The Non-Negativity of Probabilities and the Collapse of State

The dynamical equation, being the combination of Schrödinger and Liouville equations, produces noncausal evolution when the initial state of interacting quantum and classical mechanical systems is as it is demanded in discussions regarding the problem of measurement. It is found that state of quantum mechanical system instantaneously collapses due to the non-negativity of probabilities.     

On the Quantum Boltzmann Equation

We give a nonrigorous derivation of the nonlinear Boltzmann equation from the Schrödinger evolution of interacting fermions. The argument is based mainly on the assumption that a quasifree initial state satisfies a property called restricted quasifreenessin the weak coupling limit at any later time. By definition, a state is called restricted quasifree if the four-point and the eight-point functions of the state factorize in the same manner as in a quasifree state.     

Preferred states of the apparatus

A simple one-dimensional model for the system–apparatus interaction is analysed. The system is a spin-1/2 particle, and its position and momentum degrees constitute the apparatus. An analysis involving only unitary Schrödinger dynamics illustrates the nature of the correlations established in the system–apparatus entangled state. It is shown that even in the absence of any environment-induced decoherence, or any other measurement model, certain initial states of the apparatus – like localized Gaussian wavepackets – are preferred over others, in terms of measurementlike one-to-one correlations in the pure system–apparatus entangled state.     

Formal exact solution for the Heisenberg spin system in a time-dependent magnetic field and Aharonov-Anandan phase

We study the time evolution of the Heisenberg spin system in a time-dependent magnetic field. By a unitary transformation we obtain the formal exact solutions of the Schrödinger equation for this system. Based on the formal exact solutions, the Aharonov-Anandan phases are worked out.     

Exact solutions of the Korteweg-de Vries equation with a self-consistent source

Exact solutions are found for the Korteweg-de Vries equation with a source satisfying the stationary Schrödinger equation. Each solution describes the evolution of the initial moving wave with one phase velocity to the final moving wave with another phase velocity. The conditions are pointed out under which the phase velocities of these waves may differ in sign. The obtained results are relevant to some problems of hydrodynamics, plasma physics, solid state physics, etc.     

Analytical solution of the Schrödinger equation in the sudden perturbation approximation for an atom by attosecond and shorter electromagnetic pulses

An exact solution of the Schrödinger equation has been obtained in the sudden perturbation approximation for the case of the interaction of attosecond and shorter electromagnetic pulses with multielectron atoms. This solution makes it possible to exactly take into account the spatial inhomogeneity of the field of an ultrashort pulse. The result has been presented in an analytical form.     

Justification of the discrete nonlinear Schrödinger equation from a parametrically driven damped nonlinear Klein–Gordon equation and numerical comparisons

We consider a damped, parametrically driven discrete nonlinear Klein–Gordon equation, that models coupled pendula and micromechanical arrays, among others. To study the equation, one usually uses a small-amplitude wave ansatz, that reduces the equation into a discrete nonlinear Schrödinger equation with damping and parametric drive. Here, we justify the approximation by looking for the error bound with the method of energy estimates. Furthermore, we prove the local and global existence of solutions to the discrete nonlinear Schrödinger equation. To illustrate the main results, we consider numerical simulations showing the dynamics of errors made by the discrete nonlinear equation. We consider two types of initial conditions, with one of them being a discrete soliton of the nonlinear Schrödinger equation, that is expectedly approximate discrete breathers of the nonlinear Klein–Gordon equation.     

Modified Schrödinger dynamics with attractive densities

The linear Schrödinger equation does not predict that macroscopic bodies should be located at one place only, or that the outcome of a measurement shoud be unique. Quantum mechanics textbooks generally solve the problem by introducing the projection postulate, which forces definite values to emerge during measurements; many other interpretations have also been proposed. Here, in the same spirit as the GRW and CSL theories, we modify the Schrödinger equation in a way that efficiently cancels macroscopic density fluctuations in space. Nevertheless, we do not assume a stochastic dynamics as in GRW or CSL theories. Instead, we propose a deterministic evolution that includes an attraction term towards the averaged density in space of the de Broglie-Bohm position of particles, and show that this is sufficient to ensure macroscopic uniqueness and compatibility with the Born rule. The state vector can then be seen as directly related to physical reality.     

Two kinds of upper hybrid soliton bistability

The basic model employed to describe nonlinear upper hybrid wave structures is the generalized nonlinear Schrödinger equation including second and fourth order dispersive effects as well as local and nonlocal nonlinearity. For two kinds of such an equation the existence of two stable solitons with the same plasmon number but with different spatial scales and amplitudes is shown as two qualitatively different kinds of upper hybrid soliton bistability. An integral relation for an arbitrary nonlinear upper hybrid wave packet evolution is derived taking into account higher order dispersive effects. Necessary conditions for soliton formation from arbitrary wave packets and the impossibility of wave packet collapse are demonstrated taking into account higher order dispersive effects.     

Collapse dynamics of a self-magnetized Langmuir plasma

We show that the set of generalized Zakharov equations which take into account the self-magnetic field reduce under certain circumstances to a generalized non-linear Schrödinger equation. We also that in the same approximation the self-magnetic field helps rather than hinders the Langmuir collapse.     

On the distribution of energy of localized solutions of the Schrödinger equation that propagate along symmetric quantum graphs

From the point of view of applications to quantum mechanics, it is natural to pose a question concerning the distribution of energy of localized solutions of a nonstationary Schrödinger equation over the graph (in other words, the probability to find a quantum particle in a given area). This problem is apparently very complicated for general graphs, because the energy distribution is much more sensitive to the form of boundary conditions and to the initial state than the asymptotic behavior of the number of localized functions. Below, we present initial results concerning the distribution of energy in the case of symmetric quantum graphs (this means that the Schrödinger operators on different edges have the same structure). For general local self-adjoint boundary conditions, we describe the process of onestep scattering of the localized solutions and obtain a simple general result of the distribution of energy. Some special cases and specific examples are discussed.     

Quantum plasma description and Landau damping

The Schrödinger equation is shown to replace the Vlasov equation to simulate a classical collisionless plasma, provided we take a sophisticated initial condition. The Landau damping is then recovered within a high degree of accuracy.     

Stochastic electrodynamics. III. Statistics of the perturbed harmonic oscillator-zero-point field system

In this third paper in a series on stochastic electrodynamics (SED), the nonrelativistic dipole approximation harmonic oscillator-zero-point field system is subjected to an arbitrary classical electromagnetic radiation field. The ensemble-averaged phase-space distribution and the two independent ensemble-averaged Liouville or Fokker-Planck equations that it satisfies are derived in closed form without furtner approximation. One of these Liouville equations is shown to be exactly equivalent to the usual Schrödinger equation supplemented by small radiative corrections and an explicit radiation reaction (RR) vector potential that is similar to the Crisp-Jaynes semiclassical theory (SCT) RR potential. The wave function in this SED Schrödinger equation is shown to have thea priori significance of position probability amplitude. The other Liouville equation has no counterpart in ordinary quantum mechanics, and is shown to restrict initial conditions such that (i) The Wigner-type phase-space distribution is always positive, (ii) in the absence of an applied field, the only allowed solution of both equations is the quantum ground state, and (iii) if a previously applied field is suddenly turned off, then spontaneous transitions occur, with no need for a triggering perturbation as in SCT, until the system is in the ground state. It is also shown that the oscillator energy is a fluctuating quantity that must take on a continuum of values, with average value equal to the quantum ground-state energy plus a contribution due to the applied classical field.     

New extended auxiliary equation method for finding many new Jacobi elliptic function solutions of three nonlinear Schrödinger equations

In this paper, we construct many new types of Jacobi elliptic function solutions of nonlinear evolution equations using the so-called new extended auxiliary equation method. The effectiveness of this method is demonstrated by applications to three higher order nonlinear evolution equations, namely, the higher order nonlinear Schrödinger equation with derivative non-Kerr nonlinear terms, the higher order dispersive nonlinear Schrödinger equation and the generalized nonlinear Schrödinger equation. The solitary wave solutions and periodic solutions are obtained from the Jacobi elliptic function solutions. Comparing our new results and the well-known results are given.     

A Schrödinger equation for collective motion from the generator coordinate method

It is shown that if the collective wave function occuring in the Griffin-Hill-Wheeler integral equation is slowly varying, the equation can be transformed into a Schrödinger equation. The conditions under which this can be done, are studied in detail.     

Spin-dependent potentials in the linearized collective Schrödinger equation for nuclear quadrupole vibrations

The linearized collective Schrödinger equation for nuclear quadrupole surface vibrations incorporates a new spin degree of freedom with a spin value of 3/2. We use this equation to describe the low energy spectrum of certain even-odd Ir nuclei which have a spin 3/2 in their ground state. For that purpose we explicitly introduce collective spin-dependent potentials which simulate the interaction of the valence nucleon with the core. The linearized Schrödinger equation is transformed into an effective Schrödinger equation with collective spin-dependent potentials. Already collective spin-orbit couplings of SO(3) and SO(5) type are sufficient to reproduce the lowest excited states of even-odd Ir nuclei.     

Relation between Bethe-Salpeter and Schrödinger wave functions for many-particel systems

The connection is made between a many-time approach to S-matrix elements and energy eigenvalues, which naturally arises from a field theoretical point of view, and the single time Schrödinger- and Breit-like formalism often used in detailed calculations for many-particle systems, such as many-electron atoms. Specifically, the many-particle Bethe-Salpeter equation is expressed in terms of the corresponding Schrödinger equation for the non-relativistic case in which the Bethe-Salpeter kernel consists of only two-particle local static interactions. Also, the one-photon transition matrix element for this case in the Bethe-Salpeter formalism is shown to be equivalent to the corresponding well-known Schrödinger result. The treatment developed is well suited to systematic relativistic generalization.