Exact solutions of the harmonic oscillator plus non-polynomial interaction

Abstract

The exact solutions to a one-dimensional harmonic oscillator plus a non-polynomial interaction a x 2 + b x 2 /(1 + c x 2 ) ( a > 0, c > 0) are given by the confluent Heun functions H c ( α , β , γ , δ , η ; z ). The minimum value of the potential well is calculated as V min ( x ) = − ( a + | b | − 2 a | b | ) / c at x = ± [ ( | b | / a − 1 ) / c ] 1 / 2 (| b | > a ) for the double-well case ( b < 0). We illustrate the wave functions through varying the potential parameters a , b , c and show that they are pulled back to the origin when the potential parameter b increases for given values of a and c . However, we find that the wave peaks are concave to the origin as the parameter | b | is increased.

Bound State Solution Schrödinger Equation for Extended Cornell Potential at Finite Temperature

“…where m is the bare mass of a heavy quark. Using expression (39) for the energy spectrum in ( 51) we get the following equation for heavy quarkonia mass at finite temperature:…”

“…The exact solution of the Schrödinger equation for the new anharmonic oscillator, double ring-shaped oscillator, and quantum system with a nonpolynomial oscillator potential related to the isotonic oscillator was also widely studied in Refs. [37][38][39]. The relativistic Levinson theorem was also studied in detail in Ref.…”

Application of Eckart-Hellmann potential to study selected diatomic molecules using Nikiforov-Uvarov-Functional analysis method

“…The development of these methods allows one to derive the analytic eigen-solutions of the relativistic and non-relativistic wave equations which play a crucial role in interpreting the behavior of quantum mechanical systems. The frequently used analytical methods are the Nikiforov-Uvarov method (NU) , Asymptotic iterative method (AIM) [31], Laplace transformation approach [32], ansatz solution method [33], super-symmetric quantum mechanics approach (SUSYQM) [34,35], exact and proper quantization methods [36,37], the series expansion method [38][39][40][41][42][43][44][45], and the recent study via the Heun function approach has been used widely to study those soluble quantum systems which could not be solved before, such as the systems including the Mathieu potential, rigid rotor problem, sextictype problem, or the Konwent potential, to name a few [46][47][48][49][50][51][52][53][54].…”

Schrödinger equation based analytic assessment of thermal properties of confined electrons under vector and scalar fields

“…[15][16][17] Therefore, the exact solutions of the Schrödinger, Dirac-Weyl and Dirac equations have become the essential part from the beginning of quantum mechanics, 18 and such solutions are also very useful in the field of the atomic, nuclear physics and nanostructures, molecular physics and condensed matter physics. [19][20][21][22][23][24][25][26][27][28][29][30][31][32][33] For example, the hydrogen atom, the harmonic oscillator, as charge carries into low-dimensional semiconducting structures 34 and graphene. 35 In fact, the exact solutions of these equations, expressed in analytical form, describing oneelectron atoms and few-body systems are fundamental in studying the atomic structure theory and more areas.…”