Classial and quantum solitons in the transverse field Ising model

We investigate the shape and the dynamics of domain walls in the one-dimensional Ising model with spin S, exchange constant J and external transverse field Γ using numerical calculations up to S = 20 and analytical approximations. For $\tfrac{\Gamma } {{JS}}$ \] we describe classical domain walls as strongly localized excitations, which have either central spin or central bond symmetry. These symmetries are identified also in the quantum case, when solitary excitations develop into energy bands. In the classical limit S → ∞ localization results from the exponential vanishing of the bandwidth for the lowest bands. We describe the relation between the spectrum of moving classical solitons and the quantum band structure.     

Discrete accidental symmetry for a particle in a constant magnetic field on a torus

A classical particle in a constant magnetic field undergoes cyclotron motion on a circular orbit. At the quantum level, the fact that all classical orbits are closed gives rise to degeneracies in the spectrum. It is well-known that the spectrum of a charged particle in a constant magnetic field consists of infinitely degenerate Landau levels. Just as for the 1/r and r² potentials, one thus expects some hidden accidental symmetry, in this case with infinite-dimensional representations. Indeed, the position of the center of the cyclotron circle plays the role of a Runge-Lenz vector. After identifying the corresponding accidental symmetry algebra, we re-analyze the system in a finite periodic volume. Interestingly, similar to the quantum mechanical breaking of CP invariance due to the θ-vacuum angle in non-Abelian gauge theories, quantum effects due to two self-adjoint extension parameters θ_x and θ_y explicitly break the continuous translation invariance of the classical theory. This reduces the symmetry to a discrete magnetic translation group and leads to finite degeneracy. Similar to a particle moving on a cone, a particle in a constant magnetic field shows a very peculiar realization of accidental symmetry in quantum mechanics.     

The semiclassical limit of a quantum Fermi accelerator

A classical Fermi accelerator model (FAM) is known to show chaotic behavior. The FAM is defined by a free particle bouncing elastically from two rigid walls, one fixed and the other oscillating periodically in time. The central aim of this paper is to connect the quantum and the classical solutions to the FAM in the semiclassical limit. This goal is accomplished using a finite inverted parametric oscillator (FIPO), confined to a box withfixed walls, as an alternative representation of the FAM. In the FIPO representation, an explicit correspondence between classical and quantum limits is accomplished using a Husimi representation of the quasienergy eigenfunctions.     

Quantum to classical correspondence for the Fermi-acceleration model

A quantum particle which is confined to the interior of a box with infinitely high but periodically oscillating walls can have an unusual semiclassical limit: For the special case of a one-dimensional linear wall motion we show that the semiclassical domain corresponds to a classical motion in phase space where the initial momentum depends on the particle's position in the box. Another result is that quantum states which correspond to classical cycle-1 fixed points have maximum stability against the boundary induced perturbation (caused by the moving wall). Higher cycle-n fixed points are calculated by numerical bookkeeping up to n = 20. The classical motion is marginally stable. We show how a slight change in the boundary condition will lead to chaotic motion.     

Strong-coupling limit for one-dimensional polarons in a finite box

The strong coupling limit is studied for a Pekar-Fröhlich polaron confined to a one-dimensional (1D) structure. The non-linear effective Schrödinger equation is solved exactly in the case of two different external potentials which imitate a finite size 1D sample: an infinite and a finite deep rectangular well. The ground state and excited states are calculated. We found that taking the limit of a finite size box to an infinitely large box leads to additional solutions which are not found in a treatment on an infinite axis. The additional solutions, which have a 1/n ² discrete spectrum, correspond to polaron states in which the wave function is split up in identical parts which are infinitely apart from each other.     

Probability density in the complex plane

The correspondence principle asserts that quantum mechanics resembles classical mechanics in the high-quantum-number limit. In the past few years, many papers have been published on the extension of both quantum mechanics and classical mechanics into the complex domain. However, the question of whether complex quantum mechanics resembles complex classical mechanics at high energy has not yet been studied. This paper introduces the concept of a local quantum probability density ρ(z) in the complex plane. It is shown that there exist infinitely many complex contours C of infinite length on which ρ(z) dz is real and positive. Furthermore, the probability integral is finite. Demonstrating the existence of such contours is the essential element in establishing the correspondence between complex quantum and classical mechanics. The mathematics needed to analyze these contours is subtle and involves the use of asymptotics beyond all orders.     

Strong-coupling limit for one-dimensional polarons in a finite box

The strong coupling limit is studied for a Pekar-Fröhlich polaron confined to a one-dimensional (1D) structure. The non-linear effective Schrödinger equation is solved exactly in the case of two different external potentials which imitate a finite size 1D sample: an infinite and a finite deep rectangular well. The ground state and excited states are calculated. We found that taking the limit of a finite size box to an infinitely large box leads to additional solutions which are not found in a treatment on an infinite axis. The additional solutions, which have a 1/n ² discrete spectrum, correspond to polaron states in which the wave function is split up in identical parts which are infinitely apart from each other.     

QCD2 and the classical correspondence in the large-N limit

It is shown that the large-N limit of quantum chromodynamics in twodimensions is determined by classical equations with boundary conditions. The nonperturbative quantum spectrum of mesonic bound states is obtained from a classical equation with a simple N-dependent boundary condition on the local charge density. The simplicity of the classical correspondence is shown to be directly tied to the simplicity of the space of gauge invariant operators of the theory. Implications for other large-N models are discussed.     

Quantum-mechanical confinement with the robin condition

The energy spectrum of a nonrelativistic quantum particle in the confinement state in a closed spatial volume at general boundary confinement conditions (the Robin conditions) is investigated. It is shown that the properties of such a state are substantially more nontrivial compared with particle confinement using the potential barrier. It is also shown for a hydrogen-like atom arranged in a spherical cavity with radius R that if the surface layer with nonzero depth d plays the role of the boundary of the confinement region, all the energy levels of a discrete spectrum of the atom have a finite limit at R → 0, while the R-dependence of the lower layer at physically substantial parameters of the surface layer contains a deep well-pronounced minimum, in which the binding energy is considerably higher than for the lower 1s level of a free atom.     

Inertial and gravitational mass in quantum mechanics

We show that in complete agreement with classical mechanics, the dynamics of any quantum mechanical wave packet in a linear gravitational potential involves the gravitational and the inertial mass only as their ratio. In contrast, the spatial modulation of the corresponding energy wave function is determined by the third root of the product of the two masses. Moreover, the discrete energy spectrum of a particle constrained in its motion by a linear gravitational potential and an infinitely steep wall depends on the inertial as well as the gravitational mass with different fractional powers. This feature might open a new avenue in quantum tests of the universality of free fall.     

Dissipative behavior of a quantum system interacting with a macroscopic medium

The modified Coleman-Hepp (AgBr) model describes the interaction between an ultrarelativistic quantum mechanical particle Q and an N-spin array D (a macroscopic medium in the N → ∞ limit). We prove that the energy operator for D essentially behaves as a Wiener process in the weak-coupling, macroscopic limit, in a restricted state space. No assumptions are made on the spectrum of the Hamiltonian of the macroscopic system D. The mechanism of appearance of such a stochastic process and its relevance to issues like dissipation and irreversibility are briefly discussed.     

The A → X emission spectra of PbS and PbSe in neon matrices: Enhanced emission from isotopes and matrix sites

The A → X emission band systems induced by laser excitation are reported for the PbS and PbSe molecules in solid neon matrices. The spectra are characterized by sharp zero-phonon lines corresponding to the 0 → V″ transitions for V″ from 0 to about 14. The PbS spectrum also exhibits V′ → V″ vibronic bands for V′ up to 8. In some of the PbSe bands 15 isotopic species can easily be resolved. In freshly deposited matrices, the molecules occupy a number of different matrix sites, all but one of which can be partially quenched by annealing the matrix. In several cases of close coincidence between laser and 0 → V′ frequencies, much of the emission energy is confined to particular isotopic molecules in particular matrix sites.     

Electron self-energy in nonlocal field theory

In the framework of quantum electrodynamics with the nonlocal interaction it is shown that the correspondence principle holds in the problem on the self-mass of an electron and a particle with arbitrary spin. It appears that the second order of perturbation theory in the limit ? → 0 gives just the classical expression for the electron self-energy, and all higher order corrections are zero.     

Runge-Lenz vector, accidental SU(2) symmetry, and unusual multiplets for motion on a cone

We consider a particle moving on a cone and bound to its tip by 1/r or harmonic oscillator potentials. When the deficit angle of the cone divided by 2π is a rational number, all bound classical orbits are closed. Correspondingly, the quantum system has accidental degeneracies in the discrete energy spectrum. An accidental SU(2) symmetry is generated by the rotations around the tip of the cone as well as by a Runge-Lenz vector. Remarkably, some of the corresponding multiplets have fractional “spin” and unusual degeneracies.     

Wave function of the quantum black hole

We show that the Wald Noether-charge entropy is canonically conjugate to the opening angle at the horizon. Using this canonical relation, we extend the Wheeler–DeWitt equation to a Schrödinger equation in the opening angle, following Carlip and Teitelboim. We solve the equation in the semiclassical approximation by using the correspondence principle and find that the solutions are minimal uncertainty wavefunctions with a continuous spectrum for the entropy and therefore also of the area of the black hole horizon. The fact that the opening angle fluctuates away from its classical value of 2π indicates that the quantum black hole is a superposition of horizonless states. The classical geometry with a horizon serves only to evaluate quantum expectation values in the strict classical limit.     

Nonpermuting limits (of zero-damping and infinite-system size) in an exactly solvable one-dimensional electrodynamical model

The quantum mechanics of an oscillating electron interacting with a one-dimensional electromagnetic string in the limit of zero damping (e → 0) and infinite field extension (Λ → ∞) depends on the order in which the limits are taken. If the limit e → 0 is taken after the limit Λ → ∞, one obtains gaussian wavefunctions for all stationary states (n = 0, 1, 2, …) of the particle. These gaussians do have the correct standard values for the variances 〈z²〉_n but otherwise obviously differ from the standard harmonic oscillator result. The latter arises only if the zero damping limit is taken before the infinite system limit.     

Quantum Discreteness is an Illusion

I review arguments demonstrating how the concept of “particle” numbers arises in the form of equidistant energy eigenvalues of coupled harmonic oscillators representing free fields. Their quantum numbers (numbers of nodes of the wave functions) can be interpreted as occupation numbers for objects with a formal mass (defined by the field equation) and spatial wave number (“momentum”) characterizing classical field modes. A superposition of different oscillator eigenstates, all consisting of n modes having one node, while all others have none, defines a non-degenerate “n-particle wave function”. Other discrete properties and phenomena (such as particle positions and “events”) can be understood by means of the fast but continuous process of decoherence: the irreversible dislocalization of superpositions. Any wave-particle dualism thus becomes obsolete. The observation of individual outcomes of this decoherence process in measurements requires either a subsequent collapse of the wave function or a “branching observer” in accordance with the Schrödinger equation—both possibilities applying clearly after the decoherence process. Any probability interpretation of the wave function in terms of local elements of reality, such as particles or other classical concepts, would open a Pandora’s box of paradoxes, as is illustrated by various misnomers that have become popular in quantum theory.