Nonstationary Wave Function Research Articles

Foundations of a Theory of Gravity with a Constraint and Its Canonical Quantization

The gravitational equations were derived in general relativity (GR) using the assumption of their covariance relative to arbitrary transformations of coordinates. It has been repeatedly expressed an opinion over the past century that such equality of all coordinate systems may not correspond to reality. Nevertheless, no actual verification of the necessity of this assumption has been made to date. The paper proposes a theory of gravity with a constraint, the degenerate variants of which are general relativity (GR) and the unimodular theory of gravity. This constraint is interpreted from a physical point of view as a sufficient condition for the adiabaticity of the process of the evolution of the space–time metric. The original equations of the theory of gravity with the constraint are formulated. On this basis, a unified model of the evolution of the modern, early, and very early Universe is constructed that is consistent with the observational astronomical data but does not require the hypotheses of the existence of dark energy, dark matter or inflatons. It is claimed that: physical time is anisotropic, the gravitational field is the main source of energy of the Universe, the maximum global energy density in the Universe was 64 orders of magnitude smaller the Planckian one, and the entropy density is 18 orders of magnitude higher the value predicted by GR. The value of the relative density of neutrinos at the present time and the maximum temperature of matter in the early Universe are calculated. The wave equation of the gravitational field is formulated, its solution is found, and the nonstationary wave function of the very early Universe is constructed. It is shown that the birth of the Universe was random.

Time resolved mechanism of the isotope selectivity in the ultrafast light induced dissociation in N2.

The time evolution of a vacuum ultraviolet excited N2 molecule is followed all the way from an ultrafast excitation to dissociation by a quantum mechanical simulation. The primary aim is to discern the role of the excitation by a pulse short compared to the vibrational period, to discern the different coupling mechanisms between different electronic states, nonadiabatic, spin orbit, and to analyze the origin of any isotopic effect. We compare the picture in the time and energy domains. The initial ultrafast excitation pumps the molecule to a coherent electronic wave packet to which several singlet bound electronic states contribute. The total nonstationary wave function is given as a coherent sum of nuclear wave packets on each electronic state times the stationary electronic wave function. When the wave packets on different electronic states overlap, they are coupled in a mass-dependent manner whether one uses an adiabatic or a diabatic electronic basis. A weak spin-orbit coupling acts as a bottleneck between the bound singlet part of phase space and the triplet manifold of states in which dissociation takes place. To describe the spin-orbit perturbation that is ongoing in time, an energy-resolved eigenstate representation appears to be more intuitive. In the eigenstate basis, the singlet-to-triplet population transfer is large only between those vibronic eigenstates that are quasiresonant in energy. The states in resonance are different for different excitation energy ranges. The resonances are mass dependent, which explains the control of the isotope effect through the profile of the pulse.

Probability and Flux Densities in the Center-of-Mass Frame.

For an arbitrary nonstationary wave function of a nonrelativistic closed many-body system consisting of arbitrary interacting particles, the general expressions for the time-dependent one-particle probability and flux densities in the center-of-mass frame without applying Born-Oppenheimer approximation are obtained. Even the wave function for the translation is additionally introduced; it disappears in the center-of-mass frame automatically. It is shown that for the rotational ground state the time-dependent probability and flux densities of an arbitrary particle in the center-of-mass frame are isotropic. It means that the angular dependence is absent but these densities depend on radius and time. More importantly, it is shown that the angular components of the time-dependent flux density vanish. With these statements, one can calculate the radial component of the radius- and time-dependent electronic flux density within the Born-Oppenheimer approximation via the continuity equation. Application of this theory to the pulsating or exploding "quantum bubble" of the vibrating or dissociating Na2 molecule in the rotational ground state, respectively, is found elsewhere in this issue.

A study of three-dimensional reaction dynamics by means of quantum trajectories

Equations for quantum trajectories in three dimensions are derived and applied to studying the reaction dynamics of the interaction of the chlorine atom with the hydrogen molecule. The velocities of quantum trajectories are determined from the nonstationary wave function preliminary calculated by integrating the Schrodinger equation using the grid method and an expansion in basis set. An ensemble of 60 quantum trajectories for the initial vibrational state ν 0 = 0 and rotational states j 0 = 0 and 4 is obtained. A detailed analysis of one of the reaction trajectories is conducted. The existence of the effect of tunneling in the system is confirmed.

Spin precession as a consequence of the quantum spin-flip transitions with a non-stationary wave function

In this work, a precise quantum analog of spin precession of classical theory is obtained. As an example, a neutral Dirac particle (neutron) is considered. A non-stationary wave function is a superposition of wave functions with counter spins. It is used to calculate the average value of the spin operator. It is demonstrated that classical and quantum theories lead to an adequate description of the spin precession.

Experience in numerically modelling the mixed state of superconductors, applied to a study of the nonstationary Schrödinger equation

Experience in numerically modelling superconducting systems has been used to investigate the behavior of a nonstationary wave function far from equilibrium. It is shown that normalization of the wave function plays the role of an efficient nonlocal interaction that causes the wave function to be localized in one of the wells even for an infinitesimal difference of their depths. This fundamentally differs from the solution of the Fokker-Planck equation, which in equilibrium depends exponentially on the energy and is virtually symmetrical for wells that are similar in depth. When a fluctuational (small) variation causes the relative depths to reverse, the wave function’s maximum tunnels to the other well. A transition in a one-well potential from an excited state to a lower state is also considered. It is shown that this is accompanied by the emission of a localized segment of an electromagnetic wave, which can be identified with a “photon.”

Approximate description of decaying quasi-stationary state

A simple model of the decaying quantum state composed of a one-dimensional electron moving in the potential with the quantum dot separated from the innite half-axis by the ‐-function is studied. The quasibound quantum dot state properties calculated by means of approximate methods (complex energy technique and local density of states method) are compared with results obtained with the aid of an exact analytical non-stationary wave function. The physical meaning of the emitted electron wave function peculiarities is discussed.

Formation of a quasistationary state by scattering of wave packets on a finite lattice

The formation and further evolution of the quasistationary state that accompanies the scattering of wave packets in one-dimensional finite periodic structures are studied. The contributions in the spectral integral from the saddle point and simultaneously from the poles of the stationary scattering amplitudes are estimated analytically. They can have the additive character and give different peaks in the secondary packets formed by the nonstationary wave function evolution. The lifetime of the quasistationary state is long and increases significantly with the lattice length especially near the thresholds of transmission bands.

Verification of the semiclassical method for an electron moving in a homogeneous magnetic field.

A procedure based on the semiclassical approximation for high energy levels is developed to yield solutions to the classical equation of charge motion and to the Bargmann-Michel-Telegdi spin equation. To this end, exact solutions to the Klein-Gordon and the Dirac-Pauli equations are used. The essence of the procedure under review is that the quantum state of a charged particle in a homogeneous magnetic field is represented as a superposition of states corresponding to the neighboring energy levels. As a consequence, the expectation values of the momentum and spin operators with respect to the resulting nonstationary wave function (packet) strictly obey the classical equations of charge motion and spin precession.

Spectroscopic Wave Function Imaging and Potential Inversion

An accurate imaging technique of complex nonstationary wave functions and a method of inversion of excited-state potentials is developed. The wave function imaging is accomplished by using both frequency-resolved and time-resolved fluorescence data, thereby reducing the problem to a solvable set of linear algebraic equations. The procedure is demonstrated for excited-state Na2 wave packets created by ultrashort pulse excitations. A method for evaluating the sign of the transition-dipole amplitudes from the spectral line strengths is developed. Using the transition-dipole amplitudes thus generated, plus the transition frequencies and the ground state potential, we show how to obtain excited-state potentials. Contrary to the RKR procedure, this inversion scheme is not limited to diatomic molecules and is capable of yielding potentials at energies above the dissociation thresholds. Highly accurate inversions of the and Na2(B1Πu) excited-state potentials are demonstrated.

A generalized langevin equation for wave functions. Intramolecular vibrational dephasing as a stochastic process

A generalized Langevin equation satisfied by time-evolving wave functions is derived by adapting the Mori formalism to the Schrödinger time dependent equation. The equation obtained describes both deterministic and stochastic time evolution of wave functions. Memory kernel of the ”delayed friction” term of the equation obtained consists of the temporal correlation function (in the quantum mechanical sense) among non-stationary wave function, i.e., wave packets. The memory is likely to decline rapidly essentially due to the dispersion of the wave packets, so that a “Markov” limit is justified. An ensemble of probability amplitudes of different quantum states is considered, and the probability distribution of these probability amplitudes is discussed. This probability distribution is connected with spectroscopic observables: It can be derived from excitation profiles of Raman scattering amplitudes with many different final states. By assuming the “random force” term of the Langevin equation to be Gaussian white noise, a Fokker-Planck equation is derived. The solution of this Fokker-Planck equation describes how “randomization” proceeds in intramolecular vibrational dephasing. As time goes to infinity, all the accessible states are equally populated on the average, but fluctuation around the “microcanonical equilibrium” remains.

Constants of motion and non-stationary wave functions for the damped, time-dependent harmonic oscillator

We study a quantal time-dependent oscillator in the presence of a loss mechanism. We generalize to the damped oscillator the classical exact invariant found by Lewis for the undamped one. The quantal operator Ĵ associated with this constant of motion has no constant expectation values when the fluctuations are not negligible. To investigate the role of the quantal fluctuations we study the spectral properties of Ĵ for the undamped oscillator and show that its eigenstates are critical since they keep stationary the generalized uncertainty products. The nonlinear Schrödinger equation of Kostin and Kan-Griffin is shown to be a particular realization of critical-state-preserving descriptions for the damped oscillator motion.