One-dimensional Infinite Square Research Articles

Matrix Mechanics of a Particle in a One-Dimensional Infinite Square Well

We solve the infinite-potential well problem using the method of Heisenberg’s matrix mechanics. In addition to being of educational value, the matrix mechanics allows us to deal with various unphysical issues caused by this potential in a seemingly unproblematic fashion. We also show how to treat many particles within this representation.

Supersymmetric Partners of the One-Dimensional Infinite Square Well Hamiltonian: Special Cases

In a previous paper, we used a classification of the self adjoint extensions, also called self-adjoint determinations, of the differential operator −d2/dx2 in order to obtain the whole list of Supersymmetric (SUSY) partners of those selfadjoint determinations for which the ground state has strictly positive energy. The existence of self adjoint determinations with a ground state of zero or even negative energy is a proved fact. In this paper, we analyze the possibility of constructing SUSY partners for those determinations. We also study those cases for which the ground state has a degeneracy, the study of their SUSY partners should be analyzed separately. So far, we have studied those determinations having an exactly solvable eigenvalue problem. On the present study, we also included some comments in relation to determinations not exactly solvable from this point of view. In addition, the use of self adjoint determinations for which the ground state wave function has nodes (zeroes) produces formal SUSY partners with a finite number of eigenvalues or even with a purely continuous spectrum. We give some worked examples of these situations.

Supersymmetric Partners of the One-Dimensional Infinite Square Well Hamiltonian

We find supersymmetric partners of a family of self-adjoint operators which are self-adjoint extensions of the differential operator −d2/dx2 on L2[−a,a], a>0, that is, the one dimensional infinite square well. First of all, we classify these self-adjoint extensions in terms of several choices of the parameters determining each of the extensions. There are essentially two big groups of extensions. In one, the ground state has strictly positive energy. On the other, either the ground state has zero or negative energy. In the present paper, we show that each of the extensions belonging to the first group (energy of ground state strictly positive) has an infinite sequence of supersymmetric partners, such that the ℓ-th order partner differs in one energy level from both the (ℓ−1)-th and the (ℓ+1)-th order partners. In general, the eigenvalues for each of the self-adjoint extensions of −d2/dx2 come from a transcendental equation and are all infinite. For the case under our study, we determine the eigenvalues, which are also infinite, all the extensions have a purely discrete spectrum, and their respective eigenfunctions for all of its ℓ-th supersymmetric partners of each extension.

Square-well potential in quaternionic quantum mechanics

The one-dimensional infinite square well is the simplest solution of quantum mechanics, and consequently one of the most important. In this paper, we provide this solution using the real Hilbert space approach to quaternionic quantum mechanics (). We further provide the one-dimensional finite as well and a method to generate quaternionic solutions from non-degenerate complex solutions.

Quantitative analysis of the applicability limits of the model of a one-dimensional infinite square well

Abstract We discuss the influence of several factors on the deviations from energy spectrum of an infinite square quantum well (QW) for real microscopic systems that can be approximately modelled using particle in a box. We introduce the “blurring” potential in the form of the modified Woods-Saxon potential and solve the corresponding Schrodinger equation. It is found that the increase of the degree of blurring δ of the QW leads to the increase of number of the energy levels inside it and to increase of deviations from the quadratic dependence e (n) (e is the particle energy, n is the energy level number) typical for the infinite square QW, especially, for the energy levels close to the QW “tops”. It is most surprising that for relatively “large” values of δ the difference between the levels energies of such well and the appropriate (with the same n) levels energies of the square QW with the same depth changes sign (from positive to negative) as number n increases. We also conclude that the asymmetry of the QW and non-equality min≠mout (where min and mout are the particle effective mass inside and outside the QW) play a significant role for the relatively “shallow” well near the QW top.

Entropic uncertainty relation and revival structure of quantum wave packets

We consider a wave packet in the one-dimensional infinite square well formed by a combination of ground state and the first few excited states, and identify three different methods to study its time development. The first method proceeds by monitoring the temporal evolution of the so-called autocorrelation function while the second one relies on the behavior of standard deviation based Heisenberg’s position-momentum uncertainty relation as a function of time. The third method of our interest makes use of the time variation of the Shannon entropy sum or the so-called entropic uncertainty relation. The chosen form of the wave packet allowed us to study its revivals, super-revivals and fractional revivals by using analytic methods, the inevitable numerical routine being invoked at the last stage of the game. We found that, in general, the uncertainty-based approaches furnish more detailed information on the revival structure of wave packets than those provided by the autocorrelation function. The results of the entropic uncertainty relation, however, appear to be more accurate than the corresponding predictions of the standard deviation based uncertainty relation.

Wave packet dynamics for a system with position and time-dependent effective mass in an infinite square well

The problem of a particle with position and time-dependent effective mass in a one-dimensional infinite square well is treated by means of a quantum canonical formalism. The dynamics of a launched wave packet of the system reveals a peculiar revival pattern that is discussed.

Negentropy as a source of efficiency: a nonequilibrium quantum Otto cycle

We consider a single quantum mechanical particle confined to an one-dimensional (1D) infinite square well, and propose a nonequilibrium quantum Otto cycle (NQOC). Compared with the conventional quantum Otto engine (CQOE) investigated by [T.D. Kieu, Phys. Rev. Lett. 93, 140403 (2004); T.D. Kieu, Eur. Phys. J. D 39, 115 (2006)], due to the effects of negentropy produced in the NQOC, many interesting features appear: (1) in general, the NQOC is capable of extracting more work, so it is more efficient; (2) the NQOC can operate even when T 1 = T 2 or T 1 < T 2, where T 1 (T 2) represents the temperature of hot (cold) bath; (3) in some cases, the NQOC can absorb heat from both baths and completely transforms them into work. These results demonstrate that the negentropy can be understood as an effective source of efficiency in quantum heat engines (QHEs) and meanwhile it is shown that the second law of thermodynamics is not violated. At last, we also show that the efficiency of NQOC reduces to that of classical Otto cycle in the classical limit.

Heisenberg uncertainty relation and statistical measures in the square well

A non stationary state in the one-dimensional infinite square well formed by a combination of the ground state and the first excited one is considered. The statistical complexity and the Fisher-Shannon entropy in position and momentum are calculated with time for this system. These measures are compared with the Heisenberg uncertainty relation, $\Delta x\Delta p$. It is observed that the extreme values of $\Delta x\Delta p$ coincide in time with extreme values of the other two statistical magnitudes.

On the nonlocality of the fractional Schrödinger equation

A number of papers over the past eight years have claimed to solve the fractional Schrödinger equation for systems ranging from the one-dimensional infinite square well to the Coulomb potential to one-dimensional scattering with a rectangular barrier. However, some of the claimed solutions ignore the fact that the fractional diffusion operator is inherently nonlocal, preventing the fractional Schrödinger equation from being solved in the usual piecewise fashion. We focus on the one-dimensional infinite square well and show that the purported ground state, which is based on a piecewise approach, is definitely not a solution of the fractional Schrödinger equation for the general fractional parameter α. On a more positive note, we present a solution to the fractional Schrödinger equation for the one-dimensional harmonic oscillator with α=1.

Squeezing Dynamics for One-Dimensional Infinite Square Well with a Mobile Wall

We show that the dynamics for a particle confined in one-dimensional infinite square well with a mobile boundary can be converted to the case as if the boundary is time-independent at the expense of an appropriate time-dependent Hamiltonian. The Hamiltonian is deduced by the technique of integration within an ordered product of operators.

Practical Stabilization of a Quantum Particle in a One-Dimensional Infinite Square Potential Well

We consider a nonrelativistic charged particle in a one-dimensional infinite square potential well. This quantum system is subjected to a control, which is a uniform (in space) time-depending electric field. It is represented by a complex probability amplitude solution of a Schrödinger equation on a one-dimensional bounded domain, with Dirichlet boundary conditions. We prove the almost global practical stabilization of the eigenstates by explicit feedback laws.

Quantum infinite square well with an oscillating wall

A linear matrix equation is considered for determining the time dependent wave function for a particle in a one-dimensional infinite square well having one moving wall. By a truncation approximation, whose validity is checked in the exactly solvable case of a linearly contracting wall, we examine the cases of a simple harmonically oscillating wall and a non-harmonically oscillating wall for which the defining parameters can be varied. For the latter case, we examine in closer detail the dependence on the frequency changes, and we find three regimes: an adiabatic behabiour for low frequencies, a periodic one for high frequencies, and a chaotic behaviour for an intermediate range of frequencies.

A perturbative approach to Snyder space with applications

In this paper, we present a method for studying systems in the modified formulation of quantum mechanics known as Snyder space, proposed by Snyder (1947 Phys. Rev. 71 38–41). Snyder space predicts a modified commutation algebra for position and momentum operators. The method described in this paper introduces operators satisfying the canonical commutation relations and relates them to the position and momentum operators of Snyder space, effectively mapping a problem in Snyder space into a similar problem in standard quantum mechanics. The method is applied to the simple harmonic oscillator (SHO) in one and two dimensions as well as to the one-dimensional infinite square well. The energy spectra are calculated perturbatively for the SHO. We also find an exact spectrum for the one-dimensional infinite square well potential. These results are shown to agree with similar results found elsewhere in the literature.

On the small-time local controllability of a quantum particle in a moving one-dimensional infinite square potential well

We consider a quantum charged particle in a one-dimensional infinite square potential well moving along a line. We control the acceleration of the potential well. The local controllability in large time of this nonlinear control system along the ground state trajectory has been proved recently. We prove that this local controllability does not hold in small time, even if the Schrödinger equation has an infinite speed of propagation. To cite this article: J.-M. Coron, C. R. Acad. Sci. Paris, Ser. I 342 (2006).

Revivals in an infinite square well in the presence of aδwell

We have investigated quantum revivals of wave packets in a one-dimensional infinite square well potential containing a $\ensuremath{\delta}$ well in the middle. The time-dependent Schr\"odinger equation for this composite potential admits formally exact solutions. We present analytical results for revival properties in three physically motivated approximations: wave packets containing eigenstates with large numbers in the presence of an arbitrary $\ensuremath{\delta}$ well, ``shallow'' and ``deep'' $\ensuremath{\delta}$ wells. Analytical results in the case of a ``shallow'' $\ensuremath{\delta}$ well have been tested numerically.

Quantum chaos of a particle in a square well: competing length scales and dynamical localization.

The classical and quantum dynamics of a particle trapped in a one-dimensional infinite square well with a time-periodic pulsed field is investigated. This is a two-parameter non-KAM (Kolmogorov-Arnold-Moser) generalization of the kicked rotor, which can be seen as the standard map of particles subjected to both smooth and hard potentials. The virtue of the generalization lies in the introduction of an extra parameter R, which is the ratio of two length scales, namely, the well width and the field wavelength. If R is a noninteger the dynamics is discontinuous and non-KAM. We have explored the role of R in controlling the localization properties of the eigenstates. In particular, the connection between classical diffusion and localization is found to generalize reasonably well. In unbounded chaotic systems such as these, while the nearest neighbor spacing distribution of the eigenvalues is less sensitive to the nature of the classical dynamics, the distribution of participation ratios of the eigenstates proves to be a sensitive measure; in the chaotic regimes the latter is log-normal. We find that the tails of the well converged localized states are exponentially localized despite the discontinuous dynamics while the bulk part shows fluctuations that tend to be closer to random matrix theory predictions. Time evolving states show considerable R dependence, and tuning R to enhance classical diffusion can lead to significantly larger quantum diffusion for the same field strengths, an effect that is potentially observable in present day experiments.

Chaos in a well: effects of competing length scales

A discontinuous generalization of the standard map, which arises naturally as the dynamics of a periodically kicked particle in a one-dimensional infinite square well potential, is examined. Existence of competing length scales, namely the width of the well and the wavelength of the external field, introduce novel dynamical behaviour. Deterministic chaos induced diffusion is observed for weak field strengths as the length scales do not match. This is related to an abrupt breakdown of rotationally invariant curves and in particular KAM tori. An approximate stability theory is derived wherein the usual standard map is a point of “bifurcation”.

QUANTUM EFFECTS OF A PARTICLE IN AN INFINITE SQUARE WELL WITH A MOVING WALL

The quantum behavior of a particle in a one-dimensional infinite square well potential with a moving wall is studied. The particle is assumed to be initially prepared in the coherent state (Gaussian wave packet) and although the boundary is far from the particle, it is shown that the changing of the boundary conditions can instantaneously affect the dynamical behavior of the particle. It is also shown that the initial state can evolve into a squeezed state, and in some cases the spreading of the wavepacket could be suppressed. Finally the Pancharatnam phase is also discussed.

THE BERRY PHASE OF THE QUANTUM STATE OF A HARMONIC OSCILLATOR WITH TIME-DEPENDENT FREQUENCY AND BOUNDARY CONDITIONS

The problem of a harmonic oscillator of time-dependent frequency confined in a one-dimensional infinite square well with a moving wall is studied.It is shown that the exact solution and the Lewis invariant operator of the system can be obtained by performing two consecutive gauge transformations on the time-dependent Schrdinger equation.On the basis of the exact solution the Berry phases for the system are calculated by using the geometric concepts such as the geometric distance and geometric length of the curve.