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This study provides analytical solutions for the Schrödinger equation with multiple potentials, calculating wave functions to analyze Shannon entropy and variance in diatomic molecules.

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Area of Science:

  • Quantum Mechanics
  • Theoretical Chemistry
  • Mathematical Physics

Background:

  • The Schrödinger equation is fundamental to quantum mechanics.
  • Analytical solutions are crucial for understanding molecular behavior.
  • Information entropy and variance offer insights into quantum systems.

Purpose of the Study:

  • To derive approximate analytical solutions for the 3D radial Schrödinger equation with a multiple potential function.
  • To investigate Shannon entropy and variance using the obtained wave functions.
  • To explore the behavior of these quantities with respect to equilibrium bond length and apply findings to diatomic molecules.

Main Methods:

  • Parametric Nikiforov-Uvarov method applied to the Schrödinger equation.
  • Approximation scheme for the centrifugal term.
  • Calculation of energy eigenvalues and wave functions.
  • Expectation value calculations for Shannon entropy and variance.

Main Results:

  • Analytical solutions for energy and wave functions were obtained.
  • The behavior of Shannon entropy and variance was analyzed concerning equilibrium bond length.
  • Specific analysis of the pseudoharmonic-like potential was performed.
  • Numerical results validated derived variance inequalities using the Cramer-Rao uncertainty relation.

Conclusions:

  • The study successfully obtained approximate analytical solutions for the Schrödinger equation with complex potentials.
  • Shannon entropy and variance provide valuable insights into the quantum mechanical behavior of diatomic molecules.
  • The derived variance inequalities are supported by numerical evidence, reinforcing their applicability.