Variables In Quantum Mechanics Research Articles

Smart nanomaterials to support quantum-sensing electronics

Quantum sensing electronic sensing is opening up new cutting-edge possibilities to take advantage of complex quantum mechanical variables to make incredibly sensitive assessments of an array of parameters. Concurrently, there are new prospects for quantum sensing electronics to be used to enhance the processes involved in creating, distributing, and utilizing energy. To use this technology efficiently, it is necessary to handle issues related to sensing material such as stability and operation, reliable monitoring, and precision in sensing and assessment. In this direction, the present is an overview of existing and new quantum sensing materials and methods, along with the related sensing frameworks that have been created to support their advanced translational applications. Regarding next-generation sensing technologies specifically, the realization of a previously unheard degree of sensitivity is made possible using quantum methods and materials. More specifically, this article addresses how quantum sensing materials may lead to greater efficiency while highlighting established and developing quantum sensing methods, materials, and sensing platforms. The discussion of potential chances to implement quantum technologies follows a summary of the remaining obstacles and difficulties for quantum sensor deployment.

Comment on the subtlety of defining a real-time path integral in lattice gauge theories

Abstract Recently, Hoshina, Fujii, and Kikukawa pointed out that the naive lattice gauge theory action in the Minkowski signature does not result in a unitary theory in the continuum limit, and Kanwar and Wagman proposed alternative lattice actions to the Wilson action without divergences. We show here that the subtlety can be understood from the asymptotic expansion of the modified Bessel function, which has been discussed for the path integral of compact variables in nonrelativistic quantum mechanics. The essential ingredient for defining the appropriate continuum theory is the iε prescription, and with the proper implementation of this we show that the Wilson action can be used for real-time path integrals. It is important that iε should be implemented for both timelike and spacelike plaquettes. We also suggest why iε is required for the Wilson action from the Hamiltonian formalism: it is needed to manifestly suppress the contributions from singular paths, for which the Wilson action can give different values from those of the actual continuum action.

Corrected Maslov index for complex saddle trajectories.

Saddle-point approximations, extremely important in a wide variety of physical contexts, require the analytical continuation of canonically conjugate quantities to complex variables in quantum mechanics. An important component of this approximation's implementation is arriving at the phase correction attributable to caustics, which involves determinantal prefactors. The common prescription of using the inverse of half a certain determinant's total accumulated phase sometimes leads to sign errors. The root of this problem is traced to the zeros of the determinants at complex times crossing the real time axis. Deformed complex time contours around the zeros can repair the sign errors that sometimes occur, but a much more practical way is given that links saddles back to associated real trajectories and avoids the necessity of locating the complex time zeros of the determinants.

On the discrete Heisenberg group and commutative modular variables in quantum mechanics: II. Synchronization of unitary actions and homological Abelianization

On the discrete Heisenberg group and commutative modular variables in quantum mechanics: II. Synchronization of unitary actions and homological Abelianization

On the discrete Heisenberg group and commutative modular variables in quantum mechanics: I. The Abelian symplectic shadow and integrality of area

On the discrete Heisenberg group and commutative modular variables in quantum mechanics: I. The Abelian symplectic shadow and integrality of area

A discussion about the hidden variables in quantum mechanics

In the present paper, we attempt to bring to light some of the hidden variables of quantum mechanics by continuing the work of others who introduced constant curvature to spacetime. The article is also a continuation of a previous one [S. Koutandos, Phys. Essays 33, 208 (2020)] published in this journal proving London equations. We find that there is dilation of volume and that mass may have a surface density. Unlike the case of superconductors, however these surfaces collapse to a point. This is where the difficulty in measurement comes from.

The Case of Quantum Mechanics Mathematizing Reality: The ‘Superposition’ of Mathematically Modeled and Mathematical Reality: Is There Any Room for Gravity?

A case study of quantum mechanics is investigated in the framework of the philosophical opposition “mathematical model – reality”. All classical science obeys the postulate about the fundamental difference of model and reality, and thus distinguishing epistemology from ontology fundamentally. The theorems about the absence of hidden variables in quantum mechanics imply for it to be “complete” (versus Einstein’s opinion). That consistent completeness (unlike arithmetic to set theory in the foundations of mathematics in Godel’s opinion) can be interpreted furthermore as the coincidence of model and reality. The paper discusses the option and fact of that coincidence it its base: the fundamental postulate formulated by Niels Bohr about what quantum mechanics studies (unlike all classical science). Quantum mechanics involves and develops further both identification and disjunctive distinction of the global space of the apparatus and the local space of the investigated quantum entity as complementary to each other. This results into the analogical complementarity of model and reality in quantum mechanics. The apparatus turns out to be both absolutely “transparent” and identically coinciding simultaneously with the reflected quantum reality. Thus, the coincidence of model and reality is postulated as necessary condition for cognition in quantum mechanics by Bohr’s postulate and further, embodied in its formalism of the separable complex Hilbert space, in turn, implying the theorems of the absence of hidden variables (or the equivalent to them “conservation of energy conservation” in quantum mechanics). What the apparatus and measured entity exchange cannot be energy (for the different exponents of energy), but quantum information (as a certain, unambiguously determined wave function) therefore a generalized law of conservation, from which the conservation of energy conservation is a corollary. Particularly, the local and global space (rigorously justified in the Standard model) share the complementarity isomorphic to that of model and reality in the foundation of quantum mechanics. On that background, one can think of the troubles of “quantum gravity” as fundamental, direct corollaries from the postulates of quantum mechanics. Gravity can be defined only as a relation or by a pair of non-orthogonal separable complex Hilbert space attachable whether to two “parts” or to a whole and its parts. On the contrary, all the three fundamental interactions in the Standard model are “flat” and only “properties”: they need only a single separable complex Hilbert space to be defined.

Quantum-Information Conservation. The Problem About ‘Hidden Variables’, or the ‘Conservation of Energy Conservation’ in Quantum Mechanics: A Historical Lesson for Future Discoveries

The explicit history of the “hidden variables” problem is well-known and established. The main events of its chronology are traced. An implicit context of that history is suggested. It links the problem with the “conservation of energy conservation” in quantum mechanics. Bohr, Kramers, and Slaters (1924) admitted its violation being due to the “fourth Heisenberg uncertainty”, that of energy in relation to time. Wolfgang Pauli rejected the conjecture and even forecast the existence of a new and unknown then elementary particle, neutrino, on the ground of energy conservation in quantum mechanics, afterwards confirmed experimentally. Bohr recognized his defeat and Pauli’s truth: the paradigm of elementary particles (furthermore underlying the Standard model) dominates nowadays. However, the reason of energy conservation in quantum mechanics is quite different from that in classical mechanics (the Lie group of all translations in time). Even more, if the reason was the latter, Bohr, Cramers, and Slatters’s argument would be valid. The link between the “conservation of energy conservation” and the problem of hidden variables is the following: the former is equivalent to their absence. The same can be verified historically by the unification of Heisenberg’s matrix mechanics and Schrodinger’s wave mechanics in the contemporary quantum mechanics by means of the separable complex Hilbert space. The Heisenberg version relies on the vector interpretation of Hilbert space, and the Schrodinger one, on the wave-function interpretation. However the both are equivalent to each other only under the additional condition that a certain well-ordering is equivalent to the corresponding ordinal number (as in Neumann’s definition of “ordinal number”). The same condition interpreted in the proper terms of quantum mechanics means its “unitarity”, therefore the “conservation of energy conservation”. In other words, the “conservation of energy conservation” is postulated in the foundations of quantum mechanics by means of the concept of the separable complex Hilbert space, which furthermore is equivalent to postulating the absence of hidden variables in quantum mechanics (directly deducible from the properties of that Hilbert space). Further, the lesson of that unification (of Heisenberg’s approach and Schrodinger’s version) can be directly interpreted in terms of the unification of general relativity and quantum mechanics in the cherished “quantum gravity” as well as a “manual” of how one can do this considering them as isomorphic to each other in a new mathematical structure corresponding to quantum information. Even more, the condition of the unification is analogical to that in the historical precedent of the unifying mathematical structure (namely the separable complex Hilbert space of quantum mechanics) and consists in the class of equivalence of any smooth deformations of the pseudo-Riemannian space of general relativity: each element of that class is a wave function and vice versa as well. Thus, quantum mechanics can be considered as a “thermodynamic version” of general relativity, after which the universe is observed as if “outside” (similarly to a phenomenological thermodynamic system observable only “outside” as a whole). The statistical approach to that “phenomenological thermodynamics” of quantum mechanics implies Gibbs classes of equivalence of all states of the universe, furthermore re-presentable in Boltzmann’s manner implying general relativity properly … The meta-lesson is that the historical lesson can serve for future discoveries.

Hardy's Paradox as a No-go Result for Probabilistic Hidden Variables

Hardy presented a famous argument that shows that there exist no hidden variables in quantum mechanics. His argument is called Hardy's paradox. The commentaries and discussions on Hardy's paradox have been ordinarily presented for deterministic hidden variables, and it has not been discussed whether a similar paradox will occurs in probabilistic hidden variables. So, first, I will show that Hardy's argument can also be applied to probabilistic hidden variables. After that, I discuss the common cause of a correlation, which has been actively discussed in the philosophy of science in recent years. That is because requiring the common cause of a quantum correlation is mathematically equivalent to introducing a probabilistic hidden variable to explain the correlation. Some researchers argued that in the non-existence proof of hidden variables, the existence of a common common-cause to multiple correlations is tacitly assumed, however that is unreasonably strong. According to their idea, it is sufficient if there exists a respective common-cause of each correlation. However, it will be revealed that at least as far as Hardy's paradox is concerned, merely weakening the common common-cause requirement to the respective common-cause requirement does not help to construct a local model.

Quaternionic quantization principle in general relativity and supergravity

A generalized quantization principle is considered, which incorporates nontrivial commutation relations of the components of the variables of the quantized theory with the components of the corresponding canonical conjugated momenta referring to other space–time directions. The corresponding commutation relations are formulated by using quaternions. At the beginning, this extended quantization concept is applied to the variables of quantum mechanics. The resulting Dirac equation and the corresponding generalized expression for plane waves are formulated and some consequences for quantum field theory are considered. Later, the quaternionic quantization principle is transferred to canonical quantum gravity. Within quantum geometrodynamics as well as the Ashtekar formalism, the generalized algebraic properties of the operators describing the gravitational observables and the corresponding quantum constraints implied by the generalized representations of these operators are determined. The generalized algebra also induces commutation relations of the several components of the quantized variables with each other. Finally, the quaternionic quantization procedure is also transferred to [Formula: see text] supergravity. Accordingly, the quantization principle has to be generalized to be compatible with Dirac brackets, which appear in canonical quantum supergravity.

Fundamental mechanisms of the dependence of spin states on exchange correlations in quantum dots: Feynman path integrals

The effect of permutation symmetry of nonrelativistic electron wave functions in quantum dots on the spatial correlations, energy, and distribution over the spin states is investigated. The quantitative analysis is based on ab initio calculations using the technique of Feynman path integrals of entangled quantum states in model spherically symmetrical quantum dots with the parabolic confining field. In the computer simulation, the quantum-mechanical indistinguishability and spin variable are exactly taken into account. It is demonstrated that the detailed description of the high-order spatial correlations plays a key role in the correct calculation of the equilibrium spin number and energy values.

Two-Valued Probability Measure on the Pontryagin Space

The well known Kochen-Specker’s theorem is devoted to the problem of hidden variables in quantum mechanics. The Kochen-Specker theorem says: There is no two-valued probability measure on the real Hilbert space of dimension three. In the paper we present an analogy of Kochen-Specker’s theorem in Pontryagin space: A Pontryagin spase H of dimension greater than or equal to three has a two-valued probability measure if and only if H has indefinite rank one: in which case, any such two-valued probability measure on H is unique.

An uncertainty inequality

Recently, the authors derived an uncertainty relationship between noise and spatial resolution in linear computational imaging systems that is similar to the Heisenberg uncertainty principle for conjugate variables in quantum mechanics. Specifically, for linear shift-invariant systems with a fixed incident photon density, the product of noise and spatial resolution has a positive absolute lower limit which can be evaluated with the aid of the inequality ‖ g ‖ p + 1 a ( p + 1 ) ‖ | x | a g ‖ 1 n p ‖ g ‖ 1 a ( p + 1 ) + n p ≥ C ( n , p , a ) , p > 0 ; a > 0 , where g ∈ L 1 ( R n ) ∩ L 1 ( R n , | x | a ) ∩ L p + 1 ( R n ) . This inequality, which is derived in this note, also has applications in data smoothing and in variations of Heisenberg’s inequality.

Kochen-Specker Theorem in Krein Space

The well known Kochen-Specker’s theorem (Kochen and Specker J. Math. Mech. 17:59–87, 1967) is devoted to the problem of hidden variables in quantum mechanics. In the paper we present a geometric proof for an indefinite analogy of Kochen-Specker’s theorem. On the real three-dimensional Krein space there exists unique two-valued probability measure.

Dirac's equation and the nature of quantum field theory

This paper re-examines the key aspects of Dirac's derivation of his relativistic equation for the electron in order advance our understanding of the nature of quantum field theory. Dirac's derivation, the paper argues, follows the key principles behind Heisenberg's discovery of quantum mechanics, which, the paper also argues, transformed the nature of both theoretical and experimental physics vis-à-vis classical physics and relativity. However, the limit theory (a crucial consideration for both Dirac and Heisenberg) in the case of Dirac's theory was quantum mechanics, specifically, Schrödinger's equation, while in the case of quantum mechanics, in Heisenberg's version, the limit theory was classical mechanics. Dirac had to find a new equation, Dirac's equation, along with a new type of quantum variables, while Heisenberg, to find new theory, was able to use the equations of classical physics, applied to different, quantum-mechanical variables. In this respect, Dirac's task was more similar to that of Schrödinger in his work on his version of quantum mechanics. Dirac's equation reflects a more complex character of quantum electrodynamics or quantum field theory in general and of the corresponding (high-energy) experimental quantum physics vis-à-vis that of quantum mechanics and the (low-energy) experimental quantum physics. The final section examines this greater complexity and its implications for fundamental physics.

Strong correspondence principle for joint measurement of conjugate observables

It is demonstrated that the the statistics for a joint measurement of two conjugate variables in Quantum Mechanics are expressed through an equation identical to the classical one, provided that joint classical probabilities are substituted by Wigner functions and that the interaction between system and detectors is accounted for. This constitutes an extension of Ehrenfest correspondence principle, and it is thereby dubbed strong correspondence principle. Furthermore, it is proved that the detectors provide an additive term to all the cumulants, and that if they are prepared in a Gaussian state they only contribute to the first and second cumulant.

Supersymmetrical Separation of Variables in Two-Dimensional Quantum Mechanics

Two different approaches are formulated to analyze two-dimensional quantum models which are not amenable to standard separation of variables. Both methods are essentially based on supersymmetrical second order intertwining relations and shape invariance - two main ingredients of the supersymmetrical quantum mechanics. The first method explores the opportunity to separate variables in the supercharge, and it allows to find a part of spectrum of the Schr\"odinger Hamiltonian. The second method works when the standard separation of variables procedure can be applied for one of the partner Hamiltonians. Then the spectrum and wave functions of the second partner can be found. Both methods are illustrated by the example of two-dimensional generalization of Morse potential for different values of parameters.

Nonequilibrium, steady-state electron transport with N-representable density matrices from the anti-Hermitian contracted Schrödinger equation

We study molecular conductivity for a one-electron, bath-molecule-bath model Hamiltonian. The primary quantum-mechanical variable is the one-electron reduced density matrix (1-RDM). By identifying similarities between the steady-state Liouville equation and the anti-Hermitian contracted Schrödinger equation (ACSE) [D. A. Mazziotti, Phys. Rev. A 75, 022505 (2007)], we develop a way of enforcing nonequilibrium, steady-state behavior in a time-independent theory. Our results illustrate the relationship between current and voltage in molecular junctions assuming that the total number of electrons under consideration can be fixed across all driving potentials. The impetus for this work is a recent study by Subotnik et al. that also uses the 1-RDM to study molecular conductivity under different assumptions regarding the total number of electrons [J. E. Subotnik et al., J. Chem. Phys. 130, 144105 (2009)]. Unlike calculations in the previous study, our calculations result in 1-RDMs that are fully N-representable. The present work maintains N-representability through a bath-bath mixing that is related to a time-independent relaxation of the baths in the absence of the molecule, as governed by the ACSE. A lack of N-representability can be important since it corresponds to occupying energy states in the molecule or baths with more than one electron or hole (the absence of an electron) in violation of the Pauli principle. For this reason the present work may serve as an important, albeit preliminary, step in designing a 2-RDM/ACSE method for studying steady-state molecular conductivity with an explicit treatment of electron correlation.

Description of Multipole inf-Electron Systems

A systematic description of multipole degrees of freedom is discussed on the basis of the Stevens' operator-equivalent technique. The generalized Stevens' multiplicative factors are derived for all of the electric and the magnetic multipoles relevant to f-electron systems. With extensive use of the Stevens' factors, we express the spatial dependences of the electric and the magnetic fields, and the electric and the magnetic charge densities of localized f electrons. The latter is utilized to draw wave functions including their magnetic profile in addition to their shape with the charge density. The definite relation between the operators as quantum-mechanical variables in a multipole exchange model and the multipole moments in expansion of electromagnetic fields is given. The general treatments for the exchange model with the RPA susceptibility and the Ginzburg-Landau free-energy expansion are discussed, using CexLa1-xB6 as a typical example. The representative formula of the vector spherical harmonics are summarized, which are suitable basis for vector fields in the spherical expansion.

Distortion of frustrated spin cluster caused by quantum fluctuation due to magneto-elasticcoupling

We study the effects of magneto-elastic coupling on frustrated spin states of theantiferromagnetic Heisenberg model on a regular triangle and tetrahedron made of spin1/2 or 1. Displacement of spin is considered as a quantum-mechanical variable. Distortion ofclusters occurs through quantum fluctuation of normal modes caused by magneto-elasticcoupling. The nonmagnetic phase transitions of vanadium and chromium spinels arediscussed.