One-dimensional Quantum Oscillator Research Articles

Features of the radiative properties of quantum-size particles of narrow-gap semiconductors

For colloidal quantum-size particles (QP) of narrow-gap semiconductors, in contrast to quantum dots of wide-gap CdSe, in QP-PbS there take place an anomalous temperature dependence of the photoluminescence intensity. Also, in the planar microstructure containing QP-InSb, long-wavelength radiation (more than 3 μm) and photoconductivity (over 20 μm) was observed. Under certain conditions, the radiation intensity and photoconductivity demonstrate a resonance maximum. The effects were explained in the model of a one-dimensional quantum oscillator, which energy substantially depends on the effective mass of its quasi-free electron. This leads to competition between the manifestations of long-wave radiation and photoluminescence, and hence, to the anomalous temperature dependence of photoluminescence. It is assumed that QP-InSb in a planar microstructure can be sources and receivers of terahertz radiation, which properties depend on the crystal structure of quantum-sized particles determined by the parameters of their synthesis. Keywords: quantum-dimensional particle, quantum dot, narrow-gap semiconductor, effective mass, Brillouin zone, dimensional quantization, quantum oscillator, photoluminescence, long-wavelength radiation.

Hermite Functions and Fourier Series

Using normalized Hermite functions, we construct bases in the space of square integrable functions on the unit circle (L2(C)) and in l2(Z), which are related to each other by means of the Fourier transform and the discrete Fourier transform. These relations are unitary. The construction of orthonormal bases requires the use of the Gramm–Schmidt method. On both spaces, we have provided ladder operators with the same properties as the ladder operators for the one-dimensional quantum oscillator. These operators are linear combinations of some multiplication- and differentiation-like operators that, when applied to periodic functions, preserve periodicity. Finally, we have constructed riggings for both L2(C) and l2(Z), so that all the mentioned operators are continuous.

Особенности излучательных свойств квантово-размерных частиц узкозонных полупроводников

For colloidal quantum-size particles (QP) of narrow-gap semiconductors, in contrast to quantum dots of wide-gap CdSe, in QP-PbS there take place an anomalous temperature dependence of the photoluminescence intensity. Also, in the planar microstructure containing QP-InSb, long-wavelength radiation (more than 3 µm) and photoconductivity (over 20 µm) was observed. Under certain conditions, the radiation intensity and photoconductivity demonstrate a resonance maximum. The effects were explained in the model of a one-dimensional quantum oscillator, which energy substantially depends on the effective mass of its quasi-free electron. This leads to competition between the manifestations of long-wave radiation and photoluminescence, and hence, to the anomalous temperature dependence of photoluminescence. It is assumed that QP-InSb in a planar microstructure can be sources and receivers of terahertz radiation, which properties depend on the crystal structure of quantum-sized particles determined by the parameters of their synthesis.

On the spectrum-generating superalgebras of the deformed one-dimensional quantum oscillators

We investigate the dynamical symmetry superalgebras of the one-dimensional matrix superconformal quantum mechanics with inverse-square potential. They act as spectrum-generating superalgebras for the systems with the addition of the de Alfaro-Fubini-Furlan oscillator term. The undeformed quantum oscillators are expressed by 2n × 2n supermatrices; their corresponding spectrum-generating superalgebras are given by the osp(2n|2) series. For n = 1, the addition of an inverse-square potential does not break the osp(2|2) spectrum-generating superalgebra. For n = 2, two cases of inverse-square potential deformations arise. The first one produces Klein deformed quantum oscillators; the corresponding spectrum-generating superalgebras are given by the D(2, 1; α) class, with α determining the inverse-square potential coupling constants. The second n = 2 case corresponds to deformed quantum oscillators of non-Klein type. In this case, the osp(4|2) spectrum-generating superalgebra of the undeformed theory is broken to osp(2|2). The choice of the Hilbert spaces corresponding to the admissible range of the inverse-square potential coupling constants and the possible direct sum of lowest weight representations of the spectrum-generating superalgebras is presented.

The quasi-nonassociative exceptional F(4) deformed quantum oscillator

We present the deformed (for the presence of Calogero potential terms) one-dimensional quantum oscillator with the exceptional Lie superalgebra F(4) as spectrum-generating superconformal algebra. The Hilbert space is given by a 16-ple of square-integrable functions. The energy levels are 23+n, with n = 0, 1, 2, …. The ground state is 7 times degenerate. The excited states are 8 times degenerate. The (7, 8, 8, 8, …) semi-infinite tower of states is recovered from the (7; 8; 1) supermultiplet of the N=8 worldline supersymmetry. The model is unique, up to similarity transformations, and admits an octonionic-covariant formulation which manifests itself as “quasi-nonassociativity.” This means, in particular, that the Calogero coupling constants are expressed in terms of the octonionic structure constants. The associated F(4) superconformal quantum mechanics is also presented.

Sequential nonideal measurements of quantum oscillators: Statistical characterization with and without environmental coupling

A one-dimensional quantum oscillator is monitored by taking repeated position measurements. As a first con- tribution, it is shown that, under a quantum nondemolition measurement scheme applied to a system initially at the ground state, (i) the observed sequence of measurements (quantum tracks) corresponding to a single experiment converges to a limit point, and that (ii) the limit point is random over the ensemble of the experiments, being distributed as a zero-mean Gaussian random variable with a variance at most equal to the ground-state variance. As a second contribution, the richer scenario where the oscillator is coupled with a frozen (i.e., at the ground state) ensemble of independent quantum oscillators is considered. A sharply different behavior emerges: under the same measurement scheme, here we observe that the measurement sequences are essentially divergent. Such a rigorous statistical analysis of the sequential measurement process might be useful for characterizing the main quantities that are currently used for inference, manipulation, and monitoring of many quantum systems. Several interesting properties of the quantum tracks evolution, as well as of the associated (quantum) threshold crossing times, are discussed and the dependence upon the main system parameters (e.g., the choice of the measurement sampling time, the degree of interaction with the environment, the measurement device accuracy) is elucidated. At a more fundamental level, it is seen that, as an application of basic quantum mechanics principles, a sharp difference exists between the intrinsic randomness unavoidably present in any quantum system, and the extrinsic randomness arising from the environmental coupling, i.e., the randomness induced by an external source of disturbance.

The su(2) Krawtchouk oscillator model under the CP deformed symmetry

We define a new algebra, which can formally be considered as a CP deformed su(2) Lie algebra. Then, we present a one-dimensional quantum oscillator model, of which the wavefunctions of even and odd states are expressed by Krawtchouk polynomials with fixed p = 1/2, K2n(k; 1/2, 2j) and K2n(k — 1; 1/2, 2j — 2). The dynamical symmetry of the model is the newly introduced su(2)CP algebra. The model itself gives rise to a finite and discrete spectrum for all physical operators (such as position and momentum). Among the set of finite oscillator models it is unique in the sense that any specific limit reducing it to a known oscillator models does not exist.

Quantum excitations in time-dependent backgrounds

We give a technique for calculating the occupation number of quantum fields in time-dependent backgrounds by using the relation between one-dimensional quantum oscillators and two-dimensional classical oscillators. We illustrate our method by giving closed analytical results for the time-dependent spectrum of occupation numbers during gravitational collapse in any number of dimensions.

Discrete series representations for , Meixner polynomials and oscillator models

We explore a model for a one-dimensional quantum oscillator based on the Lie superalgebra . For this purpose, a class of discrete series representations of is constructed, each representation characterized by a real number β > 0. In this model, the position and momentum operators of the oscillator are odd elements of and their expressions involve an arbitrary parameter γ. In each representation, the spectrum of the Hamiltonian is the same as that of a canonical oscillator. The spectrum of a position operator can be continuous or infinite discrete, depending on the value of γ. We determine the position wavefunctions both in the continuous and the discrete case and discuss their properties. In the discrete case, these wavefunctions are given in terms of Meixner polynomials. From the embedding , it can be seen why the case γ = 1 corresponds to a paraboson oscillator. Consequently, taking the values (β, γ) = (1/2, 1) in the model yields a canonical oscillator.

A finite oscillator model related to

We investigate a new model for the finite one-dimensional quantum oscillator based upon the Lie superalgebra . In this setting, it is natural to present the position and momentum operators of the oscillator as odd elements of the Lie superalgebra. The model involves a parameter p (0 < p < 1) and an integer representation label j. In the (2j + 1)-dimensional representations Wj of , the Hamiltonian has the usual equidistant spectrum. The spectrum of the position operator is discrete and turns out to be of the form , where k = 0, 1, …, j. We construct the discrete position wavefunctions, which are given in terms of certain Krawtchouk polynomials. These wavefunctions have appealing properties, as can already be seen from their plots. The model is sufficiently simple in the sense that the corresponding discrete Fourier transform (relating position wavefunctions to momentum wavefunctions) can be constructed explicitly.

Kirkwood–Salsburg equation for a quantum lattice system of oscillators with many-particle interaction potentials

UDC 517.9 For a Gibbs system of one-dimensional quantum oscillators on a d-dimensional hypercubic lattice interacting via superstable pair and many-particle potentials of finite range, we prove the existence of a solution of the (lattice) Kirkwood–Salsburg equation for correlation functions depending on the Wiener paths. Some many-particle potentials may be nonpositive.

Solution for One-Dimensional Quantum Oscillator with Time-Dependent Frequency and Mass**The project supported by National Natural Science Foundation of China and Natural Science Foundation of Shandong Province of China

Based on the generalized linear quantum transformation theory, we present a normal ordering evolution operator for onedimensional quant urn oscillator with time-dependent frequency and mass, then give the exact expression of the evolution matrix elements, wave function and expectation value of arbitrary observable.

Analytic solution for one-dimensional quantum oscillator with a variable frequency

According to generalized linear quantum transformation theory, we give the exact expression of the evolution operator, wave function and expectation value of observable quantity for one-dimensional quantum oscillator with a variable frequency: ω(t) = (ω0, β > 0).

Exact semiclassical expansions for one-dimensional quantum oscillators

A set of rules is given for dealing with WKB expansions in the one-dimensional analytic case, whereby such expansions are not considered as approximations but as exact encodings of wave functions, thus allowing for analytic continuation with respect to whichever parameters the potential function depends on, with an exact control of small exponential effects. These rules, which include also the case when there are double turning points, are illustrated on various examples, and applied to the study of bound state or resonance spectra. In the case of simple oscillators, it is thus shown that the Rayleigh–Schrödinger series is Borel resummable, yielding the exact energy levels. In the case of the symmetrical anharmonic oscillator, one gets a simple and rigorous justification of the Zinn-Justin quantization condition, and of its solution in terms of “multi-instanton expansions.”

Meson and Baryon Families as Vibronic States in sl(2) Quantum Universal Enveloping Algebra

A mass formula of the q·deformed modified harmonic oscillator type in the sl(2) quantum universal enveloping algebra is proposed for the meson and baryon families, by taking into account the known theories as a guide. Specifying the vibronic quantum number, the deformation parameter and associated ones of the theory are determined from available data for the scalar, pseudoscalar, vector meson and baryon families. The parameters determined from totally ten families not only predict many unobserved states, but also give restrictions on the observable number of- states. The method may admit taking into account no.n·perturbative effects. mation, specified by a parameter q, into the th~ory of angular momentum. 2 ),3) The original idea of these works has been started by Faddeev 4 ) and his schoolS) from the examination of an exact solvability of classical and quantum two-dimensional models in statistical mechanics. The present author6) has pointed out that the quadratic Casimir operator and its eigenvalue discovered by Witten have many applications in the angular momentum associated problems. The purpose of this paper is to analyze the vibronic spectra of hadrons by making use of the q-deformed harmonic oscillator and its modified form. In order to con­ struct the mass formula we have taken (i) the MIT bag modeF) and a scalar ),.¢4 theory8) in two space-time dimensions as a guide, and (ii) the q-deformed version of one-dimensional oscillator and its modified form. In the former the zero-point energy is omitted by taking a normal ordering. The application of the one-dimensional quantum oscillator in a universal enveloping (UE) algebra is quite common in the analysis of the molecular spectra 9 ) under the light of the Born-Oppenheimer approxi­ mation. In order to determine the terminating quantum number for oscillation we use the analogous idea which is operative in the estimate of the dissociation energies for molecules. lO ) We consider that the total spins of baryon (meson) families belong­ ing to the oscillatory states consist of intrinsic spins of quarks (quark and antiquark), viz., with no orbital excitation. There are totally ten families suitable to analyze with spin 0, 1/2 and 1 states, for the time being, if we include all members listed in the full listings of Tables of Particle Properties.ll) The observed decay modes of ha­ drons classified as members of one and the same family are quite similar in nature except for the difference arising from the threshold effect due to mass difference. This is counted as another indication of the usefulness of our assignment on the

Single quantum oscillator with an|x|ν-type potential: Specific heat

The specific heat of a single one-dimensional quantum oscillator associated with a potential proportional to ${|x|}^{\ensuremath{\nu}}(\ensuremath{\nu}&gt;0)$ is discussed for the limiting cases $T\ensuremath{\rightarrow}0$ and $T\ensuremath{\rightarrow}\ensuremath{\infty}$, and numerically calculated for intermediate temperatures. To perform these calculations for arbitrary values of $\ensuremath{\nu}$, we use, for the energetic spectrum, an approximate formula quite superior to the standard WKB one; namely, that formula recently established by Turschner.