Quantum inverted harmonic potential

Abstract

We consider a non-interacting gas under the inverted harmonic potential and present infinitely degenerate non-stationary orthogonal states. We discuss that it has an infinite entropy at the absolute zero temperature. We show that uncertainty in position of a particle under the inverted harmonic potential can be zero as there exists a solution which asymptotes to a Dirac delta function. We obtain a new free particle wave packet using the eigenstates for the inverted harmonic potential. It has unique self-focusing feature and can be used as focusing beam without a lens in optical systems where paraxial approximation is used.

Towards the Analytical Generalization of the Transcendental Energy Equation, Group Velocity, and Effective Mass in One-Dimensional Periodic Potential Wells with a Computational Application to Common Coupled Potentials

“…Figure 1 represents the approximation of the Kronig-Penney model, the potential to which the other potentials studied will be approximated, and those that meet the condition of Equation (1). The solutions of the potentials displayed in Figure 1 of the associated time-independent SE are commonly special functions [22,23,26,31], while the zero-like potential has its solution as a linear combination of complex exponentials [3].…”

“…Figure 4 shows the energy dispersion and transcendental energy distribution for the linear-like potential with negative intensity calculated using Equations ( 18)- (22).…”

“…In solid state physics, the Kronig-Penney model stands out as the simplest and most widely known periodic potential to explain band theory [3,12], unlike other potentials, such as linear-like or parabolic-like potentials [21][22][23][24]. It should be noted that the Kronig-Penney model has an analytical solution of time-independent SE, and the solutions are real exponential-like functions.…”

Inverted Oscillator Quantum States in the Probability Representation

“…The inverted oscillator was recently discussed in [14,15] in relation to cosmological problems. Some of the inverted oscillator problems were also considered in [16][17][18][19].…”

Density prediction of self-accelerating waves

“…As a result, in Hermitian systems, self-acceleration happens exclusively for non-integrable waves. It was recently demonstrated that it happens even for integrable waves in non-Hermitian systems [2], and the new wave packets were produced using the inverted harmonic potential eigenstates in [3]. These novel wave packets contain boundless energy and a distinct self-focusing property.…”