One-particle Quantum Mechanics Research Articles

Real-time quantum calculations of phase shifts using wave packet time delays

We present a method to extract the phase shift of a scattering process using the real-time evolution in the early and intermediate stages of the collision in order to estimate the time delay of a wave packet. This procedure is convenient when using noisy quantum computers for which the asymptotic out-state behavior is unreachable. We demonstrate that the challenging Fourier transforms involved in the state preparation and measurements can be implemented in $1+1$ dimensions with current trapped ion devices and IBM quantum computers. We compare quantum computation of the time delays obtained in the one-particle quantum mechanics limit and the scalable quantum field theory formulation with accurate numerical results. We discuss the finite volume effects in the Wigner formula connecting time delays to phase shifts. The results reported involve two- and four-qubit calculations, and we discuss the possibility of larger scale computations in the near future.

Renormalons in quantum mechanics

We present a nonrelativistic one-particle quantum mechanics whose perturbative S-matrix exhibits a renormalon divergence that we explicitely compute. The potential of our model is the sum of the 2d Dirac δ-potential — known to require renormalization — and a 1d Dirac δ-potential tilted at an angle. We argue that renormalons are not specific to this example and exist for a much wider class of potentials. The ambiguity in the Borel summation of the perturbative series due to the renormalon pole is resolved by the physical condition of causality through careful consideration of the iϵ prescription. The suitably summed perturbative result coincides with the exact answer obtained through the operator formalism for scattering.

The Transition from Quantum Field Theory to One-Particle Quantum Mechanics and a Proposed Interpretation of Aharonov–Bohm Effect

In this article we demonstrate a sense in which the one-particle quantum mechanics (OPQM) and the classical electromagnetic four-potential arise from quantum field theory (QFT). In addition, the classical Maxwell equations are derived from a QFT scattering process, while both classical electromagnetic fields and potentials serve as mathematical tools to approximate the interactions among elementary particles described by QFT physics. Furthermore, a plausible interpretation of the Aharonov-Bohm (AB) effect is raised within the QFT framework. We provide a quantum treatment of the source of electromagnetic potentials and argue that the underlying mechanism in the AB effect can be understood via interactions among electrons described by QFT where the interactions are mediated by virtual photons.

Recent Progress on the Description of Relativistic Spin: Vector Model of Spinning Particle and Rotating Body with Gravimagnetic Moment in General Relativity

We review the recent results on development of vector models of spin and apply them to study the influence of spin-field interaction on the trajectory and precession of a spinning particle in external gravitational and electromagnetic fields. The formalism is developed starting from the Lagrangian variational problem, which implies both equations of motion and constraints which should be presented in a model of spinning particle. We present a detailed analysis of the resulting theory and show that it has reasonable properties on both classical and quantum level. We describe a number of applications and show how the vector model clarifies some issues presented in theoretical description of a relativistic spin: (A) one-particle relativistic quantum mechanics with positive energies and its relation with the Dirac equation and with relativistic Zitterbewegung; (B) spin-induced noncommutativity and the problem of covariant formalism; (C) three-dimensional acceleration consistent with coordinate-independence of the speed of light in general relativity and rainbow geometry seen by spinning particle; (D) paradoxical behavior of the Mathisson-Papapetrou-Tulczyjew-Dixon equations of a rotating body in ultrarelativistic limit, and equations with improved behavior.

Quantum mechanics for quantum computing

Quantum computing is a field of great interest, attracting, among others, the attention of many mathematicians. Although not all quantum mechanics is needed to successfully engage in research on quantum computing, the somewhat superficial approach usually applied by non-physicists is, in the opinion of the author of the lectures, not feasible. The following notes from lectures given at the mathematics department of George Washington University are meant to be a partial remedy to the situation, offering a very brief and slightly unorthodox introduction to one-particle quantum mechanics, and even shorter discussion of passage to multi-particle quantum mechanics, as needed for quantum computing.

Resolution of the Klein paradox

We present a resolution of the Klein paradox within the framework of one-particle relativistic quantum mechanics. Not only reflection becomes total but the vacuum remains neutral as well. This is accomplished by replacing the physical pair production process with virtual negative energy ‘incidence’ within the barrier in a similar manner to what is done with virtual sources in optics and image charges in electrostatics.

The Dirac Hamiltonian with a superstrong Coulomb field

We consider the quantum mechanical problem of a relativistic Dirac particle moving in the Coulomb field of a point charge Ze. It is often declared in the literature that a quantum mechanical description of such a system does not exist for charge values exceeding the so-called critical charge with Z = α−1 = 137 because the standard expression for the lower bound-state energy yields complex values at overcritical charges. We show that from the mathematical standpoint, there is no problem in defining a self-adjoint Hamiltonian for any charge value. Furthermore, the transition through the critical charge does not lead to any qualitative changes in the mathematical description of the system. A specific feature of overcritical charges is a nonuniqueness of the self-adjoint Hamiltonian, but this nonuniqueness is also characteristic for charge values less than critical (and larger than the subcritical charge with $$Z = (\sqrt 3 /2)\alpha ^{ - 1} = 118$$ ). We present the spectra and (generalized) eigenfunctions for all self-adjoint Hamiltonians. We use the methods of the theory of self-adjoint extensions of symmetric operators and the Krein method of guiding functionals. The relation of the constructed one-particle quantum mechanics to the real physics of electrons in superstrong Coulomb fields where multiparticle effects may be crucially important is an open question.

ON THE QUANTUM THEORY OF MASSLESS SPIN-3/2 FIELD IN MINKOWSKI SPACETIME

From the modern viewpoint and by the geometric method, this paper provides a concise foundation for the quantum theory of massless spin-3/2 field in Minkowski spacetime, which includes both the one-particle's quantum mechanics and the many-particle's quantum field theory. The explicit result presented here is useful for the investigation of spin-3/2 field in various circumstances such as supergravity, twistor programme, Casimir effect, and quantum inequality.

Fractional angular momentum in non-commutative spaces

In noncommutative spaces to maintain Bose–Einstein statistics for identical particles at the non-perturbation level described by deformed annihilation-creation operators when the state vector space of identical bosons is constructed by generalizing one-particle quantum mechanics it is explored that the consistent ansatz of commutation relations of phase space variables should simultaneously include space–space non-commutativity and momentum–momentum non-commutativity, and a new type of boson commutation relations at the deformed level is obtained. Consistent perturbation expansions of deformed annihilation-creation operators are obtained. The influence of the new boson commutation relations on dynamics is discussed. The non-perturbation and perturbation property of the orbital angular momentum of two-dimensional system are investigated. Its spectrum possesses fractional eigen values and fractional intervals.

BBGKY hierarchy underlying many-particle quantum mechanics

Recently, the one-particle quantum mechanics has been obtained in the framework of an entirely classical subquantum kinetics. In the present Letter we argue that, within the same scheme and without any additional assumption, it is possible to obtain also the n-particle non-relativistic quantum mechanics. The main goal of the present effort is to show that the classical BBGKY hierarchical equation, for the n-particle reduced distribution function, is the ancestor of the n-particle Schrödinger equation. On the other hand we show that within the scenario of the subquantum structure of quantum particle, the Fisher information measure emerges naturally in quantum mechanics.

Nonrelativistic Quantum Mechanics with Spin in the Framework of a Classical Subquantum Kinetics

Recently it has been shown that the spinless one-particle quantum mechanics can be obtained in the framework of entirely classical subquantum kinetics. In the present paper we argue that, within the same scheme and without any extra assumption, it is possible to obtain both the nonrelativistic quantum mechanics with spin, in the presence of an arbitrary external electromagnetic field, as well as the nonlinear quantum mechanics.

Coherent pulse output from Bose-Einstein condensates in Wannier-Stark systems

The pulsed output from a Bose-Einstein condensate can be described using ordinary one-particle quantum mechanics. The initial state is described in terms of Stark resonances truncated in momentum space. The states obtained in this way resemble the normalizable scattering states defined in terms of Moshinsky functions. The validity of this approach and the influence of the initial population on the pulse formation is discussed. Finally, we describe an experimental setup to manufacture and observe pulsed output in Wannier-Stark systems.

A model of population genetics and its mathematical relation to quantum theory

A basic model used in population genetics is motivated and it is shown how the resulting theory has a mathematical complexity equivalent to that of non-relativistic one-particle quantum mechanics. For the model in question, it is shown that a Schrödinger-like equation underlies the dynamics with the main difference that the time derivative term lacks a coefficient of i = ¡ -1. A number of special cases of the model are analysed, including one that makes connections with a topic of current interest in theoretical physics.

Supersymmetric quantum mechanics for inverse square potentials

In elementary particle physics, supersymmetry (SUSY) (Wess and Zumino, 1974) is a symmetry which relates half-integral spin particles (fermions) to particles with integral spin (bosons). Using the supersymmetric version of SU(5) or SO(10), it was possible to unify electromagnetic, weak, and strong interactions. There is no experimental evidence for the existence of new SUSY particles in nature. On the other hand, applications of SUSY have been found in nuclear and condensed matter physics as well. Applications of supersymmetry in nonrelativistic quantum mechanics was first discussed by Witten (1981). The aim of this paper is to analyze the inverse square potentials using supersymmetric quantum mechanics. First we discuss how to construct supersymmetric algebra in quantum mechanics. Introducing SUSY quantum mechanics, then we analyze inverse square potentials in that context. Supersymmetric quantum mechanics has a close relationship to supersymmetric field theory. Supersymmetric one-particle quantum mechanics serves as a model for the investigation of spontaneous breaking of supersymmetry, which is supposed to occur in supersymmetric field theories. Now consider a one-particle quantum mechanical system in one dimension.

The Bethe-Maximon result.

We develop and illustrate a very simple eikonal technique in the framework of one-particle quantum mechanics for calculating radiation during scattering and apply it to derive the celebrated result of Bethe and Maximon for bremsstrahlung from a point Coulomb target with a large charge Ze. This approach is a modification of that developed in a previous paper for calculating radiation during the collision of electron-positron beam pulses (beamstrahlung). The differences in the physics of scattering from a localized target and from an extended target are clarified. A second reason for the present note is the simplicity of our method and its possible extension to other processes.

A class of explicitly soluble, local, many-center Hamiltonians for one-particle quantum mechanics in two and three dimensions. I

We derive an explicit formula for the resolvent of a class of one-particle, many-center, local Hamiltonians. This formula gives, in particular, a full description of a model molecule given by point interactions at n arbitrarily placed fixed centers in three dimensions. It also gives a three−dimensional analog of the Kronig–Penney model.

A remarkable duality in one-particle quantum mechanics between some confining potentials and (R+L ∞ϵ) potentials

The equivalence between the bound-state problem for the spatial harmonic oscillator and for the Coulomb potential is reexamined and extended: a similar duality is displayed between large classes of confining potentials and (R+L ∞ ϵ ) potentials, i.e., potentials for which the Schrödinger operator has among other properties a discrete spectrum bounded below, in (-∞, 0). The involved confining potentials U( r) must satisfy: r -αU(r) ▪g>0 r→+∞, r -2α (2+α) U (r 2 (2+α) ) − g is R + L ∞ ϵ .

Moufang plane and octonionic Quantum Mechanics

It is shown that the usual axioms of one-particle Quantum Mechanics can be implemented with projection operators belonging to the exceptional Jordan algebraJ 8 3 over real octonions. Certain lemmas on these projection operators are proved by elementary means. Use is made of the Moufang projective plane. It is shown that this plane can be orthocomplemented and that there exists a unique probability function. The result of successive, compatible experiments is shown not to depend on the order in which they are performed, in spite of the non-associativity of octonion multiplication. The algebra of observables and the action of the exceptional groupF 4 is studied, as well as a possible relation with the color group SU(3) and quark confinement.