shown that the Schrodinger equation for a large family of pairs of two-dimensional quantum potentials possess wavefuctions for which the amplitude and the phase are interchangeable, producing two different solutions which are dual to each other. This is a property of solutions with vanishing Bohm potential. These solutions can be extended to three-dimensional systems. We explicitly calculate dual solutions for physical systems, such as the repulsive harmonic oscillator and the two-dimensional hydrogen atom. These dual wavefunctions are also solutions of an analogue optical system in the …
In the first paper of this series, I introduced a non-linear, Hamiltonian, generalization of Schroedinger's theory that blocks formation of macroscopic dispersion ("cats"). But that theory was entirely deterministic, and so the origin of random outcomes in experiments such as Stern-Gerlach or EPRB was left open. Here I propose that Schroedinger's wavefunction has a random component and demonstrate that such an improvised stochastic theory can violate Bell's inequality. Repeated measurements and the back-reaction on the microsystem are discussed in a toy example. Experiments that might falsify …
It is shown that the apparent incompatibility of Bohmian mechanics with standard quantum mechanics, found by Akhavan and Golshani quant-ph/0305020, is an artefact of the fact that the initial wavefunction they use, being proportional to a $δ$-function, is not a regular wavefunction.
Internal global symmetries exist for the free non-relativistic Schrödinger particle, whose associated Noether charges--the space integrals of the wavefunction and the wavefunction multiplied by the spatial coordinate--are exhibited. Analogous symmetries in classical electromagnetism are also demonstrated.
In previous papers I expounded non-linear Schrodingerist quantum mechanics as a solution of the Measurement Problem. Here I show that NLQM is compatible with Einstein's theory of General Relativity. The extension to curved space-times presumes adoption of "space-time wavefunctions" (sometimes called "multi-time wavefunctions") and some additional algebraic structure: a "bitensor" supplementing Einstein's metric tensor. This kind of matter may violate the Strong Energy Condition even without a mass term, possibly with implications for the formation of singularities within Black Holes.
We propose a new picture, which we call the {\it moving picture}, in quantum mechanics. The Schrödinger equation in this picture is derived and its solution is examined. We also investigate the close relationship between the moving picture and the Hamilton-Jacobi theory in classical mechanics.
For time-dependent systems the wavefunction depends explicitly on time and it is not a pure state of the Hamiltonian. We construct operators for which the above wavefunction is a pure state. The method is based on the introduction of conserved quantities $Q$ and the pure states are defined by ${\hat Q}ψ=qψ$. The conserved quantities are constructed using parametrised mechanics and the Noether theorem.
The ontological models framework distinguishes $ψ$-ontic from $ψ$-epistemic wavefunctions. It is, in general, quite straightforward to categorize the wave-function of a certain quantum theory. Nevertheless, there has been a debate about the ontological status of the wave-function in the statistical interpretation of quantum mechanics: is it $ψ$-epistemic and incomplete or $ψ$-ontic and complete? I will argue that the wavefunction in this interpretation is best regarded as $ψ$-ontic and incomplete.
The speculation that gravity is the key to solving the quantum measurement problem has been alive for decades, without any convincing demonstration of a solution. One necessary factor in the relevant proposals is that the gravitational energy of mutual interaction, which scales quadratically with the mass, facilitates the spontaneous collapse of the wavefunctions in spatially separated superpositions. Relying on a simple physical input from electrodynamics, supported by robust first principle calculations, we show that the speculations connecting gravity and the hypothetical spontaneous collap…
Using quantum-classical analogies, we find that dynamical pictures of quantum mechanics have precise counterparts in classical mechanics. In particular, the Eulerian and Lagrangian descriptions of fluid dynamics in classical mechanics are the analogs of the Schroedinger and Heisenberg pictures in quantum mechanics, respectively. Similarities between classical and quantum dynamical pictures are explored within the framework of the Koopman-von Neumann formalism. These allow for a natural definition of various dynamical pictures in classical mechanics as well as the application of classical conc…
I propose to resolve the controversy over the speed of collapse of quantum-mechanical wavefunctions by means of an experimental test with a modified symmetric Mach-Zehnder atom interferometer, with non-intersecting, parallel, widely separated final beams. According to the conventional collapse scenario, the coherent twin-peak atomic wavefunction in the beams of the interferometer suffers an instantaneous collapse, at infinite speed, when the atom is captured by one of the two detectors at the ends of the beams, but it remains coherent until that instant. In contrast, according to the Hellwig-K…
We propose, as an alternative theory of quantum mechanics, a relativistically covariant variational principle (VP) capable of describing both wavefunction collapse and, as an appropriate limiting case, evolution of the wavefunction according to the standard quantum mechanical (SQM) wave equation. This results in a nonlinear, nonlocal, time-symmetric hidden-variable theory; the hidden variable is the phase of the wavefunction, which affects the dynamics via zitterbewegung. The VP is δ(A_1 + εA_2) = 0, in which A_1 and A_2 are positive definite integrals (over all spacetime) of functions of the …
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.
Wavefunction collapse is usually seen as a discontinuous violation of the unitary evolution of a quantum system, caused by the observation. Moreover, the collapse appears to be nonlocal in a sense which seems at odds with General Relativity. In this article the possibility that the wavefunction evolves continuously and hopefully unitarily during the measurement process is analyzed. It is argued that such a solution has to be formulated using a time symmetric replacement of the initial value problem in Quantum Mechanics. Major difficulties in apparent conflict with unitary evolution are identif…
The existence of probability in the sense of the frequency interpretation, i.e. probability as "long term relative frequency," is shown to follow from the dynamics and the interpretational rules of Everett quantum mechanics in the Heisenberg picture. This proof is free of the difficulties encountered in applying to the Everett interpretation previous results regarding relative frequency and probability in quantum mechanics. The ontology of the Everett interpretation in the Heisenberg picture is also discussed.
Whether the total angular momentum of the photon can be separated into spin and orbital parts has been a long-standing problem due to the constraint of transversality condition on its vector wavefunction. A careful analysis shows that the situation arises from a misuse of the constraint in quantum mechanics. To use the constraint properly, we convert the vector representation into a two-component representation in which the wavefunction is not constrained by any conditions. Upon doing so, we not only separate the spin conceptually from the orbital angular momentum but also identify the Berry g…
We attempt to pull together various lines of research whose ultimate conclusion points to the actual ``locality'' of Quantum Mechanics (QM). We note that just as John Bell discovered various errors in previous ``proofs'' of the completeness of QM, he made an error of his own in deriving the ``non-locality'' of QM. We show that QM satisfies the correct locality bound for non-commuting variables -- and is therefore local for 2 or more particles. We further show that the entangled wavefunctions that produce non-local de-Broglie-Bohm guidance equations are an artifact of First Quantization, and th…
The many-worlds interpretation (MWI) of quantum mechanics is studied from an unprecedented ontological perspective based on the reality of (semi-) deterministic parallel worlds in the interpretation. It is demonstrated that with thanks to the uncertainty principle there would be no consistent way to specify the correct ontology of the Universe, hence the MWI is subject to an inherent contradiction which claims that the world where we live in is unreal.
We assume that particles are point-like objects even when not observed. We report on the consequences of our assumption within the realm of quantum theory. An important consequence is the necessity of vacuum fields to account for particle dynamics. The zero-point energy is evidence of the reality of vacuum fields. The dynamics of vacuum fields depends on wave equations, boundary conditions and external sources. The wavefunction is an expression of vacuum fields. Particle events are described by probabilistic distribution functions obtained from vacuum fields. We solve the measurement problem b…
In this short communication, I gave a generalization of measurement postulate in quantum mechanics. It is regarding the case with partial measurement, namely, measurement on only part of a wave function. Upon a partial measurement, the wavefunction will either collapse in one of the eigenstate covered by the measured partial wavefunction; or collapses out of the measured part and shifts to the unmeasured part. An explanation in the WISE (Wavefunction Is System Entity) interpretation is given.
The von Neumann collapse of the quantum mechanical wavefunction after a position measurement is derived by a purely probabilistic mechanism in the context of Nelson's stochastic mechanics.
For a system with interacting quantum mechanical particles in a one-dimensional harmonic oscillator, a trial wavefunction with simple structure based on the solution of the corresponding two-particle system is suggested and tested numerically. With the inclusion of a scaling parameter for the distance between particles, at least for the very small systems tested here the ansatz gives a very good estimate of the ground state energy, with the error being of the order of ~1% of the gap to the first excited state.
The hydrodynamic representation of quantum mechanics describes virtual flow as if a quantum system were fluid in motion. This formulation illustrates pointlike vortices when the phase of a wavefunction becomes nonintegrable at nodal points. We study the dynamics of such pointlike vortices in the hydrodynamic representation for a two-particle wavefunction. In particular, we discuss how quantum entanglement influences vortex-vortex dynamics. For this purpose, we employ the time-dependent quantum variational principle combined with the Rayleigh-Ritz method. We analyze the vortex dynamics and esta…
We show that the quantum wavefunction, interpreted as the probability density of finding a single non-localized quantum particle, which evolves according to classical laws of motion, is an intermediate description of a material quantum particle between the quantum and classical realms. Accordingly, classical and quantum mechanics should not be treated separately, a unified description in terms of the Wigner distribution function being possible. Although defined on classical phase space coordinates, the Wigner distribution function accommodates the nonlocalization property of quantum systems, a…
We formulate necessary and sufficient conditions for a symplectic tomogram of a quantum state to determine the density state. We establish a connection between the (re)construction by means of symplectic tomograms with the construction by means of Naimark positive-definite functions on the Weyl-Heisenberg group. This connection is used to formulate properties which guarantee that tomographic probabilities describe quantum states in the probability representation of quantum mechanics.
Working entirely within the Schroedinger paradigm, meaning wavefunction only, I present a modification of his theory that prevents formation of macroscopic dispersion (MD; "cats"). The proposal is to modify the Hamiltonian based on a method introduced by Steven Weinberg in 1989, as part of a program to test quantum mechanics at the atomic or nuclear level. By contrast, the intent here is to eliminate MD without affecting the predictions of quantum mechanics at the microscopic scale. This restores classical physics at the macro level. Possible experimental tests are indicated and the difference…
The original intent of the Koopman-von Neumann formalism was to put classical and quantum mechanics on the same footing by introducing an operator formalism into classical mechanics. Here we pursue their path the opposite way and examine what transfer operators can say about quantum mechanical evolution. To that end, we introduce a physically motivated scalar wavefunction formalism for a velocity field on a 4-dimensional pseudo-Riemannian manifold, and obtain an evolution equation for the associated wavefunction, a generator for an associated weighted transfer operator. The generator of the sc…
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,…
In the first days of quantum mechanics Dirac pointed out an analogy between the time-dependent coefficients of an expansion of the Schrödinger equation and the classical position and momentum variables solving Hamilton's equations. Here it is shown that the analogy can be made an equivalence in that, in principle, systems of classical oscillators can be constructed whose position and momenta variables form time-dependent amplitudes which are identical to the complex quantum amplitudes of the coupled wavefunction of an N-level quantum system with real coupling matrix elements. Hence classical m…
The 'collapse' of the wave function in a general measuring process is analyzed by a pure quantum mechanical (QM) approach. The problem of the delayed choice and Welcher-Weg (WW) experiments is analyzed for Mach-Zehnder (MZ) interferometer. The WW effect is related to complementarity principle and orthogonality of wavefunctions although it produces small momentum changes between the electromagnetic (EM) field and the beam-splitters (BS's). For QM states for which we have a superposition of states and interference effects there is no reality before measurement, but by excluding such effects the …
The overwhelming majority of scientists still takes it for granted that classical mechanics (ClM) is nothing but a limiting case of quantum mechanics (QM). Although some physicists restrict this belief to a generalized QM as represented, e. g., by the algebra of observables, it will be shown in this contribution that the view of ClM as a mere sub-set of QM is nevertheless unfounded. The usual attempts to derive the laws of ClM from QM are either insufficient or not universally applicable. The transition from traditional to algebraic QM does not add any further insight. It is demonstrated that …
Almost a century after the development of quantum mechanics, we still do not have a consensus on the process of collapse of wavefunctions. Some theories require the intervention of a conscious observer while some see it as a stochastic process, and most theories violate energy conservation. In this paper we hypothesise that the collapse of wavefunctions can be caused by interactions with other objects (macroscopic or microscopic) and energy is conserved in that process. To test various hypotheses regarding collapse of wavefunctions, we propose a model system which is the quantum analogue of a …
We first consider the Einstein-Podolsky-Rosen (EPR) paradox for the system of two particles with spin 1/2 with entangled spins in first-quantized quantum mechanics (QM). If measurement is governed by wavefunction collapse, then gedanken experiments show that a number of fundamental principles including conservation of angular momentum and the Heisenberg uncertainty principle can be violated. We conclude that the collapse of the spin part of the wavefunction cannot happen and therefore an EPR paradox does not arise for this system. QM unitarity alone is sufficient to rule out "spooky" action at…
The implications of the relativistic space-time structure for a physical description by quantum mechanical wave-functions are investigated. On the basis of a detailed analysis of Bell's concept of local causality, which is violated in quantum theory, we argue that this is a subtle, as well as an important effort. A central requirement appearing in relativistic quantum mechanics, namely local commutativity, is analyzed in detail and possible justifications are given and discussed. The complexity of the implications of wave function reduction in connection with Minkowski space-time are illustrat…
The most puzzling issue in the foundations of quantum mechanics is perhaps that of the status of the wave function of a system in a quantum universe. Is the wave function objective or subjective? Does it represent the physical state of the system or merely our information about the system? And if the former, does it provide a complete description of the system or only a partial description? We shall address these questions here mainly from a Bohmian perspective, and shall argue that part of the difficulty in ascertaining the status of the wave function in quantum mechanics arises from the fact…
In quantum mechanics, the wavefunction predicts probabilities of possible measurement outcomes, but not which individual outcome is realised in each run of an experiment. This suggests that it describes an ensemble of states with different values of a hidden variable. Here, we analyse this idea with reference to currently known theorems and experiments. We argue that the $ψ$-ontic/epistemic distinction fails to properly identify ensemble interpretations and propose a more useful definition. We then show that all local $ψ$-ensemble interpretations which reproduce quantum mechanics violate Stati…
The manipulation of matterwave represents a milestone in the history of quantum mechanics. It was at the basis of its experimental validation through the observation of diffraction of matter on crystals, as well as grating and Young's double-slit interference with electrons, neutron, atoms and molecules. More recently matterwave manipulation has become a building block in the implementation of quantum devices such as quantum sensors and it plays an essential role in many proposals for implementing quantum computers. In this letter we coherently control the spatial extent of the wavefunction by…
I am concerned with two views of quantum mechanics that John S. Bell called ``unromantic'': spontaneous wave function collapse and Bohmian mechanics. I discuss some of their merits and report about recent progress concerning extensions to quantum field theory and relativity. In the last section, I speculate about an extension of Bohmian mechanics to quantum gravity.
Modern advances in transformation optics and electromagnetic metamaterials made possible experimental demonstrations of highly unusual curvilinear optical spaces, such as various geometries necessary for electromagnetic cloaking. Recently we demonstrated that mapping light intensity in a hyperbolic metamaterial may also model the flow of time in an effective (2+1) dimensional Minkowski spacetime. Curving such an effective spacetime creates experimental model of a toy big bang. Here we demonstrate that at low light levels this model may be used to emulate a fully covariant version of quantum me…
A quantum mechanical observer might be describable as having a reference system that is a superposition of classical inertial reference frames. The present paper suggests a possible weighting function in such superpositions, determined by the product of the observer's wavefunction and the Fourier transform of the wavefunction at each point in phase space. This suggestion is made because each of these factors has a simple interpretation as a probability density amplitude. Other forms for the weighting function may well be possible.
The exact factorization of an electron-nuclear wavefunction [A. Abedi, N. T. Maitra, and E. K. U. Gross, Phys. Rev. Lett. 105, 123002 (2010)] allows us to define the rigorous nuclear time-dependent Schrödinger equation (TDSE) with a time-dependent potential-energy surface (TDPES) that fully accounts for the coupling to the electronic motion and drives the nuclear wavepacket dynamics. Here, we study whether the propagation of multiple classical trajectories can reproduce the quantum nuclear motion in strong-field processes when their motions are governed by the quantum Hamilton-Jacobi equation …
A quantum mechanics representation based on position ($\vec{r}$), linear momentum($\vec{p}$) and energy($E$) eigenvalues is presented here. A set of equations, explicitly independent on wave function, was derived relating these observables. In this view, a particle has a known trajectory and at any point on space there is a linear momentum associated. Trajectory here, can be viewed as if a measurement were taken continuously. This picture does not change current quantum mechanics interpretation, rather it comes as a new route of calculation. Also, wave function can be retrieved performing an i…
We propose the idea that in Bohmian mechanics the wavefunction is related to a density of states and explore some of its consequences. Specifically, it allows a maximum-entropy interpretation of quantum probabilities, which creates a stronger link between it and statistical mechanics. The proposed approach also allows a range of extensions of the guidance condition in Bohmian mechanics.
Bell's theorem depends crucially on counterfactual reasoning, and is mistakenly interpreted as ruling out a local explanation for the correlations which can be observed between the results of measurements performed on spatially-separated quantum systems. But in fact the Everett interpretation of quantum mechanics, in the Heisenberg picture, provides an alternative local explanation for such correlations. Measurement-type interactions lead, not to many worlds but, rather, to many local copies of experimental systems and the observers who measure their properties. Transformations of the Heisenbe…
