Energy Expectation Value Research Articles

Is gravitational mass of a composite quantum body equivalent to its energy?

Abstract We define passive gravitational mass operator of a hydrogen atom in the post-Newtonian approximation of general relativity and show that it does not commute with energy operator, taken in the absence of gravitational field. Nevertheless, the equivalence between the expectation values of passive gravitational mass and energy is shown to survive for stationary quantum states. Inequivalence between passive gravitational mass and energy at a macroscopic level results in time dependent oscillations of the expectation values of passive gravitational mass for superpositions of stationary quantum states, where the equivalence restores after averaging over time. Inequivalence between gravitational mass and energy at a microscopic level reveals itself as unusual electromagnetic radiation, emitted by the atoms, supported and moved in the Earth gravitational field with constant velocity using spacecraft or satellite, which can be experimentally measured.

Interacting boson model with energy density functionals

A brief overview of the recent advancement in the microscopic study of the interacting boson model (IBM) is given. A new nucleon-boson mapping method has been proposed recently, that derives the IBM Hamiltonian based on the self-consistent mean-field model with the microscopic energy density functional (EDF). The mean-field total energy surface computed with a given EDF is mapped onto the analogous energy expectation value in the boson condensate, thereby the energy spectrum and the transition rates are generated. Since the EDF framework allows an accurate global description of nuclear intrinsic properties, the IBM is derived in a unified way, basically for any situation of low-energy quadrupole collective states of nuclei. The basic notion of the new mapping technique is sketched.

Effect of Isotopic Substitution on Elementary Processes in Dye-Sensitized Solar Cells: Deuterated Amino-Phenyl Acid Dyes on TiO2

We present the first computational study of the effects of isotopic substitution on the operation of dye-sensitized solar cells. Ab initio molecular dynamics is used to study the effect of deuteration on light absorption, dye adsorption dynamics, the averaged over vibrations driving force to injection (∆Gi) and regeneration (∆Gr), as well as on promotion of electron back-donation in dyes NK1 (2E,4E-2-cyano-5-(4-dimethylaminophenyl)penta-2,4-dienoic acid) and NK7 (2E,4E-2-cyano-5-(4-diphenylaminophenyl)penta-2,4-dienoic acid) adsorbed in monodentate molecular and bidentate bridging dissociative configurations on the anatase (101) surface of TiO2. Deuteration causes a red shift of the absorption spectrum of the dye/TiO2 complex by about 5% (dozens of nm), which can noticeably affect the overlap with the solar spectrum in real cells. The dynamics effect on the driving force to injection and recombination (the difference between the averaged and ∆Gi,requil at the equilibrium configuration) is strong, yet there is surprisingly little isotopic effect: the average driving force to injection and to regeneration changes by only about 10 meV upon deuteration. The nuclear dynamics enhance recombination to the dye ground state due to the approach of the electron-donating group to TiO2, yet this effect is similar for deuterated and non-deuterated dyes. We conclude that the nuclear dynamics of the C-H(D) bonds, mostly affected by deuteration, might not be important for the operation of photoelectrochemical cells based on organic dyes. As the expectation value of the ground state energy is higher than its optimum geometry value (by up to 0.1 eV in the present case), nuclear motions will affect dye regeneration by recently proposed redox shuttle-dye combinations operating at low driving forces.

3H at Next-to-Next-to-Next-to Leading Order of the Chiral Expansion

The chiral three-nucleon force (3NF) at next-to-next-to-next-to leading order (N3LO) is used to calculate the triton wave function and the doublet nucleon–deuteron scattering length. This allows us to fix the values of the low-energy constants which are free parameters of the theory. The obtained values of these parameters, the expectation values of the kinetic energy, two- and three-body potentials and individual contributions of different parts of 3NF are given.

Effects of Nuclear Vibrations on the Energetics of Polythiophene: Quantized Energy Molecular Dynamics

The effects of nuclear dynamics on the energetics of polythiophene relevant for the performance of organic solar cells are studied for the first time. Nuclear motions change the expectation values of frontier orbital energies and the band gap by ~0.1 eV versus values at the equilibrium geometry, which is expected to have a significant effect on light absorption, charge separation, and donor regeneration. A new molecular dynamics algorithm that accounts for the quantum nature of vibrations is introduced. It reproduces effects of temperature and deuteration that are lost in the standard molecular dynamics. Inclusion of quantized vibrations leads to a broadening of the band gap and of energy levels by ~20 % at 300 K, while having little effect on their expectation values (which change by up to 0.03 eV). Increase in temperature from 300 to 400 K and deuteration cause an additional broadening of the spectrum by ~26 and 21 % respectively.

The quantum Lane-Emden-type Kanai-Caldirola oscillators

We obtain the exact wave functions of two Lane-Emden-type Kanai-Caldirola (LE-KC) oscillators by using the dynamical invariant method. To do so, we analytically solve the respective Milne-Pinney equation for each oscillator. We also calculate the uncertainty product, the transition probability and the quantum-mechanical energy expectation value for each oscillator. We compare the results with those of the well-known KC oscillator. The quantum-mechanical energy expectation value of the KC oscillator goes to zero faster than that of the LE-KC oscillators, indicating that the KC system is more damped than the LE-KC ones.

A simple Hill-series approach to the linear potential

A simple Hill-series method is shown to yield accurate energy levels and expectation values for the linear potential for any angular momentum L and confinement radius R. The numerical results are verified by comparison with those from two hypervirial perturbation methods which are specially constructed to give accurate results at L = 0 for small R values and at L > 1 for R = ∞.

Quantum dynamics in a time-dependent cylindrical trap

Solutions to the Schrödinger equation are examined for a particle inside a cylindrical trap of a circular time-dependent cross-section. Analytical expressions for energy and momentum expectation values are derived with respect to the exact solutions; and the adiabatic and sudden change of the boundary are discussed. The density profile as a function of time in a given observation point, resembles the diffraction-in-time pattern observed in a suddenly released particle but with an enhanced fringe visibility. Numerical computations are presented for both contracting and expanding boxes.

Diagrammatics for SU(2) invariant matrix product states

We report on a systematic implementation of su(2) invariance for matrix product states (MPS) with concrete computations cast in a diagrammatic language. As an application we present a variational MPS study of a spin-1/2 quantum chain. For efficient computations, we make systematic use of the su(2) symmetry at all steps of the calculations: (i) the matrix space is set up as a direct sum of irreducible representations, (ii) the local matrices with state-valued entries are set up as superposition of su(2) singlet operators, (iii) products of operators are evaluated algebraically by making use of identities for 3j and 6j symbols. The remaining numerical computations like the diagonalization of the associated transfer matrix and the minimization of the energy expectation value are done in spaces free of symmetry degeneracies. The energy expectation value is a strict upper bound of the true ground-state energy and yields definite conclusions about the accuracy of density-matrix renormalization group results reported in the literature. Furthermore, we present explicit results with accuracy better than 10−4 for nearest- and next-nearest-neighbour spin correlators and for general dimer–dimer correlators in the thermodynamical limit of the spin- Heisenberg chain with frustration.

Nonthermal Statistics in Isolated Quantum Spin Clusters after a Series of Perturbations

We show numerically that a finite isolated cluster of interacting spins 1/2 exhibits a surprising nonthermal statistics when subjected to a series of small nonadiabatic perturbations by an external magnetic field. The resulting occupations of energy eigenstates are significantly higher than the thermal ones on both the low and the high ends of the energy spectra. This behavior semiquantitatively agrees with the statistics predicted for the so-called "quantum microcanonical" ensemble, which includes all possible quantum superpositions with a given energy expectation value. Our findings also indicate that the eigenstates of the perturbation operators are generically localized in the energy basis of the unperturbed Hamiltonian. This kind of localization possibly protects the thermal behavior in the macroscopic limit.

Monte Carlo sampling of energy-constrained quantum superpositions in high-dimensional Hilbert spaces

Recent studies into the properties of quantum statistical ensembles in high-dimensional Hilbert spaces have encountered difficulties associated with the Monte-Carlo sampling of quantum superpositions constrained by the energy expectation value. A straightforward Monte-Carlo routine would enclose the energy constrained manifold within a larger manifold, which is easy to sample, for example, a hypercube. The efficiency of such a sampling routine decreases exponentially with the increase of the dimension of the Hilbert space, because the volume of the enclosing manifold becomes exponentially larger than the volume of the manifold of interest. The present paper explores the ways to optimise the above routine by varying the shapes of the manifolds enclosing the energy-constrained manifold. The resulting improvement in the sampling efficiency is about a factor of five for a 14-dimensional Hilbert space. The advantage of the above algorithm is that it does not compromise on the rigorous statistical nature of the sampling outcome and hence can be used to test other more sophisticated Monte-Carlo routines. The present attempts to optimise the enclosing manifolds also bring insights into the geometrical properties of the energy constrained manifold itself.

CORRELATION ENERGY AND ENTANGLEMENT GAP IN CONTINUOUS MODELS

Our goal is to clarify the relation between entanglement and correlation energy in a bipartite system with infinite dimensional Hilbert space. To this aim, we consider the completely solvable Moshinsky's model of two linearly coupled harmonic oscillators. Also, for small values of the couplings, the entanglement of the ground state is nonlinearly related to the correlation energy, involving logarithmic or algebraic corrections. Then, looking for witness observables of the entanglement, we show how to give a physical interpretation of the correlation energy. In particular, we have proven that there exists a set of separable states, continuously connected with the Hartree–Fock state, which may have a larger overlap with the exact ground state, but also a larger energy expectation value. In this sense, the correlation energy provides an entanglement gap, i.e. an energy scale, under which measurements performed on the 1-particle harmonic sub-system can discriminate the ground state from any other separated state of the system. However, in order to verify the generality of the procedure, we have compared the energy distribution cumulants for the 1-particle harmonic sub-system of the Moshinsky's model with the case of a coupling with a damping Ohmic bath at 0 temperature.

Exact and variational calculations of a hydrogenic impurity binding energy in a multilayered spherical quantum dot

The binding energy of a central or an off-central impurity located in a multilayered spherical quantum dot has been calculated. Different effective masses and dielectric constants in the different regions of the MSQD are taken into consideration. New analytical forms for the exact solution of the radial Schrödinger equation in the absence of the impurity and in the presence of a central impurity have been introduced. The analytical forms in the presence of the central impurity have the advantage that they are valid for any electron energy negative or positive unlike the forms used hitherto. The variational method has also been applied for both central and off-central impurities. The trial wavefunction has been taken in a new form in addition to the conventional form used in previous treatments. The new form has the advantage of satisfying the boundary conditions at the interfaces in the case of different masses. It further raises the results of binding energy so that they approach the exact solution from below. The two forms have yielded new analytical expressions for the energy expectation value and binding energy.

Interpreting quantum states of electromagnetic field in time-dependent linear media

Recently, Pedrosa and Rosas [Phys. Rev. Lett. 103, 010402 (2009)] investigated the quantum states of an electromagnetic field in time-dependent linear media using a Hermitian linear invariant. The wave function obtained by them is represented in terms of an arbitrary weight function $g({\ensuremath{\lambda}}_{l})$. Since the type of wave function varies depending on the choice of $g({\ensuremath{\lambda}}_{l})$ in their problem, it may be a difficult task to construct a coherent state that resembles the classical state from their theory. We suggest, on the basis of a non-Hermitian linear invariant, another quantum state that is a kind of coherent state. The expectation value of canonical variables in this alternate state follows an exact classical trajectory. For a simple case in which the time dependence of the parameters $\ensuremath{\epsilon}(t)$, $\ensuremath{\mu}(t)$, and $\ensuremath{\sigma}(t)$ disappears, we showed that the quantum energy expectation value in the alternate quantum state recovers exactly to the classical energy in the limit $\ensuremath{\hbar}\ensuremath{\rightarrow}0$. This alternate state leads to the correspondence between the quantum and the classical behaviors of physical observables in a high-energy limit.

ChemInform Abstract: Energy Component Analysis of the Jahn-Teller Effect in the Methane Radical Cation.

Extensive configuration‐interaction calculations with double‐zeta plus polarization and near triple‐zeta plus polarization basis sets are used to analyze the Jahn–Teller (JT) effect in the methane radical cation. Energy component analysis shows that the Jahn–Teller effect leads to a decrease in the expectation value of the electron‐nuclear attraction energy, an increase in the expectation value of the interelectronic repulsion energy, and an increase in the internuclear repulsion energy. These observations are consistent with a contraction of the electron cloud. The dominant factor in the −0.0550 hartree Jahn–Teller distortion (Td→C2v) in CH+4 is the −0.5262 hartree change in the electron–nuclear attraction energy. The differences in all energy components are large in relation to the JT distortion. Interelectronic repulsion plays a dominant role in determining the relative energies of the possible JT distorted structures, but electron correlation effects are relatively unimportant.

An efficient recursive algorithm to compute wave function optimization gradients for the graphically contracted function method

AbstractAn efficient recursive algorithm is presented to compute orbital‐level Hamiltonian matrices for wave functions expanded in a basis of graphically contracted functions (GCF). Each GCF depends on a nonlinear set of parameters called arc factors. The orbital‐level Hamiltonian matrices characterize the dependence of the energy on the wave function changes associated with a subset of these nonlinear parameters corresponding to an individual molecular orbital. From these Hamiltonian matrices, gradients with respect to the nonlinear arc factor parameters may be computed and other arc factor optimization algorithms may be used. The recursive algorithm allows the orbital‐level Hamiltonian matrices to be computed with O(Nωn4) total effort, where NGCF is the dimension of the GCF basis, n is the dimension of the orbital basis, and where the scale factor ω depends on the number of electrons N and ranges from O(N0) to O(N2) depending on the complexity of the underlying Shavitt graph. This effort is about two to five times that required to compute an energy expectation value for a given set of arc factors; thus the energy and gradient have the same scaling behavior with increasing molecule size, NGCF dimension, and orbital basis size. Timings are given for wave functions that correspond to configuration state function expansions over 1073 in length, many orders of magnitude larger than can be considered using traditional electronic structure methods. © 2010 Wiley Periodicals, Inc. Int J Quantum Chem, 2010

Hydrogenic impurity in a coaxial quantum well wire: Effect of different masses of wires and barriers

A variational calculation is given within the effective mass approximation of the binding energy of an axial or an off-axial impurity located in coaxial quantum well wire. A uniform magnetic field is applied along the axis of the wire. The different effective masses in the different regions of the CQWW are taken into consideration. The Schrödinger radial equation in the absence of the impurity has been solved analytically in the different regions instead of using the numerical methods employed in earlier treatments. In the presence of the impurity, the trial wavefunction has been taken in a new form in addition to the conventional form used in previous work. The new form has the advantage of satisfying the boundary conditions at all the interfaces in the case of different masses. It tends to raise the results of binding energy so that they approach the exact solution from below. Moreover, some new analytical expressions for the energy expectation value and for the binding energy have been derived

Relativistic diffusion with friction on a pseudo-Riemannian manifold

We study a relativistic diffusion equation on the Riemannian phase space defined by Franchi and Le Jan. We discuss stochastic Ito (Langevin) differential equations as a perturbation by noise of the geodesic equation. We show that the expectation values of the angular momentum and the energy grow exponentially fast. We discuss drifts leading to an equilibrium. As an example we consider a particle in the de Sitter universe. It is shown that the relativistic diffusion of momentum in the de Sitter space is the same as the relativistic diffusion on the Minkowski mass-shell with the temperature proportional to the de Sitter radius. We study a diffusion process with a drift corresponding to the Jüttner or quantum equilibrium distributions. We show that such a diffusion has bounded expectation values of angular momentum and energy. The energy and the angular momentum tend exponentially fast to their equilibrium values.

Application of pseudo-Hermitian quantum mechanics to a complex scattering potential with point interactions

We present a generalization of the perturbative construction of the metric operator for non-Hermitian Hamiltonians with more than one perturbation parameter. We use this method to study the non-Hermitian scattering Hamiltonian H = p2/2m + ζ−δ(x + α) + ζ+δ(x − α), where ζ± and α are respectively complex and real parameters and δ(x) is the Dirac delta function. For regions in the space of coupling constants ζ± where H is quasi-Hermitian and there are no complex bound states or spectral singularities, we construct a (positive-definite) metric operator η and the corresponding equivalent Hermitian Hamiltonian h. η turns out to be a (perturbatively) bounded operator for the cases where the imaginary part of the coupling constants have the opposite sign, ℑ(ζ+) = −ℑ(ζ−). This in particular contains the -symmetric case: ζ+ = ζ*−. We also calculate the energy expectation values for certain Gaussian wave packets to study the nonlocal nature of h or equivalently the non-Hermitian nature of H. We show that these physical quantities are not directly sensitive to the presence of the -symmetry.

Exact quantum motion of a single trapped ion interacting with standing laser pulses in Lamb-Dicke regime

We investigate the secular motion of a single Paul-trapped ion in the Lamb-Dicke regime, which interacts with a sequence of standing laser pulses. By using the ansatz method, we get an exact quantum solution of the system. Based on the wave-packet trains described by the exact solution, we find that: i) The center, height and width of the wave-packet trains depend on the strength of laser pulses, the deformation and spread of the wave-packet trains can be controlled by adjusting the strength of laser. ii) Energy expectation values of the ion show jumps at the instantaneous switching on of the laser pulses. In the time intervals when the laser pulses are switched off some narrow energy bands are generated. iii) When the strength of laser pulses reaches a critical value, the system changes its stability.