Critical assessment of the Schrödinger picture of quantum mechanics

We provide an example in which the Heisenberg and the Schrödinger pictures of quantum mechanics give different results, thus confirming the statement of P.A.M. Dirac that the two pictures may lead to inequivalent results. We consider a one-dimensional nonrelativistic charged harmonic oscillator (frequency ω₀ and mass m), and take into account the action of the radiation reaction and the vacuum electromagnetic forces on the charged oscillator. We show that the Heisenberg picture gives the correct value, ℏω₀/2, for the ground state energy of the harmonic oscillator in both cases of classical and quantized vacuum fields. In the case of the Schrödinger picture, considering classical vacuum fields, and using a simple calculation for the classical radiation reaction force that is valid in the limit of large mass (mc²⪢ℏω₀), we obtain the value ℏω₀ for the ground state energy of the harmonic oscillator. We show that the vacuum electromagnetic forces play a very important role in the understanding of this discrepancy.     

The dynamical-quantization approach to open quantum systems

The dynamical-quantization approach to open quantum systems does consist in quantizing the Brownian motion starting directly from its stochastic dynamics under the framework of both Langevin and Fokker–Planck equations, without alluding to any model Hamiltonian. On the ground of this non-Hamiltonian quantization method, we can derive a non-Markovian Caldeira–Leggett quantum master equation as well as a non-Markovian quantum Smoluchowski equation. The former is solved for the case of a quantum Brownian particle in a gravitational field whilst the latter for a harmonic oscillator. In both physical situations, we come up with the existence of a non-equilibrium thermal quantum force and investigate its classical limit at high temperatures as well as its quantum limit at zero temperature. Further, as a physical application of our quantum Smoluchowski equation, we take up the tunneling phenomenon of a non-inertial quantum Brownian particle over a potential barrier. Lastly, we wish to point out, corroborating conclusions reached in our previous paper [A. O. Bolivar, Ann. Phys. 326 (2011) 1354], that the theoretical predictions in the present article uphold the view that our non-Hamiltonian quantum mechanics is able to capture novel features inherent in quantum Brownian motion, thereby overcoming shortcomings underlying the Caldeira–Leggett Hamiltonian model.     

Quanta and coherent states

The quantum harmonic oscillator can be considered as a composite system of indistinguishable Bose-Einstein symmetric two-level-systems (quanta). In analogy to the classical Poisson limit theorem, we show that a coherent state is the limit of a sequence of homogeneous product states (coherent spin states) and discuss statistical properties of the quanta in classical and nonclassical states.     

Quantum corrected cell model for an anharmonic generalized Lennard-Jones solid

A simple method is proposed to dispose the quantum effect and anharmonic effect at the same time. Considering the quantum effect is remarkable only at low temperature, and tends to zero at high temperature, the potential energy of an atom is expanded harmonically to consider the quantum effect of solids within the harmonic oscillator framework. The anharmonic effect is remarkable only at high temperature, and tends to zero at low temperature, it was disposed by using a classical approximation. The universal formalism is applied to the generalized Lennard-Jones solid. The comparison shows that the results with and without anharmonic effect are in agreement with each other at some low temperature, to which the Einstein model is applicable. The results without anharmonic effect become divergent at slightly higher temperatures; however, the results including anharmonic effect are in good agreement with the experimental data of solid xenon. The method proposed in this paper can be extended to other potentials to develop practical molecular thermodynamic equations of state for solids.     

Does accidental degeneracy imply a symmetry group?

The relation between the appearance of accidental degeneracy in the energy levels of a given Hamiltonian and its symmetry group is probed. This is done by analyzing the very simple problem of an oscillator to which a particular spin-orbit and centrifugal force are added. The operators that connect all the states of given energy as well as their corresponding observables in the classical limit are found. The Poisson bracket relations between these observables leads to a Lie algebra U(3) × SU(2), but it does not translate into a Lie algebra for the commutators of the corresponding operators, as some matrix elements of commutators, corresponding to Poisson brackets that are zero, do not vanish. Thus while accidental degeneracy in the quantum problem may lead to a larger group in the classical limit, it is not always given by the dimensions of the irreducible representations of this group.     

Exact classical and quantum solutions for a covariant oscillator near the black hole horizon in Stueckelberg-Horwitz-Piron theory

We found exact solutions for canonical classical and quantum dynamics for general relativity in Horwitz general covariance theory. These solutions can be obtained by solving the generalized geodesic equation and Schrödinger-Stueckelberg-Horwitz-Piron (SHP) wave equation for a simple harmonic oscillator in the background of a two dimensional dilaton black hole spacetime metric. We proved the existence of an orthonormal basis of eigenfunctions for generalized wave equation. This basis functions form an orthogonal and normalized (orthonormal) basis for an appropriate Hilbert space. The energy spectrum has a mixed spectrum with one conserved momentum p according to a quantum number n. To find the ground state energy we used a variational method with appropriate boundary conditions. A set of mode decomposed wave functions and calculated for the Stueckelberg-Schrodinger equation on a general five dimensional blackhole spacetime in Hamilton gauge.     

Coherent States and Modified de Broglie-Bohm Complex Quantum Trajectories

This paper examines the nature of classical correspondence in the case of coherent states at the level of quantum trajectories. We first show that for a harmonic oscillator, the coherent state complex quantum trajectories and the complex classical trajectories are identical to each other. This congruence in the complex plane, not restricted to high quantum numbers alone, illustrates that the harmonic oscillator in a coherent state executes classical motion. The quantum trajectories we consider are those conceived in a modified de Broglie-Bohm scheme. Though quantum trajectory representations are widely discussed in recent years, identical classical and quantum trajectories for coherent states are obtained only in the present approach. We may note that this result for standard harmonic oscillator coherent states is not totally unexpected because of their holomorphic nature. The study is extended to coherent states of a particle in an infinite potential well and that in a symmetric Poschl-Teller potential by solving for the trajectories numerically. For the Gazeau-Klauder coherent state of the infinite potential well, almost identical classical and quantum trajectories are obtained whereas for the Poschl-Teller potential, though classical trajectories are not regained, a periodic motion results as t→∞. Similar features were found for the SUSY quantum mechanics-based coherent states of the Poschl-Teller potential too, but this time the pattern of complex trajectories is quite different from that of the previous case. Thus we find that the method is a potential tool in analyzing the properties of generalized coherent states.     

Universal and deterministic manipulation of the quantum state of harmonic oscillators: a route to unitary gates for Fock state qubits

We present a simple quantum circuit that allows for the universal and deterministic manipulation of the quantum state of confined harmonic oscillators. The scheme is based on the selective interactions of the referred oscillator with an auxiliary three-level system and a classical external driving source, and enables any unitary operations on Fock states, two by two. One circuit is equivalent to a single qubit unitary logical gate on Fock states qubits. Sequences of similar protocols allow for complete, deterministic, and state-independent manipulation of the harmonic oscillator quantum state.     

Quantum mechanical barrier problems: II. Dissipative tunnelling at finite temperatures for the unbiased oscillator

The decay rate for an unbiased local quasi-equilibrium oscillator, leaking through a quartic potential energy barrier into free space and weakly coupled to an ohmic thermal environment, is calculated in the low temperature quantum tunnelling regime. The imaginary part of the free energy is determined by a bound kink-antikink pair. It is found that the limits of zero dissipation (λ → 0) and zero temperature (T → 0) do not commute. In the limit λ → 0 (for nonzero T) one recovers the finite nondissipative quantum tunnelling rate obtained in a previous paper (I), whereas in the limit T → 0 (for nonzero λ) a freezing phenomenon occurs.     

QUANTUM BROWNIAN MOTION AND NUCLEAR FISSION

Based on a simplified Hamiltonian model the transition from quantum to classical diffusion behaviours for a system has been shown.The approximation of a locally harmonic oscillator and reduced density operator in the harmonic oscillator are generalized to calculate the escape tate over the barrier for the system in a fission potential consisting of a ground state well and barrier.     

Efficient Scheme for One-Way Quantum Computing in Thermal Cavities

We propose a practical scheme for one-way quantum computing based on efficient generation of 2D cluster state in thermal cavities. We achieve a controlled-phase gate that is neither sensitive to cavity decay nor to thermal field by adding a strong classical field to the two-level atoms. We show that a 2D cluster state can be generated directly by making every two atoms collide in an array of cavities, with numerically calculated parameters and appropriate operation sequence that can be easily achieved in practical Cavity QED experiments. Based on a generated cluster state in Box⁽⁴⁾ configuration, we then implement Grover’s search algorithm for four database elements in a very simple way as an example of one-way quantum computing.     

Quantum Harmonic Oscillator Systems with Disorder

We study many-body properties of quantum harmonic oscillator lattices with disorder. A sufficient condition for dynamical localization, expressed as a zero-velocity Lieb-Robinson bound, is formulated in terms of the decay of the eigenfunction correlators for an effective one-particle Hamiltonian. We show how state-of-the-art techniques for proving Anderson localization can be used to prove that these properties hold in a number of standard models. We also derive bounds on the static and dynamic correlation functions at both zero and positive temperature in terms of one-particle eigenfunction correlators. In particular, we show that static correlations decay exponentially fast if the corresponding effective one-particle Hamiltonian exhibits localization at low energies, regardless of whether there is a gap in the spectrum above the ground state or not. Our results apply to finite as well as to infinite oscillator systems. The eigenfunction correlators that appear are more general than those previously studied in the literature. In particular, we must allow for functions of the Hamiltonian that have a singularity at the bottom of the spectrum. We prove exponential bounds for such correlators for some of the standard models.     

Nonclassical states: An observable criterion

An observable criterion is derived that allows one to distinguish nonclassical states of the harmonic oscillator from those having a classical counterpart. A quantum state is shown to have no classical counterpart if and only if the characteristic functions of the quadrature distributions or the s-parametrized phase-space distributions exhibit a slower decay than for the ground state of the oscillator. This renders it possible to experimentally check the failure of the P function to be a probability measure.     

Activation-Like Processes at Zero Temperature

We examine the possibility that a metastable quantum state could experiment a phenomenon similar to thermal activation but at zero temperature. To do that we study the real-time dynamics of the reduced Wigner function in a simple open quantum system: an anharmonic oscillator with a cubic potential linearly interacting with an environment of harmonic oscillators. Our results suggest that this activation-like phenomenon exists indeed as a consequence of the fluctuations induced by the environment and that its associated decay rate is comparable to the tunneling rate as computed by the instanton method, at least for the particular potential of the system and the distribution of frequencies for the environment considered in this paper. However, we are not able to properly deal with the term which leads to tunneling in closed quantum systems, and a definite conclusion cannot be reached until tunneling and activation-like effects are considered simultaneously.     

Semiclassical approach to static and dynamic aspects of fermions in a harmonic well

Some properties of fermions in a harmonic oscillator well; that is density distributions, momentum distributions in k-space, binding energy at zero and finite temperature, and the giant isoscalar monopole resonance energy at zero temperature are calculated in a semiclassical way and the results are compared with those obtained by quantum mechanical methods.     

Semiclassical description of inelastic atom scattering by surfaces

Inelastic scattering of atoms of moderate energies (say<5 eV) by solid surfaces is almost entirely due to energy exchange with lattice vibrations. It can give valuable information about the atom-surface interaction potential and the vibrational dynamics at surfaces. Theoretically this process represents a challenging many-body problem, calling for suitable approximation methods. Work in progress (K. Burke, L. D. Chang, and W. Kohn) is described. (1) We have solved a simple model problem in which the normal modes of the lattice are schematized by a single one-dimensional harmonic oscillator, initially in its ground state (T=0). The classical solution gives a unique energy loss. We have calculated the leading quantum correction and find a Gaussian final energy distribution whose width is proportional toh ^1/2. Our exact results are in general different from the so-called trajectory approximation. (2) We are about to propose a new type of atom-surface scattering experiment, which will provide a direct measure of the quantum corrections to classical scattering.     

Stability of the ground state of a harmonic oscillator in a monochromatic wave

The stability of the ground state of a harmonic oscillator in a monochromatic wave is studied. This model describes, in particular, the dynamics of a cold ion in a linear ion trap, interacting with two laser fields with close frequencies. The stability of the "classical ground state"-the vicinity of the point (x=0,p=0)-is analyzed analytically and numerically. For the quantum case, a method for studying a stability of the quantum ground state is developed, based on the quasienergy representation. It is demonstrated that stability of the ground state may be substantially improved by increasing the resonance number, l, where l=Omega/omega+delta, Omega and omega are, respectively, the wave frequency and the oscillator frequency, l=1,2, em leader, mid R:deltamid R:<1; or by detuning the system from exact resonance, so that delta not equal 0. The influence of a large-amplitude wave (in the presence of chaos) on the stability of the ground state is analyzed for different parameters of the model in both the quantum and classical cases. (c) 2001 American Institute of Physics.     

量子谐振子与经典谐振子的比较

运用广义线性量子变换的普遍理论求解了量子谐振子,同经典谐振子比较给出了量子谐振子趋近于经典极限的条件,并得出相干态是最理想的经典极限态．     

Thermal state of the general time-dependent harmonic oscillator

Taking advantage of dynamical invariant operator, we derived quantum mechanical solution of general time-dependent harmonic oscillator. The uncertainty relation of the system is always larger than ħ/2 not only in number but also in the thermal state as expected. We used the diagonal elements of density operator satisfying Leouville-von Neumann equation to calculate various expectation values in the thermal state. We applied our theory to a special case which is the forced Caldirola-Kanai oscillator.     

The second hyperpolarizability of systems described by the space-fractional Schrödinger equation

The static second hyperpolarizability is derived from the space-fractional Schrödinger equation in the particle-centric view. The Thomas–Reiche–Kuhn sum rule matrix elements and the three-level ansatz determines the maximum second hyperpolarizability for a space-fractional quantum system. The total oscillator strength is shown to decrease as the space-fractional parameter α decreases, which reduces the optical response of a quantum system in the presence of an external field. This damped response is caused by the wavefunction dependent position and momentum commutation relation. Although the maximum response is damped, we show that the one-dimensional quantum harmonic oscillator is no longer a linear system for $\alpha \ne 1$, where the second hyperpolarizability becomes negative before ultimately damping to zero at the lower fractional limit of $\alpha \to 1/2$.