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Physica Scripta

The sojourn time of the inverted isotropic harmonic oscillators in a tilted magnetic field

Yanjie Wang, Wei Guo, Xiao‐Yin Pan

Published Jan 14, 2020• 1 citation
Publisher

Abstract

Abstract By employing the Feynman path integral approach, we investigate the sojourn time of a two-dimensional (2D) and a three-dimensional (3D) inverted isotropic harmonic oscillator under a tilted magnetic field. It is shown that the sojourn time in the 2D case behaves quite differently from the 3D case. In the 3D case, the sojourn time is a monotonically increasing function of the strength of the magnetic field, and its value approaches a finite value as the strength of the magnetic field approaches infinity. In the 2D case, the behavior of the sojourn time as a function of the strength and the angle θ of inclination of the magnetic field is much complicated. When the angle of inclination θ > π 4 , the system is unstable and the sojourn time is always finite. However, when 0 ≤ θ < π 4 , the system possesses a stable region with infinite large sojourn time as the strength of the magnetic field varies. The reason behind is explained via stability region analysis.

DOI
10.1088/1402-4896/ab6b41

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Abstract By employing the Feynman path integral approach, we investigate the sojourn time of a two-dimensional (2D) and a three-dimensional (3D) inverted isotropic harmonic oscillator under a tilted magnetic field. It is shown that the sojourn time in the 2D case behaves quite differently from the 3D case. In the 3D case, the sojourn time is a monotonically increasing function of the strength of the magnetic field, and its value approaches a finite value as the strength of the magnetic field approaches infinity. In the 2D case, the behavior of the sojourn time as a function of the strength and the angle θ of inclination of the magnetic field is much complicated. When the angle of inclination θ > π 4 , the system is unstable and the sojourn time is always finite. However, when 0 ≤ θ < π 4 , the system possesses a stable region with infinite large sojourn time as the strength of the magnetic field varies. The reason behind is explained via stability region analysis.
In the nonrelativistic or relativistic studies of the well-known effect of an infinitely long and infinitely thin solenoid containing magnetic flux Φ, the so-called Aharonov-Bohm (AB) field, in terms of the Dirac's fundamental unit of the magnetic flux Φ 0 , the magnetic flux Φ is usually expressed as a flux quantum number, which can be decomposed into two parts: the integer and the mantissa part. It has been commonly accepted in the literature that the integer part is unimportant and only the mantissa determines all the physical effects. In this paper, by investigating the AB effect on the the sojourn time of a two-dimensional (2D) inverted harmonic oscillator under a perpendicular uniform magnetic field, we show that the sojourn time depends upon the magnitude, rather than solely the mantissa of the magnetic flux quantum number. This then provides an example that can manifest the quantum effect of the integer part of the flux quantum number, the reason behind is explained via gauge transformation.
The sojourn time of the Gaussian wavepacket that is stationed at the center of the inverted harmonic oscillator is investigated on the noncommutative plane in detail. In ordinary commutative space quantum mechanics, the sojourn time of the Gaussian wavepacket is always a monotonically decreasing function of the curvature parameter ω of the potential. However, in this paper, we find that the spatial noncommutativity makes the sojourn time a concave function of ω with a minimum at an inflection point ω0. Furthermore, if ω is larger than a certain critical value the sojourn time will become infinity. Thus, the ordinary intuitive physical picture about the relation between the sojourn time and the shape of the inverted oscillator potential is changed when the spatial noncommutativity is considered.
The time evolution of a time-dependent inverted harmonic oscillator (TDIHO) of arbitrary dimensions is investigated. Using the algebraic method, we obtain the exact orthogonal basis of solutions of a TDIHO in which the dimensionality d and angular momentum l appear as parameters, and also discuss its properties. With the wavepacket as the initial state, the general expressions of sojourn time of a TDIHO are given. The method is also applied to the inverted Caldirola–Kanai harmonic oscillator. The results show that the sojourn time appears as an increasing function of the dissipation parameter in arbitrary dimensions, but a decreasing function of the dimensionality.
An analytical study of the radial Schrodinger wave equation with magnetic field, nonlinear and inverted nonlinear isotropic harmonic oscillator potentials was carried out. A modified substitution technique was used to determine the energy eigenvalues of the nonlinear and inverted nonlinear oscillators while the Laplace and computational methods were used to determine the radial wave functions. The determination of the energies’ eigenvalues and the radial wave functions enabled the study to generate tables, plot graphs, and discuss the effect of the magnetic field, angular frequency, and nonlinear term on the thermodynamic functions and systems. Results of the study revealed that the energies’ eigenvalues of the inverted nonlinear oscillator fluctuates increasingly and decreasingly, while that of the nonlinear isotropic oscillator fluctuates decreasingly and degeneracies, completely removed. As the angular frequency is improved, expansion or increase is observed in the nonlinear and the inverse nonlinear cases in the radial wave functions. In the case of increase in the nonlinear term, saturation in the radial wave functions are observed in the inverted and non-inverted cases. For the effect of the increase in the magnetic field potential, initial decrease with saturation was observed in the both cases. The effect of the magnetic field and nonlinear term in both the nonlinear and inverted oscillators on the thermodynamic functions, attained saturation on the polyatomic molecules, while the effects of angular frequency is observed to enhance the thermodynamic functions. Generally, the contraction or saturation experienced with the increase of the magnetic field term is as a result of the small value of the electron charge, and the carbon (iv) oxide molecule is at higher potential than that of the water molecule.
In this paper we solve exactly the problem of the spectrum and Feynman propagator of a charged particle submitted to both an anharmonic oscillator in the plane and a constant and homogeneous magnetic field of arbitrary strength aligned with the perpendicular direction to the plane. As we shall see in the beginning of the letter, the Hamiltonian, being a quadratic form, is easily diagonalizable and the Classical Action can be used to construct the exact Feynman Propagator using the Stationary Phase Approximation. The result is useful for the treatment of quasi two dimensional samples in the field of magnetic effects in nano-structures and quantum optics. The presented solution, after minor extensions, can also be used for motion in three dimensions, and in fact it has been used for years in such cases. Also it can be used as a good exercise of a Feynman Path Integral that can be calculated easily with just the help of the Classical Action.
The invariants for the time-dependent one-dimensional harmonic oscillator and the time-dependent two-dimensional harmonic oscillator in a static magnetic field are derived from the real representation of the exact solution to the equation of motion.Mathematically,the orthogonal functions invariant is the angular momentum of an isotropic time-dependent two-dimensional harmonic oscillator.Based on the invariants obtained here,the wave function for time-dependent two-dimensional harmonic oscillator in a static magnetic field in cylindrical coordinate is simply derived and the dynamical and geometrical phases are easily got by expressing the wave function as the superpositions of the wave functions of time-dependent two-dimensional harmonic oscillator in rectangular coordinate.