Probability Wave Function Research Articles

Dirac Electrons of a Split-Gate Hall Bar

In this work we study several unusual properties of Klein tunneling through the abrupt and flat barriers of a split-gate Hall bar system of graphene. We show that Klein tunneling of Dirac electrons can be rather strong in such a system, and that a significant electron density can be present under the barrier. It can be shown that the probability wavefunctions for large angular momenta are identical to the probability wavefunctions of the same angular momenta in the absence of the potential barrier, i.e., it is as if the barrier does not exist and the Klein tunneling is complete. This is a unique effect in a magnetic field. We propose that STM measurements may be used to detect the presence of such a density. We have also investigated drift velocity of electrons as the center of probability wavefunction varies from outside to inside of the flat potential barrier, and find a significant deviation from the semiclassical result.

Most probable wave function of a single free-moving particle

We develop the most probable wave functions for a single free quantum particle given its momentum and energy by imposing its quantum probability density to maximize Shannon information entropy. We show that there is a class of solutions in which the quantum probability density is self-trapped with finite-size spatial support, uniformly moving, hence keeping its form unchanged.

From probabilities to mathematical structure of quantum mechanics

A large part of the mathematical formalism of quantum mechanics including the probability density, probability density current, uncertainty relations, wave function, momentum operator, kinetic energy, commutation relations and potentials is related to the probabilistic character of quantum measurements. Equations of motion of quantum mechanics can be obtained from the relativistic invariance of the space–time Fisher information. Some properties of potentials and existence of antiparticles are related to the CPT symmetry. Operators representing parallel or serial physical events in experimental setups can be represented by the sum or product of the corresponding operators.

Physical Understanding of Strain-Induced Modulation of Gate Oxide Reliability in MOSFETs

We have systematically investigated the effects of strain on the gate oxide reliability, using biaxially strained Si MOSFETs, to elucidate their physical origins. It was found that the time-dependent dielectric breakdown reliability was significantly improved in strained Si nMOSFETs but was slightly degraded in strained Si pMOSFETs. These observations could be well explained by the strain-induced modulation of the gate current, resulting from band-structure modulation in the channels. It was also found that negative bias temperature instability was degraded in the strained Si pMOSFETs. This fact was attributable to the strain-induced enhancement of hole tunneling probability and hole wave function penetration into Si-H bonding states near the MOS interface, which could enhance Si-H bond breaking.

Computation of correlation functions and wave function projections in the context of quantum trajectory dynamics

The de Broglie-Bohm formulation of the Schrodinger equation implies conservation of the wave function probability density associated with each quantum trajectory in closed systems. This conservation property greatly simplifies numerical implementations of the quantum trajectory dynamics and increases its accuracy. The reconstruction of a wave function, however, becomes expensive or inaccurate as it requires fitting or interpolation procedures. In this paper we present a method of computing wave packet correlation functions and wave function projections, which typically contain all the desired information about dynamics, without the full knowledge of the wave function by making quadratic expansions of the wave function phase and amplitude near each trajectory similar to expansions used in semiclassical methods. Computation of the quantities of interest in this procedure is linear with respect to the number of trajectories. The introduced approximations are consistent with approximate quantum potential dynamics method. The projection technique is applied to model chemical systems and to the H+H(2) exchange reaction in three dimensions.

Can we predict Λ for the non-SUSY sector of the landscape?

We propose a new selection criterion for predicting the most probable wavefunction of the universe that propagates on the string landscape background, by studying its dynamics from a quantum cosmology view. Previously, we applied this proposal to the SUSY sector of the landscape. In this work, the dynamic selection criterion is applied to the investigation of the non-SUSY sector. In the absence of detailed information about its structure, it is assumed that this sector has a stochastic distribution of vacua energies. The calculation of a distribution probability for the cosmological constants Λeff, obtained from the density of states ρ, indicates that the most probable wavefunction is peaked around universes with zero Λeff. In contrast to the extended wavefunction solutions found for the SUSY sector with N-vacua and peaked around , wavefunctions residing on the non-SUSY sector exhibit Anderson localization. Although minisuperspace is a limited approach it presently provides a dynamical quantum selection rule for the most probable vacua solution from the landscape.

Generalized Schrödinger equation and quantum measurement

A generalized Schrödinger equation for describing irreversible processes is introduced. It reveals that the wave function probability for an open system is not necessarily conservative, and enables a number of irreversible kinetic equations to be derived. One example in the field of quantum measurement is presented—the problem of wave function collapse with the example of using oscillators as measurement device.

A POINT-LIKE PICTURE OF THE HYDROGEN ATOM

A point-like picture of the Schrödinger solution for hydrogen atom is worked to emphasize that "point-like particles" may describe as "probability wave function". In each case, the three-dimensional shape of the |Ψnlm(rn, cos θ)|2 is plotted and the paths of the point-like electron (it is better to say reduced mass of the pair particles) are described in each closed shell. Finally, the orbital shape of the molecules are given according to the present simple model. In our opinion, "interpretations of the Correspondence Principle", which is a basic principle in all elementary quantum text, seems to be reviewed again!

Resonances in the double-ionization signal of two-electron model atoms

Using numerical simulations, we examine the wave-function probability density of a model two-electron atom exposed to intense femtosecond laser pulses. We determine numerically the lowest-lying energies and eigenstates of the model atom. This information is then used to identify resonance features within the double-ionization yield, defined as probability density in which both electrons are far from the nucleus, and the atomic levels responsible for these features. This resonance behavior is confirmed using two distinct procedures.

Low-scale quantum gravity and double nucleon decay

In models with a low quantum gravity scale, one might expect sizable effects from nonrenormalizable interactions that violate the global symmetries of the standard model. While some mechanism must be invoked in such theories to suppress higher-dimension operators that contribute to proton decay, operators that change baryon number by two units are less dangerous and may be present at phenomenologically interesting levels. Here we focus on Delta B=2 operators that also change strangeness. We demonstrate how to compute explicitly a typical nucleon-nucleon decay amplitude, assuming a nonvanishing six-quark cluster probability and MIT bag model wave functions. We then use our results to estimate the rate for other possible modes. We find that such baryon-number-violating decays may be experimentally accessible if the operators in question are present and the Planck scale is less than ~ 400 TeV.

One-dimensional crystal with a complex periodic potential

A one-dimensional crystal model is constructed with a complex periodic potential. A wave function solution for the crystal model is derived without relying on Bloch functions. The new wave function solution of this model is shown to correspond to the solution for the probability amplitude of a two-level system. The energy discriminant is evaluated using an analytic formula derived from the probability amplitude solution, and based on an expansion parameter related to the energy and potential amplitude. From the wave function energy discriminant the crystal band structure is derived and related to standard energy bands and gaps. It is also shown that several of the properties of the two-level system apply to the one-dimensional crystal model. The two-level system solution which evolves in time is shown to manifest as a spatial configuration of the one-dimensional crystal model. The sensitivity of the wave function probability density is interpreted in the context of the new solution. The spatial configuration of the wave function, and the appearance of a long wavelength in the wave function probability density is explained in terms of the properties of Bessel functions.

Experimental evidence for spin-triplet superconductivity in Sr2RuO4

We review the experimental evidence for spin-triplet superconductivity in Sr2RuO4. The Knight shift experiments decisively demonstrated that the Cooper-pair symmetry is spin triplet. We discuss the most probable wave function of the Cooper pairs on the basis of the results of a number of key experiments. We point out that a simple picture of the unitary state with the isotropic gap is not compatible with the observed behavior of the specific heat, and that a profound modification of the pairing wave function is needed. We also present the in-plane angular dependence of the upper critical field, with the field applied exactly parallel to the quasi-two-dimensional plane. We discuss the possible implication of the observed fourfold anisotropy in connection with the proposed second superconducting state with a line-node gap.

Quantum Mechanics Based on Probability Wave Functions Induced by the Minimum Mean Deviation from Statistical Equilibrium. I

The maximum entropy probability distributionsare generally used as probabilistic models fordescribing statistical equilibrium subject to given meanvalues of some random variables. The paper deals with the construction of probability wave functionsby minimizing the mean deviation from statisticalequilibrium subject to generalized correlations, inducedby random fluctuations, whose values are determined looking for stationary points of the meanenergy of the system. The results are applied to thestudy of several quantum systems (the free particle ina box, two independent particles in a box, the harmonic oscillator, and the hydrogen atom, withouthaving to solve the corresponding Schrodingerequation.

Spectral density correlations and eigenfunction fluctuations in one-dimensional quasi-periodic systems

The authors study the scaling and correlation-fluctuation properties for the spectra and wavefunctions of a simple one-dimensional quasi-periodic system that displays an Anderson metal-insulator transition. This system and its extensions are models for studying localization phenomena, and their usefulness for the description of the Anderson transition as well as insulating and metallic phases in disordered systems is exploited. They present numerical work on the critical behaviour of the spectrum and wavefunctions, which display multifractal fluctuations. Appropriate probability densities are studied, and it is demonstrated that: (i) the subband energy width statistics, which may express spectral correlations, is consistent with linear level repulsion at the critical point and a Poisson distribution in the insulating regime: and (ii) the critical wavefunction probability amplitude distributions approach a universal function as the system size increases. It is concluded that certain critical properties of quasi-periodic models are similar to what is expected for electronic states in weakly disordered metals, and others show a striking similarity to mobility edge behaviour in three dimensions.

Quantum dynamics of the three-dimensional F+H2 reaction. II. Scattering wave function density and flux analysis

Further analysis of the quantum dynamics of the three-dimensional F+H2→FH+H reaction for total angular momentum J=0 focuses on the properties of the total scattering wave function. After assembling the scattering wave function (as outlined in the previous paper), we study throughout the interaction region the wave function probability density and flux. In order to present 3D figures, two different 2D projections of the wave function are utilized—constant s planes and constant m planes. Salient topological features are noted and compared at system energies below, on, and above the 0.36 eV resonance. Quantum whirlpools (which are quite sensitive to the total energy) are located in the reactant and product valleys.

Quantum ergodicity for time-dependent wave-packet dynamics

The dynamics of nonstationary wave packets in model quantum-mechanical systems are studied for the properties of ergodicity and instability. The long-time-average Wigner phase-space distribution is calculated for each wave packet and analyzed for uniformity on a surface of section. A mild transition from nonergodic to ergodic behavior is observed with increasing energy. These results are compared to long-time averages of the wave-function probability density. The stability of each packet is also analyzed by several methods, including the survival of the initial state and a new measure of separation of similar wave packets. The relationships between the wave-packet ergodicity and instability and the corresponding classical dynamics are developed. Comparisons to other quantum-ergodic studies, and implications for intramolecular energy transfer and classical-ergodic studies are discussed.

Fermi-Liquid Parameters ofHe3in the Brueckner Theory

Functional differentiation of the Brueckner-theory equations gives a prescription for calculating the effective interaction in a Fermi liquid. The many terms that result from the differentiation are calculated for the liquid-${\mathrm{He}}^{3}$ system using \O{}stgaard's reaction matrices. ${\mathrm{He}}^{3}$ is strongly interacting in the sense that the defect wave-function probability $\ensuremath{\kappa}$ is large, of the order 0.56. As a result, many of the terms contribute substantially to the total. Of the Landau parameters ${f}_{0}$, ${f}_{1}$, and ${z}_{0}$, only ${z}_{0}$ is in reasonable agreement with experiment. The parameter ${f}_{0}$ is too small. This is known to be due in part to the neglect of three-body clusters. We find a small value for ${f}_{1}$, implying that the effective mass ${m}^{*}$ is close to one. Larger effective masses could be obtained by treating low-momentum-transfer processes more carefully and by summing higher-order diagrams.

Interaction of Plasmons

The nonlinear interaction between plasma oscillation normal modes is studied in a hydrodynamic approximation. A modified quasilinear treatment is used, in which the normal modes are taken to affect each other through the slow nonlinear temporal and spatial changes they induce in the (equilibrium) background quantities. Plasmons, i.e., wave packets of plasma oscillations, are found in this approximation to have an interaction similar in form to the electromagnetic interaction. The plasmon interaction is describable in terms of scalar and vector potentials which satisfy wave equations of the form satisfied by their electromagnetic counterparts. Here, quantities proportional to the slow changes induced in the background number and current densities play the roles of the scalar and vector potentials, with the thermal velocity corresponding to the speed of light. The source terms in the potential wave equations are quadratic in the normal mode density perturbation. The normal mode density perturbation in turn satisfies a Klein-Gordon-like equation, so that, normalized, it would appear to play a role similar to the quantum-mechanical probability wave function Ψ. The nonlinear interaction leads to an instability with a growth rate proportional to the square root of the ratio of the plasma oscillation energy to the background thermal energy.

Nuclear Magnetic Resonance Studies of the Metallic Transition in Doped Silicon

The Si: P and Si: B systems have been studied using the methods of pulse and cw nuclear magnetic resonance. The purpose of this study is to investigate the transition of an impurity system in a solid from an array of isolated paramagnetic atoms or clusters of atoms to a superlattice of impurity atoms having strong wave-function overlap and metallic character. Knight shifts, line shapes, and nuclear spin relaxation times were measured for ${\mathrm{Si}}^{29}$ and ${\mathrm{B}}^{11}$ in $p$-type silicon and ${\mathrm{Si}}^{29}$ and ${\mathrm{P}}^{31}$ in $n$-type silicon. Phosphorus concentrations vary from ${10}^{17}$ to ${10}^{20}$ impurities/${\mathrm{cm}}^{3}$ and the temperature range investigated extends from 1.4 to 300\ifmmode^\circ\else\textdegree\fi{}K. Onset of metallic behavior in $n$-type silicon at 4\ifmmode\times\else\texttimes\fi{}${10}^{18}$ phosphorus impurities/${\mathrm{cm}}^{3}$ is indicated by the ${\mathrm{Si}}^{29}$ ${T}_{1}$ becoming proportional to ${T}^{\ensuremath{-}1}$ between 1.4 and 4.2\ifmmode^\circ\else\textdegree\fi{}K and by the existence of a Knight shift for ${\mathrm{Si}}^{29}$. Above a phosphorus concentration of approximately 3\ifmmode\times\else\texttimes\fi{}${10}^{19}$ ${\mathrm{cm}}^{\ensuremath{-}3}$, ${\mathrm{Si}}^{29}$ ${T}_{1}'\mathrm{s}$ and Knight shifts obey the Korringa relation. Broadening of the ${\mathrm{Si}}^{29}$ resonance line by 5 times the dipolar width and of the ${\mathrm{P}}^{31}$ resonance line by 100 times the dipolar width at concentrations of 1.4\ifmmode\times\else\texttimes\fi{}${10}^{20}$ ${\mathrm{cm}}^{\ensuremath{-}3}$ is shown to be caused by fluctuations of the local Knight shift about the average Knight shift value. Such fluctuations are explained by a model of a Poisson distribution for the local ${\mathrm{P}}^{31}$ impurity density with a threshold local density of 3\ifmmode\times\else\texttimes\fi{}${10}^{19}$ ${\mathrm{cm}}^{\ensuremath{-}3}$ for transition to metallic properties. This model agrees with the observed ${\mathrm{P}}^{31}$ resonance line shape and explains the transition to metallic behavior in $n$-type silicon. In $p$-type silicon, ${\mathrm{B}}^{11}$ and ${\mathrm{Si}}^{29}$ Knight shifts are measured for boron concentrations greater than 1\ifmmode\times\else\texttimes\fi{}${10}^{19}$ ${\mathrm{cm}}^{\ensuremath{-}3}$. The ${\mathrm{B}}^{11}$ ${T}_{1}'\mathrm{s}$ and Knight shifts agree with the Korringa relation within a 15% experimental error. However, both the ${\mathrm{B}}^{11}$ ${T}_{1}$ and Knight shift are independent of concentration for boron concentrations between 2\ifmmode\times\else\texttimes\fi{}${10}^{19}$ ${\mathrm{cm}}^{\ensuremath{-}3}$ and 8.5\ifmmode\times\else\texttimes\fi{}${10}^{19}$ ${\mathrm{cm}}^{\ensuremath{-}3}$. Such concentration independence may be explained by postulating a clustering of boron atoms at an average local density in a cluster greater than 8.5\ifmmode\times\else\texttimes\fi{}${10}^{19}$ ${\mathrm{cm}}^{\ensuremath{-}3}$. Wave function probability densities are calculated from Knight shifts with a free carrier density of states assumed valid. To facilitate comparison, wave-function densities are normalized per unit volume of the crystal and are 2600 ${\mathrm{cm}}^{\ensuremath{-}3}$ at ${\mathrm{P}}^{31}$ and 100 ${\mathrm{cm}}^{\ensuremath{-}3}$ at ${\mathrm{Si}}^{29}$ in $n$-type silicon and 80 ${\mathrm{cm}}^{\ensuremath{-}3}$ at ${\mathrm{Si}}^{29}$ in $p$-type silicon.

Statistical Theory of the Error in Approximate Wavefunctions

The theory of how a probability distribution may be estimated by sampling is applied to the problem of using approximate wavefunctions for quantum mechanical systems and estimating the errors involved. The most probable wavefunction of given mean energy is considered and some of its properties found. The energy variance, which gives a lower bound to the true energy, can be calculated. The theory is applied to the simple harmonic oscillator and the predicted relation between mean energy and variance is compared with the actual relations found using two different types of approximate wavefunction.