Quantum Mechanical Counterpart Research Articles

Real Virtuality and Actual Transitions: Historical Reflections on Virtual Entities before Quantum Field Theory

Abstract This paper studies the notion of virtuality in the Bohr-Kramers-Slater theory of 1924. We situate the virtual entities of BKS within the tradition of the correspondence principle and the radiation theory of the Bohr model. We show how, in this context, virtual oscillators emerged as classical substitute radiators and were used to describe the otherwise elusive quantum transitions. They played an effective role in the quantum theory of radiation while remaining categorically distinct and ontologically separated from the quantum world of the Bohr model. The notion of virtuality thus differs markedly from its counterpart in quantum mechanics or QFT.

Spin Evolution in the Born‐Oppenheimer Approximation

AbstractThe first part of this article attempts to highlight in chronological order some prominent papers that have direct connexion with the Born‐Oppenheimer approximation. This timeline successively points to the role of level crossing, the notions of gauge field and geometrical or Berry phase, early experiments that were accounted for with that concept, and its major role in understanding anomalous transport in solids. The second part analyzes the time evolution of states under the adiabatic approximation when the states are spin degenerate. The geometrical phase thus found entails an electron spin evolution. Then, we examine the case of moleclar dynamics. Starting from the Born‐Oppenheimer approximation, we obtain a spin evolution by replacing the classical nuclear positions by their quantum mechanical counterparts in the terms that describe electrostatic interactions between nuclei and electrons.

Determinism In Quantum Slit-Experiments

A mathematical model for the slit experiments in the heart of quantum mechanics is developed to gain insight into quantum ‎theory. The proposed system-theoretical model is entirely based on commutative mathematics, i.e. convolution, and integral ‎transformations, and starts with spacetime functions with inherent energy-based cause and effect relations of the state-‎function Ѱ in the complex Hilbert space. The benefits of his approach are as 1-Invariance in time reversal. 2-Deterministic ‎result functions in the model in line with the outcome of slit experiments. 3- Separation of causality and cross-correlations of ‎attained states. 4- Disappearance of a posteriori probability of quantum states. 5- Quantum a priori fixed states after ‎causality interactions have ended, (even) when quanta are (light-years) separated. The model predicts the patterns in the ‎experiments with mathematical functions of the energy distributions. The quantum mechanical counterpart description of the ‎physical reality of slit experiments thus may be considered complete in A. Einstein’s definition. The patterns in double slit ‎experiments are found to be an effect of energy (amplitude-) modulation. An equivalent double-slit pattern can be retrieved ‎from an input modulated 1-slit experiment excluding interference interpretations. The system-theoretical model uses generic ‎properties of quanta and evolves into determinism in ‎quantum mechanics slit experiments. The mathematics in the model ‎handles beables by treatment ‎of momentum p in system theoretical I/O relations of the transformed functions and allows the ‎proposed description by the avoidance of a direct addressing of the individual quanta through variables. The following ‎method yields exact, non-probabilistic results.‎

Zitterbewegungof massless particles

Zitterbewegung of massless particles with an arbitrary spin is analyzed in various representations. Dynamics of the group velocity of a massless particle as a whole and of the corresponding radius vector is determined. This radius vector defines any fixed point of the envelope of the moving wave packet characterizing the particle and its group velocity differs from the group velocities of any points of the wavefront. We consider free massless scalar and Dirac particles, the photon, and massive and massless particles with an arbitrary spin and describe them in different representations. For particles with an arbitrary spin, the generalized Feshbach-Villars representation and the Foldy-Wouthuysen one are used. Zitterbewegung takes place in any representation except for the Foldy-Wouthuysen one. Formulas describing the motion of a ``trembling'' free particle are the same in any representation. In the Foldy-Wouthuysen representation, the operators of the velocity and momentum of a free particle are proportional and Zitterbewegung does not take place. Since the radius vector (position) and velocity operators are the quantum-mechanical counterparts of the classical position and velocity just in the Foldy-Wouthuysen representation, Zitterbewegung is not observable. The same conclusion has been previously made for free massive particles. For relativistic massive particles with spins 0, 1/2, 1 and massless particles with spins 0, 1/2 in arbitrarily strong electromagnetic fields, independent of the external fields Zitterbewegung does not appear in the Foldy-Wouthuysen representation either. This conclusion is made for leading terms in the Hamiltonian proportional to the zero and first powers of the Planck constant and for such terms proportional to ${\ensuremath{\hbar}}^{2}$ which describe contact interactions.

Benchmarking quantum Stirling and Otto cycles for an interacting spin system

A properly selected quantum heat engine consumes resources efficiently. Also, there are many parameters to determine the right option. In this study, we construct and compare the quantum mechanical counterparts of Stirling and Otto cycles for an experimentally achievable quantum working medium where two spins 1/2 interact through isotropic Heisenberg interaction in the tunable external magnetic field. We consider two comparison cases that operate heat cycles with the same system parameters or equal efficiencies. The quantum Stirling cycle produces more work than that of the Otto cycle under the same parameters or equal efficiency conditions. The quantum Stirling engine operates a wider range of coupling strengths. The work performed in the cycles is enhanced by coupling strength. On the other hand, we investigate the role of coupling strength on the performance of the non-regenerative quantum Stirling heat cycle under various magnitudes of the magnetic field. The distance between the applied magnetic fields at the initial and final points of the isothermal stage is found to be an enhancer for the work output and efficiency of the quantum Stirling cycle.

Dirac Electrons in a Planar Potential Well in a Magnetic Field

We study the Dirac equation in two spatial dimensions for quasiparticles in a potential well in graphene in a homogeneous magnetic field. It is shown that, at some critical value of the potential strength, the lowest empty energy level crosses a filled negative energy level leading to the instability of the system. The critical potential strength decreases with decrease of a quasiparticle gap and becomes zero in the gapless case. It is argued that the magnetically driven instability of a charged center can be considered as a quantum mechanical counterpart of the magnetic catalysis phenomenon in graphene.

On the Stochastic Mechanics Foundation of Quantum Mechanics

Among the famous formulations of quantum mechanics, the stochastic picture developed since the middle of the last century remains one of the less known ones. It is possible to describe quantum mechanical systems with kinetic equations of motion in configuration space based on conservative diffusion processes. This leads to the representation of physical observables through stochastic processes instead of self-adjoint operators. The mathematical foundations of this approach were laid by Edward Nelson in 1966. It allows a different perspective on quantum phenomena without necessarily using the wave-function. This article recaps the development of stochastic mechanics with a focus on variational and extremal principles. Furthermore, based on recent developments of optimal control theory, the derivation of generalized canonical equations of motion for quantum systems within the stochastic picture are discussed. These so-called quantum Hamilton equations add another layer to the different formalisms from classical mechanics that find their counterpart in quantum mechanics.

Exactly Solvable Model of Classical and Quantum Oscillators of Time Dependent Complex Frequencies: Squeezing Properties of Coherent Field

The exact analytical solution for a classical oscillator with time-dependent frequency is explored. We establish that the closed form solution for such an oscillator is possible provided the frequency is complex and obeying some stringent conditions. The complex nature of the frequency very often occurs in several physical problems. Therefore, the present model calculations are of relevant in fundamental and in practical situations. The solution of the classical oscillator of time-dependent complex frequency is used to obtain the solution of its quantum mechanical counterpart. The possibilities of getting the squeezed states and the nonclassical properties of the initial coherent radiation field coupled to the oscillator are discussed.

Weyl points and topological surface states in a three-dimensional sandwich-type elastic lattice

Following the realization of Weyl semimetals in quantum electronic materials, classical wave analogues of Weyl materials have also been theorized and experimentally demonstrated in photonics and acoustics. Weyl points in elastic systems, however, have been a much more recent discovery. In this study, we report on the design of an elastic fully-continuum three-dimensional material that, while offering structural and load-bearing functionalities, is also capable of Weyl degeneracies and surface topologically-protected modes in a way completely analogous to its quantum mechanical counterpart. The topological characteristics of the lattice are obtained by ab initio numerical calculations without employing any further simplifications. The results clearly characterize the topological structure of the Weyl points and are in full agreement with the expectations of surface topological modes. Finally, full field numerical simulations are used to confirm the existence of surface states and to illustrate their extreme robustness towards lattice disorder and defects.

Position and spin in relativistic quantum mechanics

The problem of the position and spin in relativistic quantum mechanics is analyzed in detail. It is definitively shown that the position and spin operators in the Foldy-Wouthuysen representation (but not in the Dirac one) are quantum-mechanical counterparts of the classical position and spin variables. The probabilistic interpretation is valid only for Foldy-Wouthuysen wave functions. The relativistic spin operators are discussed. The spin-orbit interaction does not exist for a free particle if the conventional operators of the orbital angular momentum and the rest-frame spin are used. Alternative definitions of the orbital angular momentum and the spin are based on noncommutative geometry, do not satisfy standard commutation relations, and can allow the spin-orbit interaction.

Zitterbewegung of Bosons

Zitterbewegung of massive and massless scalar bosons and a massive Proca (spin-1) boson is analyzed. The equations describing the evolution of the velocity and position of the scalar boson in the generalized Feshbach-Villars representation and the corresponding equations for the massive Proca particle in the Sakata-Taketani representation are equivalent to each other and to the well-known equations for the Dirac particle. However, Zitterbewegung does not appear in the Foldy-Wouthuysen representation. Since the position and velocity operators in the Foldy-Wouthuysen representation and their transforms to other representations are the quantum-mechanical counterparts of the corresponding classical variables, Zitterbewegung is not observable.

Maxwell's Demon in Quantum Mechanics.

Maxwell’s Demon is a thought experiment devised by J. C. Maxwell in 1867 in order to show that the Second Law of thermodynamics is not universal, since it has a counter-example. Since the Second Law is taken by many to provide an arrow of time, the threat to its universality threatens the account of temporal directionality as well. Various attempts to “exorcise” the Demon, by proving that it is impossible for one reason or another, have been made throughout the years, but none of them were successful. We have shown (in a number of publications) by a general state-space argument that Maxwell’s Demon is compatible with classical mechanics, and that the most recent solutions, based on Landauer’s thesis, are not general. In this paper we demonstrate that Maxwell’s Demon is also compatible with quantum mechanics. We do so by analyzing a particular (but highly idealized) experimental setup and proving that it violates the Second Law. Our discussion is in the framework of standard quantum mechanics; we give two separate arguments in the framework of quantum mechanics with and without the projection postulate. We address in our analysis the connection between measurement and erasure interactions and we show how these notions are applicable in the microscopic quantum mechanical structure. We discuss what might be the quantum mechanical counterpart of the classical notion of “macrostates”, thus explaining why our Quantum Demon setup works not only at the micro level but also at the macro level, properly understood. One implication of our analysis is that the Second Law cannot provide a universal lawlike basis for an account of the arrow of time; this account has to be sought elsewhere.

Dynamical stability of the weakly nonharmonic propeller-shaped planar Brownian rotator.

Dynamical stability is a prerequisite for control and functioning of desired nanomachines. We utilize the Caldeira-Leggett master equation to investigate dynamical stability of molecular cogwheels modeled as a rigid, propeller-shaped planar rotator. To match certain expected realistic physical situations, we consider a weakly nonharmonic external potential for the rotator. Two methods for investigating stability are used. First, we employ a quantum-mechanical counterpart of the so-called "first passage time" method. Second, we investigate time dependence of the standard deviation of the rotator for both the angle and angular momentum quantum observables. A perturbationlike procedure is introduced and implemented to provide the closed set of differential equations for the moments. Extensive analysis is performed for different combinations of the values of system parameters. The two methods are, in a sense, mutually complementary. Appropriate for the short time behavior, the first passage time exhibits a numerically relevant dependence only on the damping factor as well as on the rotator size. However, the standard deviations for both the angle and angular momentum observables exhibit strong dependence on the parameter values for both short and long time intervals. Contrary to our expectations, the time decrease of the standard deviations is found for certain parameter regimes. In addition, for certain parameter regimes nonmonotonic dependence on the rotator size is observed for the standard deviations and for the damping of the oscillation amplitude. Hence, nonfulfillment of the classical expectation that the size of the rotator can be reduced to the inertia of the rotator. In effect, the task of designing the desired protocols for the proper control of the molecular rotations becomes an optimization problem that requires further technical elaboration.

Representation and Reality by Language: How to Make a Home Quantum Computer?

A set theory model of reality, representation and language based on the relation of completeness and incompleteness is explored. The problem of completeness of mathematics is linked to its counterpart in quantum mechanics. That model includes two Peano arithmetics or Turing machines independent of each other. The complex Hilbert space underlying quantum mechanics as the base of its mathematical formalism is interpreted as a generalization of Peano arithmetic: It is a doubled infinite set of doubled Peano arithmetics having a remarkable symmetry to the axiom of choice. The quantity of information is interpreted as the number of elementary choices (bits). Quantum information is seen as the generalization of information to infinite sets or series. The equivalence of that model to a quantum computer is demonstrated. The condition for the Turing machines to be independent of each other is reduced to the state of Nash equilibrium between them. Two relative models of language as game in the sense of game theory and as ontology of metaphors (all mappings, which are not one-to-one, i.e. not representations of reality in a formal sense) are deduced.

Fundamental operators in Dirac quantum mechanics

Old achievements and more recent results in a solution of problem of the position and spin in relativistic quantum mechanics are considered. It is definitively shown that quantum-mechanical counterparts of the classical position and spin variables are the position and spin operators in the Foldy-Wouthuysen representation (but not in the Dirac one). The probabilistic interpretation is valid only for Foldy-Wouthuysen wave functions.

Zitterbewegung in quantum mechanics of Proca particles

Zitterbewegung of a massive Proca (spin-1) boson is analyzed. The equations of motion of a massive Proca particle in the Sakata-Taketani representation are equivalent to the corresponding equations for the Dirac particle. However, Zitterbewegung does not appear in the Foldy-Wouthuysen representation. Zitterbewegung is not observable because the quantum-mechanical counterparts of the classical position and velocity are the position and velocity operators in the Foldy-Wouthuysen representation and their transforms to other representations.

The quantum thermodynamic force responsible for quantum state transformation and the flow and backflow of information

Why do quantum evolutions occur and why do they stop at certain points? In classical thermodynamics affinity was introduced to predict in which direction an irreversible process proceeds. In this paper the quantum mechanical counterpart of the classical affinity is found. It is shown that the quantum version of affinity can predict in which direction a process evolves. A new version of the second law of thermodynamics is derived through quantum affinity for energy-incoherent state interconversion under thermal operations. we will also see that the quantum affinity can be a good candidate to be responsible, as a force, for driving the flow and backflow of information in Markovian and non-Markovian evolutions. Finally we show that the rate of quantum coherence can be interpreted as the pure quantum mechanical contribution of the total thermodynamic force and flow. Thus it is seen that, from a thermodynamic point of view, any interaction from the outside with the system or any measurement on the system may be represented by a quantum affinity.

Covariant projective representation of symplectic group on discrete phase space

The phase point operator Δ(q,p) is the quantum mechanical counterpart of the classical phase point (q,p). The discrete form of Δ(q,p) was formulated for an odd number of lattice points by Cohendet et al. and for an even number of lattice points by Leonhardt. Both versions have symplectic covariance, which is of fundamental importance in quantum mechanics. However, an explicit form of the projective unitary representation of the symplectic group that appears in the covariance relation is not yet known. We show in this paper the existence and uniqueness of the representation, and describe a method to construct it using the Euclidean algorithm.

Quantum backflow in solutions to the Dirac equation of the spin-1 2 free particle

It was known that a free, non-relativistic particle in a superposition of positive momenta can, in certain cases, bear a negative probability current — hence termed quantum backflow. Here, it is shown that more variations can be brought about for a free Dirac particle, particularly when negative-energy solutions are taken into account. Since any Dirac particle can be understood as an antiparticle that acts oppositely (and vice versa), quantum backflow is found to arise in the superposition (i) of a well-defined momentum but different signs of energies, or more remarkably (ii) of different signs of both momenta and energies. Neither of these cases has a counterpart in non-relativistic quantum mechanics. A generalization by using the field-theoretic formalism is also presented and discussed.

Semiclassical transition state theory based on fourth order vibrational perturbation theory: Model system studies beyond symmetric Eckart barrier.

The following one-dimensional model potential barriers are studied to compute the tunneling probabilities using semiclassical transition state theory (SCTST) based on fourth order vibrational perturbation theory (VPT4): (i) Morse barrier, (ii) asymmetric Eckart barrier, (iii) adjacent well barrier, and (iv) first order approximation to the symmetric Eckart barrier. The current investigation extends a previous study performed for the symmetric Eckart barrier. The performance of VPT4-SCTST is investigated for a broad range of energies, including the very deep tunneling regime (approximately 1% of the barrier height), and the results are compared to the exact quantum mechanical treatment, as well as to the popular second order vibrational perturbation theory (VPT2)-SCTST approach. While the VPT4-SCTST results are in excellent agreement with their quantum mechanical counterparts for a wide range of energies above and below the barrier, it is found for systems (ii) and (iv) that higher order terms (in ) in energy expansion are important for similarly excellent agreement in the deep tunneling regime. For the asymmetric Eckart barrier, the convergence properties of SCTST are analyzed in terms of its analytically known Siegert eigenvalues. A simple empirical modification of VPT2-SCTST advocated in the aforementioned previous work for the symmetric Eckart barrier is also tested here for the new model systems, and it is shown not to provide a uniformly systematic improvement.