Exact solutions of the harmonic oscillator plus non-polynomial interaction

Area of Science:

• Quantum mechanics
• Mathematical physics

Background:

• The one-dimensional harmonic oscillator is a fundamental model in quantum mechanics.
• Non-polynomial potentials introduce complexities not found in simpler models.
• Confluent Heun functions provide exact solutions for specific complex potentials.

Purpose of the Study:

• To find exact solutions for a one-dimensional harmonic oscillator with a specific non-polynomial interaction.
• To analyze the behavior of wave functions under variations of potential parameters.
• To investigate the characteristics of the potential well and wave peaks.

Main Methods:

• Solving the Schrödinger equation for the given potential using confluent Heun functions.
• Calculating the minimum potential well value.
• Illustrating and analyzing wave function behavior by varying parameters a, b, and c.

Main Results:

• Exact solutions obtained using confluent Heun functions H(α, β, γ, δ, η; z).
• Minimum potential well value calculated for the double-well case (b < 0).
• Wave functions are shown to be pulled towards the origin as parameter b increases.
• Wave peaks exhibit concavity towards the origin with increasing |b|.

Conclusions:

• The study successfully provides exact solutions for a complex quantum mechanical system.
• Parameter variations significantly alter wave function localization and shape.
• The findings offer insights into the behavior of quantum systems with non-polynomial interactions.