Formulation Of Quantum Mechanics Research Articles

Minkowski Space from Quantum Mechanics

Penrose’s Spin Geometry Theorem is extended further, from SU(2) and E(3) (Euclidean) to E(1, 3) (Poincaré) invariant elementary quantum mechanical systems. The Lorentzian spatial distance between any two non-parallel timelike straight lines of Minkowski space, considered to be the centre-of-mass world lines of E(1, 3)-invariant elementary classical mechanical systems with positive rest mass, is expressed in terms of E(1, 3)-invariant basic observables, viz. the 4-momentum and the angular momentum of the systems. An analogous expression for E(1, 3)-invariant elementary quantum mechanical systems in terms of the basic quantum observables in an abstract, algebraic formulation of quantum mechanics is given, and it is shown that, in the classical limit, it reproduces the Lorentzian spatial distance between the timelike straight lines of Minkowski space with asymptotically vanishing uncertainty. Thus, the metric structure of Minkowski space can be recovered from quantum mechanics in the classical limit using only the observables of abstract quantum mechanical systems.

Q-Deformed quantum mechanics related to the Tamm-Dancoff oscillator algebra and some physical applications

In this paper, we consider a system of the q-deformed bosonic Tamm-Dancoff oscillators, whose spectrum has some exponential cutoff factors at high energies. We first investigate the q-calculus in the Tamm-Dancoff (TD) boson algebra, and within this framework, the q-derivative, q-integral and q-exponential function are introduced. Using these properties, we construct a new formalism for the q-deformed quantum mechanics, which accordingly involve the q-adjoint operator and the q-Hermitian operator properties. We then derive the q-deformed Heisenberg relation, and develop the q-Hermitian momentum operator. The q-deformed Schrödinger equation is introduced, and as applications, we study the momentum eigenfunction and one-dimensional box problem. Another application of the TD type deformation onto lattice oscillations is also discussed through a model of the q-deformed Debye solid. Finally, other potential applications of the TD-oscillators gas model are concisely pointed out.

Foundations of Quantum Information for Physical Chemistry.

Quantum information, a field in which great advances have been made in the past decades, now presents opportunities for advanced chemistry. One roadblock to progress, especially for experimental chemical science, is that new concepts and technical definitions need to be learned. In this paper, we review some basic, but sometimes misunderstood, concepts of quantum information based on the mathematical formulation of quantum mechanics that will be useful for chemists interested in discovering ways that chemistry can contribute to the quantum information field. We cover topics including qubits and their density matrix formalism, quantum measurement as a quantum operation, information theory, and entanglement. We focus on the difference between the concepts in the quantum context and the classic context. We also discuss the relation and distinction among entanglement, correlation, and coherence. We aim to clarify the rigorous definition of these concepts and then indicate some examples in physical chemistry.

Dirac’s Approach to Quantum Mechanics in Physics Teacher Education: From Linear to Circular Polarisation

The teaching/learning of quantum mechanics via two-state systems is continuously spreading in secondary schools due to its promising results in physics education research. One possibility is the use of polarisation states of photons. This paper reports on a polarisation-based introduction to quantum mechanics in physics teacher education. A widely used school material prepares teacher trainees for their future work and also improves their conceptual knowledge. This part includes statistical calculations using only secondary school mathematics and a new formulation of the uncertainty relation, using only real numbers. The second step is to prepare the formalism of quantum mechanics using real two-dimensional vectors and matrices. Considering that students may not learn complex linear algebra, we offer a new way to introduce the complete formalism of two-state systems via circular polarisation providing a step-by-step exploration of complex quantum states. This points out the advantage of using complex linear algebra via a physical example, providing the opportunity to reach the elements of advanced quantum physics and quantum computing while deepening the physics background of the secondary school material.

О расширенном пространстве Гильберта

The construction of the extended Hilbert space is presented in the form of a direct sum of the spaces of vectors of finite and infinite norms as the main space in the mathematical formalism of quantum mechanics of a many-electron atom. On the example of constructing the analytical structure of the probability amplitude 1s-3p of photoexcitation of a neon atom, the implementation of the EHS-construct for solving the equations of the self-consistent Hartree-Fock field is given.

On the emergent “Quantum” theory in complex adaptive systems

We explore the concept of emergent quantum-like theory in complex adaptive systems, and examine in particular the concrete example of such an emergent (or “mock”) quantum theory in the Lotka–Volterra system. In general, we investigate the possibility of implementing the mathematical formalism of quantum mechanics on classical systems, and what would be the conditions for using such an approach. We start from a standard description of a classical system via Hamilton–Jacobi (HJ) equation and reduce it to an effective Schrodinger-type equation, with a (mock) Planck constant ▪ , which is system-dependent. The condition for this is that the so-called quantum potential VQ, which is state-dependent, is canceled out by some additional term in the HJ equation. We consider this additional term to provide for the coupling of the classical system under consideration to the ‘environment’. We assume that a classical system could cancel out the VQ term (at least approximately) by fine tuning to the environment. This might provide a mechanism for establishing a stable, stationary states in (complex) adaptive systems, such as biological systems. In particular, we present a general argument as to why the non-equilibrium dynamics of a classical system could lead to a mock quantum description that ensures stability compatible with adaptability. In this context we emphasize the state dependent nature of the mock quantum dynamics and we also introduce the new concept of the mock quantum, state dependent, statistical field theory. We also discuss some universal features of the quantum-to-classical as well as the mock-quantum-to-classical transition found in the turbulent phase of the hydrodynamic formulation of our proposal. In this way we re-frame the concept of decoherence into the concept of ‘quantum turbulence’, i.e. that the transition between quantum and classical could be defined in analogy to the transition from laminar to turbulent flow in hydrodynamics.

Instability and quantization in quantum hydrodynamics

We show how the quantum hydrodynamical formulation of quantum mechanics converts the nonlocality in the standard wave-like description of quantum systems by an instability of the quantum system, which opens the door to a new way for studying quantum systems based on known methodologies for studying the stability of fluids. As a second result, we show how the Madelung equations describe quantized energies without any external quantization conditions.

The quantum density of states and distribution functions of the helium-3: Wigner approach in path integral Monte Carlo simulations

A new path integral representation of the quantum density of states (DOS) and distribution functions of strongly correlated fermions are derived in the Wigner formulation of quantum mechanics. A new path integral Monte Carlo approach to calculate DOS and thermodynamic functions is suggested. Using 3 He as an interesting example, we calculate the DOS, internal energy distribution (IED), momentum distribution functions (MDFs), spin-resolved radial distribution functions at different densities and temperatures. The physical meaning and parameters of exchange-correlation holes, the quantum oscillations of IEDs and DOS as well as the high-momentum asymptotics (‘quantum tails’) of MDFs have been considered and explained.

The mean square displacement of a ballistic quantum particle

A commonly used quantum mechanical formulation of the mean square displacement δ x 2 ( t ) is based on the quantum correlation function of the position operator. While this quantity yields the classically expected result δ x 2 ( t ) ∝ t 2 for a ballistic particle in the interval topology, it is found here to diverge at essentially all times, when evaluated on infinitely large rings. A somewhat different formulation of the mean square displacement in quantum mechanics was proposed in a previous work (R. Marquardt, Mol. Phys. 119, e1971315 (2021)) and yielded the result δ x 2 ( t ) ∝ t for the ballistic particle on an infinitely large ring. Here, it is shown analytically that that formulation yields δ x 2 ( t ) ≡ 0 for the ballistic particle on an infinitely long interval. The two formulations define two different, topology dependent quantities that have the dimension of an area and that could in principle be determined from the same experiment, following a measurement idea proposed in the aforementioned work. That idea is critically reviewed here.

Phase-space path-integral representation of the quantum density of states: Monte Carlo simulation of strongly correlated soft-sphere fermions.

The Wigner formulation of quantum mechanics is used to derive a path-integral representation of the quantum density of states (DOS) of strongly correlated fermions in the canonical ensemble. A path-integral Monte Carlo approach for the simulation of DOS and other thermodynamic functions is suggested. The derived Wigner function in the phase space resembles the Maxwell-Boltzmann distribution but allows for quantum effects. We consider a three-dimensional quantum system of strongly correlated soft-sphere fermions at different densities and temperatures. The calculated properties include the DOS, momentum distribution functions, spin-resolved radial distribution functions, potentials of mean force, and related energy levels obtained from the Bohr-Sommerfeld condition. We observe sharp peaks on DOS and momentum distribution curves, which are explained by the appearance of fermionic bound states.

Homological quantum mechanics

We provide a formulation of quantum mechanics based on the cohomology of the Batalin-Vilkovisky (BV) algebra. Focusing on quantum-mechanical systems without gauge symmetry we introduce a homotopy retract from the chain complex of the harmonic oscillator to finite-dimensional phase space. This induces a homotopy transfer from the BV algebra to the algebra of functions on phase space. Quantum expectation values for a given operator or functional are computed by the function whose pullback gives a functional in the same cohomology class. This statement is proved in perturbation theory by relating the perturbation lemma to Wick’s theorem. We test this method by computing two-point functions for the harmonic oscillator for position eigenstates and coherent states. Finally, we derive the Unruh effect, illustrating that these methods are applicable to quantum field theory.

Quantum Evolution and Genetic Mutations

One of the important problems in quantum biology has been explained in this work. It is the main reason for genetic mutation, which is known as the origin of evolution in organisms. An approach to clarify the main reason for mutation is quantum mechanics. A brief history of genetics, new-Darwinism, and Mendelian inheritance from ancient times to the twentieth century has been presented. The meaning and location of chromosomes and genes in cells, DNA structure, the replication mechanism, the genetic code, tautomeric forms, and the application of quantum mechanics have been described. Finally, we present the importance of quantum tunnelling in playing a main role in the mutation.

Probabilistic Interpretation of Entrepreneurial Opportunity Using the Many-Worlds Model

This research engages with the uncertainties and unknowable in entrepreneurship with literature on the philosophy of quantum mechanics in an analogical way. Often characterized as a journey into an uncertain future, entrepreneurship stands out as a dynamic and unpredictable domain driven by a compelling and indefinable force. This research’s central argument revolves around the notion that within the entrepreneurial landscape, many latent propensities exist, each carrying its distinct probability of manifesting into various outcomes. Unlike a deterministic process with a single predetermined outcome, entrepreneurship is a complex interplay of potentialities. To capture this multifaceted perspective, we leverage the principles and formalism of quantum mechanics, renowned for its emphasis on potentiality over actuality until subjected to observation and action. This philosophical framework describes the inherent potentialities embedded within entrepreneurial opportunities. This study introduces a groundbreaking Many-Worlds model inspired by the interpretations of quantum mechanics. In this model, the act of observation serves as a catalyst, unraveling a spectrum of potential outcomes, each characterized by its probability. These potentialities materialize based on entrepreneurs’ intentional actions and decisions, reflecting their commitment to shaping the future of their ventures. In summary, our study bridges the gap between entrepreneurship and quantum mechanics by proposing a novel framework that portrays entrepreneurship as an exploration of a quantum-like landscape of possibilities. By emphasizing the significance of intentionality and action, our research sheds new light on entrepreneurship’s dynamic and unpredictable nature, illustrating how entrepreneurs play a pivotal role in transforming potential into reality.

QUICK3$^3$ ‐ Design of a Satellite‐Based Quantum Light Source for Quantum Communication and Extended Physical Theory Tests in Space

AbstractModern quantum technologies have matured such that they can now be used in space applications, e.g., long‐distance quantum communication. Here, the design of a compact true single photon source is presented that can enhance the secure data rates in satellite‐based quantum key distribution scenarios compared to conventional laser‐based light sources. The quantum light source is a fluorescent color center in hexagonal boron nitride. The emitter is off‐resonantly excited by a diode laser and directly coupled to an integrated photonic processor that routes the photons to different experiments performed directly on‐chip: i) the characterization of the single photon source and ii) testing a fundamental postulate of quantum mechanics, namely the relation of the probability density and the wave function (known as Born's rule). The described payload is currently being integrated into a 3U CubeSat and scheduled to launch in 2024 into low Earth orbit. Therefore the feasibility of true single photon sources and reconfigurable photonic circuits in space can be evaluated. This provides a promising route toward a high‐speed quantum network.

Power-duality in path integral formulation of quantum mechanics

Power duality in Feynman's path integral formulation of quantum mechanics is investigated. The power duality transformation consists of a change in coordinate and time variables, an exchange of energy and coupling, and a classical angular momentum replacement. Two physical systems connected by the transformation form a power-dual pair. The propagator (Feynman's kernel) expressed by Feynman's path integral cannot be form-invariant under the transformation, whereas the promotor constructed by modifying Feynman's path integral is found form-invariant insofar as the angular momentum is classical. Upon angular quantization, the power duality breaks down. To save the notion of power duality, the idea of quasi power duality is proposed, which constitutes of an ad hoc angular momentum replacement. The power-dual invariant promotor leads to the quasi-dual invariant Green function. A formula is proposed, which determines the Green function for one of a dual pair by knowing the Green function for the other. As examples, the Coulomb-Hooke dual pair and a family of two-term confinement potentials for a zero-energy state are discussed.

Nonadiabatic Field on Quantum Phase Space: A Century after Ehrenfest.

Nonadiabatic transition dynamics lies at the core of many electron/hole transfer, photoactivated, and vacuum field-coupled processes. About a century after Ehrenfest proposed "Phasenraum" and the Ehrenfest theorem, we report a conceptually novel trajectory-based nonadiabatic dynamics approach, nonadiabatic field (NAF), based on a generalized exact coordinate-momentum phase space formulation of quantum mechanics. It does not employ the conventional Born-Oppenheimer or Ehrenfest trajectory in the nonadiabatic coupling region. Instead, in NAF the equations of motion of the independent trajectory involve a nonadiabatic nuclear force term in addition to an adiabatic nuclear force term of a single electronic state. A few benchmark tests for gas phase and condensed phase systems indicate that NAF offers a practical tool to capture the correct correlation of electronic and nuclear dynamics for processes where the states remain coupled all the time as well as for the asymptotic region where the coupling of electronic states vanishes.

Earthquake Quantization

In this homage to Einstein's 144th birthday we propose a novel quantization prescription, where the paths of a path-integral are not random, but rather solutions of a geodesic equation in a random background. We show that this change of perspective can be made mathematically equivalent to the usual formulations of non-relativistic quantum mechanics. To conclude, we comment on conceptual issues, such as quantum gravity coupled to matter and the quantum equivalence principle.

Integrable scattering theory with higher derivative Hamiltonians

We discuss how a standard scattering theory a of multi-particle theory generalises to systems based on Hamiltonians that involve higher-order derivatives in their quantum mechanical formulation. As concrete examples, we consider Hamiltonian systems built from higher-order charges of Calogero and Calogero-Moser systems. Exploiting the integrability of these systems, we compute the classical phase shifts and briefly comment on the quantum versions of these types of theories.

Berry’s Phase and Quantum Mechanical Formulation of Anomalous Hall Effect

Berry’s Phase and Quantum Mechanical Formulation of Anomalous Hall Effect

Quantum mechanical lateral force on an atom due to matter wave

The area of trapping the atoms or molecules using light has advanced tremendously in the last few decades. In contrast, the idea of controlling (not only trapping) the movement of atomic-sized particles using matter waves is a completely new emerging area of particle manipulation. Though a single previous report has suggested the pulling of atoms based on matter-wave tractor beams, an attempt is yet to be made to produce a lateral force using this technique. This article demonstrates an asymmetric setup that engenders reversible lateral force on an atom due to the interaction energy of the matter wave in the presence of a metal surface. Several full-wave simulations and analytical calculations were performed on a particular set-up of Xenon scatterers placed near a Copper surface, with two counter-propagating plane matter waves of Helium impinging in the direction parallel to the surface. By solving the time-independent Schrödinger equation and using the solution, quantum mechanical stress tensor formalism is applied to compute the force acting on the particle. The simulation results are in excellent agreement with the analytical calculations. The results for the adsorbed scatterer case find this technique to be an efficient cleaning procedure similar to electron-stimulated desorption for futuristic applications.