Quantum Mechanical Oscillator Research Articles

ON THE SENSITIVITY OF GRAVITATIONAL WAVE RESONANT BAR DETECTORS

Different theoretical estimates of the sensitivity of gravitational wave resonant bar detectors, which have been published in the last decades, are reviewed and discussed. The "classical" cross-section estimate is obtained considering the bar as a classical or quantum oscillator, whose initial thermal state is that of a single oscillator driven by a single external stochastic force. Other theoretical studies computed a much larger cross-section, using a variety of quantum-mechanical arguments. The review of the existing literature shows that there is no well established model for the response of a resonant detector to gravitational waves. The resonant, yet random, nature of the Brownian thermal motion may justify considering the bar response at the fundamental longitudinal eigenfrequency as that of a large number of effective quantum mechanical oscillators. Assuming this hypothesis, quantum coherence effects, as first suggested by Weber, lead to a much larger cross-section than that "classically" predicted. The reduction of this amplification due to thermal noise itself is also computed.

Trial introduction of a complex potential function for perturbation that causes quantum mechanical transition in a one‐dimensional harmonic oscillator

AbstractSome results of computer simulation of the behavior of a one‐dimensional quantum mechanical oscillator are reported in this article. This harmonic oscillator comprises a particle trapped within a hyperbolic potential V(x) = x2. Further, a perturbation potential function V′(x, t) was superposed upon the hyperbolic potential in order to induce a quantum mechanical transition. This perturbation function V′(x, t) is a function of both of space and time variables, and is set to represent a wave packet that is enveloped by a Gaussian bell‐shaped curve. A wave that probably has an appropriate wave number and angular frequency was inputted into the expression for the wave packet. In the initial phase, while the harmonic oscillator was allowed to oscillate almost freely, the wave packet was allowed to approach the harmonic oscillator. In the middle phase, the wave packet passes through the harmonic oscillator, affecting the shape of the quantum mechanical wave that represents the physical state of the system. In the last phase, when the wave packet left the system of the harmonic oscillator, the system settled onto an energetically stable state. The main objective of the simulation was to simulate the instance of a quantum mechanical transition from one eigenstate to another. After several trials, it was found that the perturbation function consisting of a complex function was, at least superficially, able to cause one desired transition, that is, a transition from one eigenstate to another eigenstate. By using such a complex perturbation function, a transition from the first excited state to the ground state was observed to occur. © 2004 Wiley Periodicals, Inc. Int J Quantum Chem, 2004

Dipole oscillator strengths, dipole properties and dispersion energies for SiF4

A recommended isotropic dipole oscillator strength distribution (DOSD) has been constructed for the silicon tetrafluoride (SiF4) molecule through the use of quantum mechanical constraint techniques and experimental dipole oscillator strength data. The constraints are furnished by experimental molar refractivity data and the Thomas-Reiche-Kuhn sum rule. The DOSD is used to evaluate a variety of isotropic dipole oscillator strength sums, logarithmic dipole oscillator strength sums and mean excitation energies for the molecule. A pseudo-DOSD for SiF4 is also presented which is used to obtain reliable results for the isotropic dipole-dipole dispersion energy coefficients C6, for the interaction of SiF4 with itself and with 43 other species and the triple-dipole dispersion energy coefficient C9 for (SiF4)3.

Three-dimensional harmonic oscillator and time evolution in quantum mechanics

The problem of defining time (or phase) operator for three-dimensional harmonic oscillator has been analyzed. A new formula for this operator has been derived. The results have been used to demonstrate a possibility of representing quantum-mechanical time evolution in the framework of an extended Hilbert space structure. Physical interpretation of the extended structure has been discussed shortly, too.

Extended Scale Relativity, p-Loop Harmonic Oscillator, and Logarithmic Corrections to the Black Hole Entropy

An extended scale relativity theory, actively developed by one of the authors, incorporates Nottale's scale relativity principle where the Planck scale is the minimum impassible invariant scale in Nature, and the use of polyvector-valued coordinates in C-spaces (Clifford manifolds) where all lengths, areas, volumes⋅ are treated on equal footing. We study the generalization of the ordinary point-particle quantum mechanical oscillator to the p-loop (a closed p-brane) case in C-spaces. Its solution exhibits some novel features: an emergence of two explicit scales delineating the asymptotic regimes (Planck scale region and a smooth region of a quantum point oscillator). In the most interesting Planck scale regime, the solution recovers in an elementary fashion some basic relations of string theory (including string tension quantization and string uncertainty relation). It is shown that the degeneracy of the first collective excited state of the p-loop oscillator yields not only the well-known Bekenstein–Hawking area-entropy linear relation but also the logarithmic corrections therein. In addition we obtain for any number of dimensions the Hawking temperature, the Schwarschild radius, and the inequalities governing the area of a black hole formed in a fusion of two black holes. One of the interesting results is a demonstration that the evaporation of a black hole is limited by the upper bound on its temperature, the Planck temperature.

Dipole oscillator strength properties and dispersion energies for SiH4

A recommended isotropic dipole oscillator strength distribution (DOSD) has been constructed for the silane (SiH4) molecule through the use of quantum mechanical constraint techniques and experimental dipole oscillator strength data. The constraints are furnished by experimental molar refractivity data and the Thomas–Reiche–Kuhn sum rule. The DOSD is used to evaluate a variety of isotropic dipole oscillator strength sums, logarithmic dipole oscillator strength sums, and mean excitation energies for the molecule. A pseudo-DOSD for SiH4 is also presented which is used to obtain reliable results for the isotropic dipole–dipole dispersion energy coefficients C6, for the interaction of silane with itself and with forty-four other species, and the triple–dipole dispersion energy coefficient C9 for ðSiH4Þ 3 . 2002 Elsevier Science B.V. All rights reserved.

Dipole oscillator strength properties and dispersion energies for CI2

A recommended isotropic dipole oscillator strength distribution (DOSD) has been constructed for the chlorine molecule through the use of quantum mechanical constraint techniques and experimental dipole oscillator strength and molar refractivity data. It has been used to evaluate a variety of dipole oscillator strength sums, logarithmic dipole oscillator strength sums, and mean excitation energies for the molecule. A pseudo-DOSD for C12 is also presented which is used to obtain reliable results for the isotropic dipole-dipole dispersion energy coefficients C6, for the interaction of Cl2 with itself and forty-two other species, and the triple-dipole dispersion energy coefficient C9 for (Cl2)3.

The tau-oscillator

The tau-oscillator is a one-dimensional quantum-mechanical oscillator which can exist only below a critical temperature Tmax. Its quantized energy levels have the usual form but, unusually, these degeneracies involve the energy levels. A generalized equipartition law is also found and the constant volume heat capacity can exhibit a Schottky anomaly.

One-dimensional multiple-well oscillators: A time-dependent quantum mechanical approach

Time-dependent Schrodinger equation (TDSE) is solved numerically to calculate the ground- and first three excited-state energies, expectation values x 2 j j 1 2 6, and probability densities of quantum mechanical multiple-well oscillators. An imaginary-time evolution technique, coupled with the minimization of energy expectation value to reach a global minimum, subject to orthogonality constraint (for excited states) has been employed. Pseudodegeneracy in symmetric, deep multiple-well potentials, probability densities and the effect of an asymmetry parameter on pseudodegeneracy are discussed.

Chirality tunneling in a ferromagnetic spin chain

The quantum-mechanical oscillations between two domain walls of opposite chirality in a ferromagnetic spin chain with easy and hard axis anisotropies are studied. The equations of motion of the instantons that connect the degenerate ground states in imaginary time are obtained and it is shown that there are two equivalent instantons that contribute to the expression of the splitting of the degenerate energy level. The instanton solutions and the energy splittings are obtained numerically for different values of the anisotropy parameters. It is found that the hard axis anisotropy inhibits chirality tunneling, in contrast to what happens in the single spin problem where the hard axis anisotropy favors tunneling. This behavior is explained as a crossover, driven by the increase of the hard axis anisotropy, from the ferromagnetic spin chain model to the sine-Gordon model.

Influence of acceleration on the evolution of a quantum mechanical harmonical oscillator

Wave functions of stationary states for a uniformly accelerated harmonical oscillator and its energy spectrum are obtained. The oscillator excitation due to the acceleration is studied. Based on the obtained results a descriptive interpretation of fulling- Unruh effect is given.

Fractional revivals of coherence in quantum mechanical oscillators

A case study is made of the delocalisation and revival dynamics of a continuously driven quantum pendulum in integrable and near integrable regimes, utilising the Husimi phase-space distribution function, and an entropy function which measures the degree of localisation. The numerical results can be well understood in terms of a simple model description based on a quasi-energy representation. Two important time-scales of the evolution are derived, and the model description appears to remain highly accurate in the region of classical KAM-tori, even in the driven case.

Above-barrier ionization and quantum interference in strong laser fields

Classical predictions of field ionization of hydrogen and of Coulomb scattering in strong laser fields are compared with corresponding exact numerical solutions of the time-dependent Schrodinger equation. In the quasi-static approximation, ionization rates are significantly lower than the quasi-classical tunneling rates in the regime of above-barrier ionization. This reduction is consistent with a simple classical barrier-suppression model. In time-periodic fields, the major differences between the classical and quantum-mechanical treatments arise from quantum-mechanical interference oscillations. These are typical for tunneling barriers in an applied field and for scattering by an oscillating potential.

Harmonic oscillators in relativistic quantum mechanics

For the Dirac equation in one space dimension, we examine the possibility of harmonic oscillator (HO) potentials. A HO potential is such that it has only bound states and the energy levels are equally spaced. Regarding their Lorentz transformation properties, there are three types of potentials for the Dirac equation: vector, scalar, and pseudoscalar. HO potentials are possible for the scalar and pseudoscalar types, but not for the vector type. We show how HO potentials can be constructed by means of the ``inverse scattering method.'' We also examine the behavior of a wave packet in the HO potentials so constructed. The behavior is, in most of the cases, very similar to that of Schr\odinger's coherent wave packet of the nonrelativistic harmonic oscillator.

On the square integrability of the q-Hermite functions

Overlap integrals over the full real line −∞< x<∞ for a family of the q-Hermite functions H n( sinκx¦q) e −x 2 2 , 0< q=e −2 κ 2<1 are evaluated. In particular, an explicit form of the squared norms for these q-extensions of the Hermite functions (or the wave functions of the linear harmonic oscillator in quantum mechanics) is obtained. The classical Fourier-Gauss transform connects the q-Hermite functions with different values 0< q<1 and q>1 of the parameter q. An explicit expansion of the q-Hermite polynomials H n( sinκx¦q) in terms of the Hermite polynomials H n ( x) emerges as a by-product.

Lie-series approach to the evolution of resonant and nonresonant anharmonic oscillators in quantum mechanics

A version of a normal-form method of Kummer and Gompa, analogous to the Lie-series methods of classical mechanics, is used to rigorously construct uniform long-time approximations to the quantum evolution of a system of resonant or nonresonant anharmonic oscillators. The Hamiltonian Hε governing the system is a self-adjoint operator in the Hilbert space H=L2(Rν) (ν⩾2), given formally by H0+εV. Here, H0 denotes the Hamiltonian of ν one-dimensional simple harmonic oscillators and εV their anharmonic coupling, ε⩾0 being a small parameter and V an operator of multiplication by a smooth function of polynomial growth at infinity. We treat the case in which an arbitrary number ρ⩾1 of the frequencies of these oscillators are rationally independent, imposing a standard diophantine condition on the independent frequencies if ρ⩾2. Under these assumptions, stated precisely in the paper, an Nth-order approximant ψN(t,ε) to the exact solution ψ(t,ε) of the Schrödinger equation idψ(t,ε)/dt=Hεψ(t,ε), satisfying the initial condition ψ(0,ε)=ψ0, is constructed recursively, where the initial state ψ0 belongs to a family of smooth functions dense in ℋ. The main result is that ψN(t,ε) differs in ℋ-norm from ψ(t,ε) by ⩽const εN+1|t| for t∈R and ε in an arbitrary compact interval [0,ε0]. This result is obtained by an approach simpler and quite different from that of Kummer and Gompa, and extends their work to oscillators with nonpolynomial couplings, under very general resonance conditions.

Integral equations and the simple harmonic oscillator

The energy eigenstates for the simple harmonic oscillator in quantum mechanics are found through the solution of an integral equation.

The finite temperature real-time ℏ 2 corrections in quantum mechanics

We study non-perturbative real-time correlation functions at finite temperature. In order to see whether the classical term gives a good approximation in the high temperature limit T ⪢ ℏω, we consider the first ℏ 2 quantum corrections. We find that for the simplest non-trivial case, the quantum mechanical anharmonic oscillator, the classical result is reliable only for moderately large times: after some time t ∗ the classical approximation breaks down even at high temperatures. Moreover, the result for the first quantum corrections cannot, in general, be reproduced by modifying the parameters of the classical theory.

The Central Two‐Point Connection Problem in Quantum Mechanics

AbstractExponentially small effects in physical systems have attained much attention in recent literature. They not only are difficult to understand from the viewpoint of classical (Poincaré) asymptotics but also are hard to calculate. In the following we give a brief account of an approach to the central two‐point connection problem of the triconfluent case of Heun's differential equation. It does not seem to be recognized that the solution of this problem opens the possibility to calculate exponentially small effects. A famous example in mathematical physics is the effect of avoided crossings of eigenvalue curves in the case of the double‐well quartic oscillator in quantum mechanics. On the basis of the above‐mentioned singularity‐analytical approach we have carried out the numerical calculation of the effect with a rather low calculatory effort.

Perturbative expansion around the Gaussian effective action: The background field method

We develop a systematic method of the perturbative expansion around the Gaussian effective action based on the background field method. We show, by applying the method to the quantum mechanical anharmonic oscillator problem, that even the first non-trivial correction terms greatly improve the Gaussian approximation.