Expectation Values Of Position Research Articles

Echoes and revival echoes in systems of anharmonically confined atoms

We study echoes and what we call 'revival echoes' for a collection of atoms that are described by a single quantum wavefunction and are confined in a weakly anharmonic trap. The echoes and revival echoes are induced by applying two, successive temporally localized potential perturbations to the confining potential, one at time $t=0$, and a smaller one at time $t=\tau$. Pulse-like responses in the expectation value of position $<x(t)>$ are predicted at $t \approx n\tau$ ($n=2,3,...$) and are particularly evident at $t \approx 2\tau$. A novel result of our study is the finding of 'revival echoes'. Revivals (but not echoes) occur even if the second perturbation is absent. In particular, in the absence of the second perturbation, the response to the first perturbation dies away, but then reassembles, producing a response at revival times $mT_x$ ($m=1,2,...$). Including the second perturbation at $t=\tau$, we find temporally localized responses, revival echoes, both before and after $t\approx mT_x$, e.g., at $t\approx m T_x-n \tau$ (pre-revival echoes) and at $t\approx mT_x+n\tau$, (post-revival echoes), where $m$ and $n$ are $1,2,...$ . Depending on the form of the perturbations, the 'principal' revival echoes at $t \approx T_x \pm \tau$ can be much larger than the echo at $t \approx 2\tau$. We develop a perturbative model for these phenomena, and compare its predictions to the numerical solutions of the time-dependent Schr\"odinger Equation. The scaling of the size of the various echoes and revival echoes as a function of the symmetry and size of the perturbations applied at $t=0$ and $t=\tau$ is investigated. We also study the presence of revivals and revival echoes in higher moments of position, $<x^p (t)>$, $p>1$, and the effect of atom-atom interactions on these phenomena.

Coherent states for the nonlinear harmonic oscillator

Wave packets for the quantum nonlinear oscillator are considered in the generalized coherent state framework. To first order in the nonlinearity parameter the coherent state behaves very similar to its classical counterpart. The position expectation value oscillates in a simple harmonic manner. The energy-momentum uncertainty relation is time independent as in a harmonic oscillator. Various features (such as the squeezed state nature) of the coherent state have been discussed.

Complexified coherent states and quantum evolution with non-Hermitian Hamiltonians

The complex geometry underlying the Schrödinger dynamics of coherent states for non-Hermitian Hamiltonians is investigated. In particular, two seemingly contradictory approaches are compared: (i) a complex WKB formalism, for which the centres of coherent states naturally evolve along complex trajectories, which leads to a class of complexified coherent states; (ii) the investigation of the dynamical equations for the real expectation values of position and momentum, for which an Ehrenfest theorem has been derived in a previous paper, yielding real but non-Hamiltonian classical dynamics on phase space for the real centres of coherent states. Both approaches become exact for quadratic Hamiltonians. The apparent contradiction is resolved building on an observation by Huber, Heller and Littlejohn, that complexified coherent states are equivalent if their centres lie on a specific complex Lagrangian manifold. A rich underlying complex symplectic geometry is unravelled. In particular, a natural complex structure is identified that defines a projection from complex to real phase space, mapping complexified coherent states to their real equivalents.This article is part of a special issue of Journal of Physics A: Mathematical and Theoretical devoted to ‘Coherent states: mathematical and physical aspects’.

Dipole induced transitions in an anharmonic oscillator: A dynamical mean field model

We calculate transition probabilities between discrete states of an anharmonic oscillator induced by a time dependent dipole. Our prototype oscillator includes a quadratic term n2 in the quantum number n. The corresponding unperturbed Hamiltonian is expressed in terms of deformed ladder operators. To consider the perturbation due to the dipole interaction we introduce a dynamical mean field model Hamiltonian for the oscillator part. The resulting system is a parametric harmonic oscillator with a time dependent frequency determined self-consistently. We present results for the time dependence of transition probabilities for different pulses, as well as the expectation values of position and momentum.

Noncommutative quantum mechanics of a test particle under linearized gravitational waves

We consider the quantum dynamics of a test particle in noncommutative space under the influence of linearized gravitational waves in the long wave-length and low-velocity limit. A prescription for quantizing the classical Hamiltonian for the interaction of gravitational wave with matter in noncommutative space is proposed. The Hamiltonian (and hence the system) is then exactly solved by using standard algebraic methods. The solutions show prominent signatures of the noncommutative nature of space. Computation of the expectation value of the particle's position reveals the inherent quantum nature of spacetime noncommutativity.

On the Ehrenfest theorem of quantum mechanics

We give a mathematically rigorous derivation of Ehrenfest’s equations for the evolution of position and momentum expectation values under general and natural assumptions which include atomic and molecular Hamiltonians with Coulomb interactions.

A Systematic Approach for Computing Zero-Point Energy, Quantum Partition Function, and Tunneling Effect Based on Kleinert's Variational Perturbation Theory.

In this paper, we describe an automated integration-free path-integral (AIF-PI) method, based on Kleinert's variational perturbation (KP) theory, to treat internuclear quantum-statistical effects in molecular systems. We have developed an analytical method to obtain the centroid potential as a function of the variational parameter in the KP theory, which avoids numerical difficulties in path-integral Monte Carlo or molecular dynamics simulations, especially at the limit of zero-temperature. Consequently, the variational calculations using the KP theory can be efficiently carried out beyond the first order, i.e., the Giachetti-Tognetti-Feynman-Kleinert variational approach, for realistic chemical applications. By making use of the approximation of independent instantaneous normal modes (INM), the AIF-PI method can readily be applied to many-body systems. Previously, we have shown that in the INM approximation, the AIF-PI method is accurate for computing the quantum partition function of a water molecule (3 degrees of freedom) and the quantum correction factor for the collinear H(3) reaction rate (2 degrees of freedom). In this work, the accuracy and properties of the KP theory are further investigated by using the first three order perturbations on an asymmetric double-well potential, the bond vibrations of H(2), HF, and HCl represented by the Morse potential, and a proton-transfer barrier modeled by the Eckart potential. The zero-point energy, quantum partition function, and tunneling factor for these systems have been determined and are found to be in excellent agreement with the exact quantum results. Using our new analytical results at the zero-temperature limit, we show that the minimum value of the computed centroid potential in the KP theory is in excellent agreement with the ground state energy (zero-point energy) and the position of the centroid potential minimum is the expectation value of particle position in wave mechanics. The fast convergent property of the KP theory is further examined in comparison with results from the traditional Rayleigh-Ritz variational approach and Rayleigh-Schrödinger perturbation theory in wave mechanics. The present method can be used for thermodynamic and quantum dynamic calculations, including to systematically determine the exact value of zero-point energy and to study kinetic isotope effects for chemical reactions in solution and in enzymes.

Time development in quantum mechanics using a reduced Hilbert space approach

We have created a suite of open source programs that numerically calculate and visualize the evolution of arbitrary initial quantum-mechanical bound states. The calculations are based on the expansion of an arbitrary wave function in terms of basis vectors in a reduced Hilbert space. The approach is stable, fast, and accurate at depicting the long-time dependence of complicated bound states. Several real-time visualizations, such as the position and momentum expectation values and the Wigner quasiprobability distribution for the position and momentum, can be shown. We use these computational tools to study the time-dependent properties of quantum-mechanical systems and discuss the effectiveness of the algorithm.

FFT-split-operator code for solving the Dirac equation in 2+1 dimensions

The main part of the code presented in this work represents an implementation of the split-operator method [J.A. Fleck, J.R. Morris, M.D. Feit, Appl. Phys. 10 (1976) 129–160; R. Heather, Comput. Phys. Comm. 63 (1991) 446] for calculating the time-evolution of Dirac wave functions. It allows to study the dynamics of electronic Dirac wave packets under the influence of any number of laser pulses and its interaction with any number of charged ion potentials. The initial wave function can be either a free Gaussian wave packet or an arbitrary discretized spinor function that is loaded from a file provided by the user. The latter option includes Dirac bound state wave functions. The code itself contains the necessary tools for constructing such wave functions for a single-electron ion. With the help of self-adaptive numerical grids, we are able to study the electron dynamics for various problems in 2+1 dimensions at high spatial and temporal resolutions that are otherwise unachievable. Along with the position and momentum space probability density distributions, various physical observables, such as the expectation values of position and momentum, can be recorded in a time-dependent way. The electromagnetic spectrum that is emitted by the evolving particle can also be calculated with this code. Finally, for planning and comparison purposes, both the time-evolution and the emission spectrum can also be treated in an entirely classical relativistic way. Besides the implementation of the above-mentioned algorithms, the program also contains a large C++ class library to model the geometric algebra representation of spinors that we use for representing the Dirac wave function. This is why the code is called “Dirac++”. Program summary Program title: Dirac++ or (abbreviated) d++ Catalogue identifier: AEAS_v1_0 Program summary URL: http://cpc.cs.qub.ac.uk/summaries/AEAS_v1_0.html Program obtainable from: CPC Program Library, Queen's University, Belfast, N. Ireland Licensing provisions: Standard CPC licence, http://cpc.cs.qub.ac.uk/licence/licence.html No. of lines in distributed program, including test data, etc.: 474 937 No. of bytes in distributed program, including test data, etc.: 4 128 347 Distribution format: tar.gz Programming language: C++ Computer: Any, but SMP systems are preferred Operating system: Linux and MacOS X are actively supported by the current version. Earlier versions were also tested successfully on IRIX and AIX Number of processors used: Generally unlimited, but best scaling with 2–4 processors for typical problems RAM: 160 Megabytes minimum for the examples given here Classification: 2.7 External routines: FFTW Library [3,4], Gnu Scientific Library [5], bzip2, bunzip2 Nature of problem: The relativistic time evolution of wave functions according to the Dirac equation is a challenging numerical task. Especially for an electron in the presence of high intensity laser beams and/or highly charged ions, this type of problem is of considerable interest to atomic physicists. Solution method: The code employs the split-operator method [1,2], combined with fast Fourier transforms (FFT) for calculating any occurring spatial derivatives, to solve the given problem. An autocorrelation spectral method [6] is provided to generate a bound state for use as the initial wave function of further dynamical studies. Restrictions: The code in its current form is restricted to problems in two spatial dimensions. Otherwise it is only limited by CPU time and memory that one can afford to spend on a particular problem. Unusual features: The code features dynamically adapting position and momentum space grids to keep execution time and memory requirements as small as possible. It employs an object-oriented approach, and it relies on a Clifford algebra class library to represent the mathematical objects of the Dirac formalism which we employ. Besides that it includes a feature (typically called “checkpointing”) which allows the resumption of an interrupted calculation. Additional comments: Along with the program's source code, we provide several sample configuration files, a pre-calculated bound state wave function, and template files for the analysis of the results with both MatLab and Igor Pro. Running time: Running time ranges from a few minutes for simple tests up to several days, even weeks for real-world physical problems that require very large grids or very small time steps.

Quantum Signatures of Chaos in Adiabatic Interaction Between a Trapped Ion and a Laser Standing Wave

We study quantum motion around a classical heteroclinic point of a single trapped ion interacting with a strong laser standing wave. We construct a set of exact coherent states of the quantum system and from the exact solutions reveal that quantum signatures of chaos can be induced by the adiabatic interaction between the trapped ion and the laser standing wave, where the quantum expectation values of position and momentum correspond to the classically chaotic orbit. The chaotic region on the phase space is illustrated. The energy crossing and quantum resonance in time evolution and the exponentially increased Heisenberg uncertainty are found. The results suggest a theoretical scheme for controlling the unstable regular and chaotic motions.

A Coherent Extension of the Transport Equations in Semiconductors Incorporating the Quantum Correction—Part I: Single-Particle Dynamics

The aim of the investigation is to consistently incorporate quantum corrections in the transport model for applications to nanoscale semiconductor devices. This paper is comprised of two parts. Part I derives a set of two semiclassical equations in which the dynamics of the dispersion of the single-particle wave function is accounted for in addition to that of the expectation value of position. The model is founded on an approximate description of the wave function that eliminates the need of the Ehrenfest approximation. This leads to a set of two Newton-like single-particle equations for position and dispersion. In Part II, it will be shown that the Lagrangian form of the single-particle equations naturally lends itself to the incorporation of such extended dynamics into the statistical framework. The theory is suitable for different levels of applications: description of the single-particle ballistic dynamics, solution of the generalized Boltzmann equation by the Monte Carlo method or other methods, and solution of the continuity equations in the position-dispersion space.

A Coherent Extension of the Transport Equations in Semiconductors Incorporating the Quantum Correction—Part II: Collective Transport

For part I see ibid., vol.4, p.495 (2005). The aim of this investigation is to consistently incorporate quantum corrections in the transport model for applications to nanoscale semiconductor devices. This paper is made of two parts. In Part I, a set of two semiclassical equations were derived, in which the dynamics of the dispersion of the single-particle wave function is accounted for in addition to that of the expectation value of position. The model is founded on an approximate description of the wave function that eliminates the need for the Ehrenfest approximation. This leads to a set of two Newton-like single-particle equations for position and dispersion. Here, in Part II, it is shown that the Lagrangian form of the single-particle equations naturally lends itself to the incorporation of such extended dynamics into the statistical framework. The theory is suitable for different levels of applications: description of the single-particle ballistic dynamics, solution of the generalized Boltzmann equation by Monte Carlo or other methods, and solution of the continuity equations in the position-dispersion space.

Exact Results for `Bouncing' Gaussian Wave Packets

We consider time-dependent Gaussian wave packet solutions of the Schrödinger equation, with arbitrary initial central position, x0, and momentum, p0, for an otherwise free particle, but with an infinite wall at x = 0, so-called bouncing wave packets. We show how difference or mirror solutions of the form ψ(x,t) − ψ(−x,t) can, in this case, be normalized exactly, allowing for the evaluation of a number of time-dependent expectation values and other quantities in closed form. For example, we calculate ⟨p2⟩t explicitly which illustrates how the free-particle kinetic (and hence total energy) is affected by the presence of the distant boundary. We also discuss the time dependence of the expectation values of position, ⟨x⟩t, and momentum, ⟨p⟩t, and their relation to the impulsive force during the `collision' with the wall. Finally, the x0, p0 → 0 limit is shown to reduce a special case of a non-standard free-particle Gaussian solution. The addition of this example to the literature then expands of the relatively small number of Gaussian solutions to quantum mechanical problems with familiar classical analogs (free particle, uniform acceleration, harmonic oscillator, unstable oscillator, and uniform magnetic field) available in closed form.

A complete description of the spin force

Exact three-dimensional solutions of the Dirac equation describing an electron driven by anarbitrarily intense laser field are shown. Those solutions present new features of thewavepackets and constitute the first exact evidence of the spin-induced force that deviatesthe position expectation value from the classical electron trajectory according to the spin ofthe electron.

The Force Exerted by the Walls of an Infinite Square Well on a Wave Packet: Ehrenfest Theorem, Revivals and Fractional Revivals

The temporal evolution of a wave packet ψ(x,t) confined to an infinite square well (occupying the region 0 ≤ x ≤ L), as manifested by fractional revivals of the probability density P(x,t) ≡ | ψ(x,t) |2 and the time-dependence of the expectation values of position, momentum and force (X(t) ≡ ⟨x⟩t, Π(t) ≡ ⟨ p̂ ⟩t and Φ(t) ≡ ⟨F̂⟩t, respectively) are studied, using wave packets for which Π(0) = 0. In the region accessible to the wave packet (0 < x < L), p̂ and the Hamiltonian Ĥ commute; still, p̂ is, in general, not a constant of the motion, and the three expectation values are all time-dependent, except when the initial packet has a definite parity, that is ψ(x,0) = ±ψ(L - x, 0), which implies X(t) = L/2 and Π(t) = 0 = Φ(t). For a packet with no definite parity, Y(t) ≡ X(t) - L/2 stays, if the packet is sufficiently narrow at t = 0 close to zero nearly all the time, but executes large excursions in the neighbourhood of instants at which the probability density experiences a mirror revival or a revival. This behaviour, with no classical counterpart, can be understood by paying attention to the presence of infinite potential barriers (at x = 0 and x = L) and taking account of the net force felt by an initially off-centred wave packet. It is shown that though plots (against time) of expectation values catch a glimpse of the revival structure of a wave packet, such plots fail to reveal some fractional revivals.

Dynamical theory of molecular photoionization: electron ejection kinematics and angular distributions from molecules fixed in space

Photoionization of molecules and other atomic aggregates fixed in space is studied from a dynamical perspective employing solutions of the time-dependent Schrödinger equation. The wave functions and associated kinematical behaviors of excited and ejected electrons are determined explicitly in short- and long-time limits, generalizing early results of Bethe for central potentials to anisotropic targets. Ehrenfest’s theorem is employed to clarify the origins of the transient forces and consequent kinematical behaviors of excited and ionized K-shell electrons obtained from explicit solutions of the time-dependent Schrödinger equation, revealing hybrid classical-quantal behaviors in wave functions and expectation values of position and momentum operators. In addition to providing pedagogical insights into time-resolved aspects of these fundamental photoprocesses, and a corresponding basis for theoretical studies of dynamical target relaxation, associated photoelectron-ion correlation, and other specifically time-dependent post-collision interaction phenomena, the new development resolves technical issues associated with the irreducibly infinite degeneracy of the scattering continuum for the anisotropic potentials characteristic of polyatomic molecules. Photoionization cross sections differential in electron ejection angles are derived from the formalism for fixed-in-space molecules of arbitrary complexity which are applicable in both time-resolved and steady-state situations, and an invariant subspace of excitation and ionization functions required in computational applications of the approach is constructed employing Lanczos–Krylov sequences of L 2 vectors and previously devised Stieltjes–Akhiezer methods, without reference to or calculations of the underlying individual discrete or continuum eigenstates. Electron angular-distribution data for fixed-in-space molecules measured over the full (4 π) range of emission angles for all molecular orientations are shown to be highly redundant, and to provide invariant physical information in the form of no more than three irreducible-tensor body-frame complex angular emission amplitudes at a given photon energy. These issues are illustrated with applications to ionization of fixed-in-space diatomic molecules for linear and circularly polarized incident radiation, in which cases the minimal set of molecular configurations and polarizations required to determine the invariant body-frame emission amplitudes is described in detail.

Study of Relations between Position and Momentum Expectation Values for Molecules

Various established relations between position and momentum expectation values based on the maximum-entropy principle have been tested for molecular systems. The results show that most of these bounds tested here are not very good, and that more work is needed to obtain better ones. It is hoped that this paper will stimulate new derivations. Atomic units (e = m = ℏ = 1) are used throughout the paper.

Electron helicity-dependence in (e,e′p) reactions with polarized nuclei and the fifth response function

The cross section for electron-induced proton knockout reactions with polarized beam and target is computed in a DWIA model. The analysis is focused on the electron helicity-dependent (HD) piece of the cross section. For the case of in-plane emission, a new symmetry is found in the HD cross section, which takes opposite values for specific opposing pairs of nuclear orientations, explaining why the fifth response (involving polarized electrons, but unpolarized nuclei) is zero for this choice of kinematics. Even when this symmetry breaks down in the case of nucleon emission out of the scattering plane, it is shown that it is still present in the separate response functions. A physical explanation of these results is presented using a semiclassical model of the reaction based on the PWIA and on the concept of a nucleon orbit with its implied expectation values of position and spin.

Influence of quantum-well structural parameters on capacitance–voltage characteristics

The effects of quantum well (QW) parameters on the apparent carrier distribution (ACD) are investigated using a self-consistent simulation and the capacitance–voltage (C–V) profiling techniques. The simulation results show that the change of position expectation value of two-dimensional electrons (Δx̄2D) completely determines the full width at half maximum (FWHM) of 100 K ACD peak of In0.23Ga0.77As/GaAs single QW for the well width tw⩾125 Å. The value of Δx̄2D is insensitive to the variation of conduction band offset ΔEc for ΔEc⩾120 meV. The influence of the Debye averaging effects on the ACDs increases as the values of tw and ΔEc are decreased. In multi-QW structure, the Debye averaging effects influence the FWHM of ACD peaks mostly in the first QW. Extraction of QW parameters directly from the measured C–V profile is finally discussed.

Electron distribution and capacitance–voltage characteristics of n-doped quantum wells

The effect of multiple quantum well (MQW) parameters on the free carrier distribution (FCD) and the apparent carrier distribution (ACD) obtained from capacitance–voltage (C–V) profiling has been investigated using a self-consistent simulation technique and the C–V profiling technique. The FCD in MQW structures reveals large outer peaks and small inner peaks even when all the parameters of each quantum wells (QWs) are the same. Such a feature becomes more apparent as the barrier layer becomes thinner or the barrier doping level becomes lower. These characteristics are found to originate from the fact that the density of carriers confined in each well is mostly determined by the depletion region formed alongside the well via the charge neutrality condition. The ACD is found to vary drastically as the thickness or the doping level of barrier changes. When the Debye averaging process is prominent, the ACD peaks are broader and smaller than the FCD peaks and are displaced toward the bottom layer side. The ACD inner peaks even disappear completely when the Debye screening length is comparable to or larger than the QW period, while real free carriers are well confined in each well. The effect of temperature on the ACD through the Debye averaging process is also investigated. When the Debye length is much smaller than the QW period, the full width at half maximum of the ACD is determined by the change of the position expectation value of the two-dimensional differential carriers. This change of position expectation value is found to be much smaller than the well width for relatively narrow QWs. The accuracy of our simulation results is confirmed by the excellent agreements between the simulated ACDs and the measured ACDs of In0.2Ga0.8As/GaAs MQWs. As an example of extracting the qualitative informations from the measured C–V profile, the C–V profiles of partially strain relaxed InxGa1−xAs/GaAs MQWs with x=0.15 and x=0.25, grown by metal organic chemical vapor deposition, are discussed. These results show systematically how the QW parameters affect the FCD and the ACD.