Version Of Quantum Mechanics Research Articles

Forward–backward semiclassical and quantum trajectory methods for time correlation functions

Forward-backward trajectory formulations of time correlation functions are reviewed. Combination of the forward and reverse time evolution operators within the time-dependent semiclassical approximation minimizes phase cancellation, giving rise to an efficient methodology for simulating the dynamics of low-temperature fluids. A quantum mechanical version of the forward-backward formulation, based on the hydrodynamic formulation of time-dependent quantum mechanics, is also available but is practical only for small systems.

The Status of our Ordinary Three Dimensions in a Quantum Universe1

AbstractThere are now several, realist versions of quantum mechanics on offer. On their most straightforward, ontological interpretation, these theories require the existence of an object, the wavefunction, which inhabits an extremely high‐dimensional space known as configuration space. This raises the question of how the ordinary three‐dimensional space of our acquaintance fits into the ontology of quantum mechanics. Recently, two strategies to address this question have emerged. First, Tim Maudlin, Valia Allori, and her collaborators argue that what I have just called the ‘most straightforward’ interpretation of quantum mechanics is not the correct one. Rather, the correct interpretation of realist quantum mechanics has it describing the world as containing objects that inhabit the ordinary three‐dimensional space of our manifest image. By contrast, David Albert and Barry Loewer maintain the straightforward, wavefunction ontology of quantum mechanics, but attempt to show how ordinary, three‐dimensional space may in a sense be contained within the high‐dimensional configuration space the wavefunction inhabits. This paper critically examines these attempts to locate the ordinary, three‐dimensional space of our manifest image “within” the ontology of quantum mechanics. I argue that we can recover most of our manifest image, even if we cannot recover our familiar three‐dimensional space.

Can we make a Bohmian electron reach the speed of light, at least for one instant?

In Bohmian mechanics, a version of quantum mechanics that ascribes world lines to electrons, we can meaningfully ask about an electron's instantaneous speed relative to a given inertial frame. Interestingly, according to the relativistic version of Bohmian mechanics using the Dirac equation, a massive particle's speed is less than or equal to the speed of light, but not necessarily less. That is, there are situations in which the particle actually reaches the speed of light—a very nonclassical behavior. That leads us to the question of whether such situations can be arranged experimentally. We prove a theorem, Theorem 5, implying that for generic initial wave functions the probability that the particle ever reaches the speed of light, even if at only one point in time, is zero. We conclude that the answer to the question is no. Since a trajectory reaches the speed of light whenever the quantum probability current \documentclass[12pt]{minimal}\begin{document}$\overline{\psi }\gamma ^\mu \psi$\end{document}ψ¯γμψ is a lightlike 4-vector, our analysis concerns the current vector field of a generic wave function and may thus be of interest also independently of Bohmian mechanics. The fact that the current is never spacelike has been used to argue against the possibility of faster-than-light tunneling through a barrier, a somewhat similar question. Theorem 5, as well as a more general version provided by Theorem 6, are also interesting in their own right. They concern a certain property of a function \documentclass[12pt]{minimal}\begin{document}$\psi :\mathbb {R}^4\rightarrow \mathbb {C}^4$\end{document}ψ:R4→C4 that is crucial to the question of reaching the speed of light, namely being transverse to a certain submanifold of \documentclass[12pt]{minimal}\begin{document}$\mathbb {C}^4$\end{document}C4 along a given compact subset of space-time. While it follows from the known transversality theorem of differential topology that this property is generic among smooth functions \documentclass[12pt]{minimal}\begin{document}$\psi :\mathbb {R}^4\rightarrow \mathbb {C}^4$\end{document}ψ:R4→C4, Theorem 5 asserts that it is also generic among smooth solutions of the Dirac equation.

Quantum mechanics writ large

Some two centuries before the quantum revolution, Newton (1) suggested that corpuscles of light generate waves in an aethereal medium like skipping stones generate waves in water, with their motion then being affected by these aether waves. Times have changed. Light corpuscles are now known as photons, and the majority of physicists have dispensed with the notion of aether. Nevertheless, certain features of Newton's metaphor live on in one particular version of quantum mechanics. According to pilot wave theory, first proposed by de Broglie (2) and later developed by Bohm (3) with Einstein's encouragement, microscopic elements such as photons and electrons consist of both particle and wave, the former being guided by the latter. Although this physical picture has not been widely accepted, it has had some notable proponents, including Bell (4). Its principal appeal is that it restores realism and determinism to quantum mechanics, its weakness that the physical nature of the guiding wave field remains unclear. At the time that pilot wave theory was developed and then overtaken by the Copenhagen interpretation as the standard view of quantum mechanics, there was no macroscopic pilot wave analog to draw upon. Now there is.

The Quantum Measurement Problem and Cluster Separability

A modified Beltrametti-Cassinelli-Lahti model of the measurement apparatus that satisfies both the probability reproducibility condition and the objectification requirement is constructed. Only measurements on microsystems are considered. The cluster separability forms a basis for the first working hypothesis: the current version of quantum mechanics leaves open what happens to systems when they change their separation status. New rules that close this gap can therefore be added without disturbing the logic of quantum mechanics. The second working hypothesis is that registration apparatuses for microsystems must contain detectors and that their readings are signals from detectors. This implies that the separation status of a microsystem changes during both preparation and registration. A new rule that specifies what happens when these changes occur and that guarantees the objectification is formulated and discussed. A part of our result has certain similarities with ‘collapse of the wave function’.

ChemInform Abstract: Perspective on “The Transition State Method”

A perspective is provided on Wigner’s classic paper on transition-state theory (TST). After providing a brief review of the historical context of this work, we review its key contributions including Wigner’s dynamical perspective on TST, the fundamental assumption of TST, and the upper-bound property of classical TST. A discussion is also presented of subsequent progress in the field, which was stimulated by this work. This progress includes the following: 1. Demonstrations of the validity of the fundamental assumption for classical systems. 2. Further investigations into the classical foundations of TST that helped elucidate relationships between classical trajectories and TST. 3. The development of a variational form of the theory. 4. The development of variational TST into a quantitative tool for predicting rate constants. 5. The search for an “exact” quantum mechanical version of TST. 6. The development of TST-like expressions for the exact quantum mechanical rate constant. 7. The extension of TST to reactions in condensed phases.

Deterministic flows of order-parameters in stochastic processes of quantum Monte Carlo method

In terms of the stochastic process of quantum-mechanical version of Markov chain Monte Carlo method (the MCMC), we analytically derive macroscopically deterministic flow equations of order parameters such as spontaneous magnetization in infinite-range (d(= ∞)-dimensional) quantum spin systems. By means of the Trotter decomposition, we consider the transition probability of Glauber-type dynamics of microscopic states for the corresponding (d + 1)-dimensional classical system. Under the static approximation, differential equations with respect to macroscopic order parameters are explicitly obtained from the master equation that describes the microscopic-law. In the steady state, we show that the equations are identical to the saddle point equations for the equilibrium state of the same system. The equation for the dynamical Ising model is recovered in the classical limit. We also check the validity of the static approximation by making use of computer simulations for finite size systems and discuss several possible extensions of our approach to disordered spin systems for statistical-mechanical informatics. Especially, we shall use our procedure to evaluate the decoding process of Bayesian image restoration. With the assistance of the concept of dynamical replica theory (the DRT), we derive the zero-temperature flow equation of image restoration measure showing some 'non-monotonic' behaviour in its time evolution.

Variational Formulation for Quaternionic Quantum Mechanics

A quaternionic version of Quantum Mechanics is constructed using the Schwinger's formulation based on measurements and a Variational Principle. Commutation relations and evolution equations are provided, and the results are compared with other formulations.

Quantum Mechanical Version for Bessel Beam's Propagation in ABCD Optical System

By introducing the entangled state representation and the two-mode Fresnel operator we provide quantum mechanical version for Bessel beam's classical propagation in ABCD optical system. This provides the opportunity of studying various classical Fresnel transformations in the context of quantum optics.

Editorial: Special issue on quantum mechanical modeling of biological systems

In 1944 Schrodinger published a remarkable little book called “What is Life?” (Cambridge University Press). In it, the father of the wave-function version of quantum mechanics ranged widely over some of the big scientific and philosophical questions of the day and, indeed, of today. On his way to questions of the nature of consciousness and self, he introduced the notion of the genetic material being an “aperiodic solid” in which mutations could be related to some sort of quantum jump between different forms of the matter within.

Contradiction between Conservation Laws and Orthodox Quantum Mechanics

In this paper, it showed that the orthodox version of quantum mechanics contradicts the idea that conservation laws are valid in individual processes of measurement.

The classical basis for the κ-Poincaré Hopf algebra and doubly special relativity theories

Several issues concerning the quantum κ-Poincaré algebra are discussed and reconsidered here. We propose two different formulations of the κ-Poincaré quantum algebra. Firstly we present a complete Hopf algebra formula of κ-Poincaré in classical Poincaré basis. Further by adding one extra generator, which modifies the classical structure of the Poincaré algebra, we eliminate nonpolynomial functions in the κ-parameter. Hilbert space representations of such algebras make doubly special relativity (DSR) similar to Stueckelberg's version of (proper-time) relativistic quantum mechanics.

A simple proof of the Kochen‐Specker theorem on the problem of hidden variables

AbstractIn this contribution I present a simple derivation of an old result of Kochen and Specker, which is apparently unrelated to the famous work of Bell on hidden variables, but is presumably equally important. Kochen and Specker showed in 1967 that quantum mechanics cannot be embedded into a classical stochastic theory, provided the quantum theoretical probability distributions are reproduced and one additional highly desirable property is satisfied. This showed in a striking manner what were the difficulties in implementing the Einstein programme of a ‘complete’ version of quantum mechanics.

A simple proof of the Kochen-Specker theorem on the problem of hidden variables

In this talk I present a simple derivation of an old result of Kochen and Specker, which is apparently unrelated to the famous work of Bell on hidden variables, but is presumably equally important. Kochen and Specker showed in 1967 that quantum mechanics cannot be embedded into a classical stochastic theory, provided the quantum theoretical probability distributions are reproduced and one additional highly desirable property is satisfied. This showed in a striking manner what were the difficulties in implementing the Einstein programme of a `complete' version of quantum mechanics.

On physical and mathematical causality in quantum mechanics

This paper critically examines the view of quantum phenomena that has persisted since the introduction of quantum mechanics and that still remains prevalent in the foundational literature on quantum theory. According to this view, the independent behavior of quantum systems is causal, while the experimentally manifest lack of causality in observable quantum phenomena and, as a result, the probabilistic nature of our predictions concerning these phenomena are due to the disruption of this causal behavior by interfering with it through the measuring process. It appears that this view originates with P. A. M. Dirac and his work on the transformation theory (introduced by him and P. Jordan), which brought together W. Heisenberg's and E. Schrödinger's versions of quantum mechanics within a single scheme. Other founding figures of quantum theory, specifically N. Bohr, W. Heisenberg, and J. von Neumann, also advanced this view and helped to establish its prominence. The paper discusses these arguments and contends them to be insufficient to support the view that the independent behavior quantum systems is physically causal. It suggests that one can meaningfully speak of mathematical causality in quantum theory, and advocates an alternative, physically noncausal, interpretation of quantum mechanics.

NONCOMMUTATIVE QUANTUM MECHANICS AND QUANTUM COSMOLOGY

We present a phase-space noncommutative version of quantum mechanics and apply this extension to Quantum Cosmology. We motivate this type of noncommutative algebra through the gravitational quantum well (GQW) where the noncommutativity between momenta is shown to be relevant. We also discuss some qualitative features of the GQW such as the Berry phase. In the context of quantum cosmology we consider a Kantowski-Sachs cosmological model and obtain the Wheeler-DeWitt (WDW) equation for the noncommutative system through the ADM formalism and a suitable Seiberg-Witten (SW) map. The WDW equation is explicitly dependent on the noncommutative parameters, θ and η. We obtain numerical solutions of the noncommutative WDW equation for different values of the noncommutative parameters. We conclude that the noncommutativity in the momenta sector leads to a damped wave function implying that this type of noncommmutativity can be relevant for a selection of possible initial states for the universe.

De Broglie-Bohm, delayed-choice, double-slit experiment, and density matrix

The de Broglie-Bohm version of quantum mechanics is applied to the delayed-choice double-slit experiment. The role of the density matrix is considered.

Bohmian Mechanics and Quantum Information

Many recent results suggest that quantum theory is about information, and that quantum theory is best understood as arising from principles concerning information and information processing. At the same time, by far the simplest version of quantum mechanics, Bohmian mechanics, is concerned, not with information but with the behavior of an objective microscopic reality given by particles and their positions. What I would like to do here is to examine whether, and to what extent, the importance of information, observation, and the like in quantum theory can be understood from a Bohmian perspective. I would like to explore the hypothesis that the idea that information plays a special role in physics naturally emerges in a Bohmian universe.

On foundation of quantum physics

Some aspects of the interpretation of quantum theory are discussed. It is emphasized that quantum theory is formulated in the Cartesian coordinate system; in other coordinates the result obtained with the help of the Hamiltonian formalism and commutator relations between “canonically conjugated” coordinate and momentum operators leads to a wrong version of quantum mechanics. The origin of time is analyzed by the example of atomic collision theory in detail; it is shown that the time-dependent Schrödinger equation is meaningless since in the high-impact-energy limit it transforms into an equation with two time-like variables. Following the Einstein-Rozen-Podolsky experiment and Bell’s inequality, the wave function is interpreted as an actual field of information in the elementary form. The concept “measurement” is also discussed.

Adaption of Collins formula to fractional Fourier transform studied in the entangled state representation of quantum optics

Adaption of Collins diffraction formula to the fractional Fourier transform in complex form (FFT-C) is demonstrated in entangled state |η⟩ representation of quantum optics. The additivity property of FFT-C corresponding to Collins formula has been proved by using group multiplication of the Fresnel operator and the quantum mechanical version of the Collins formula.