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Tadanori Hyouguchi1, Satoshi Adachi, Masahito Ueda

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A novel semiclassical method enhances the Schrödinger equation by incorporating quantum corrections. This approach yields a new quantization condition providing exact solutions for specific potentials.

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Area of Science:

  • Quantum mechanics
  • Computational physics
  • Mathematical physics

Background:

  • The Schrödinger equation is fundamental in quantum mechanics.
  • Existing semiclassical methods like WKB and Thomas-Fermi have limitations.

Purpose of the Study:

  • To develop a new semiclassical approach for the one-dimensional Schrödinger equation.
  • To improve the accuracy of wave function expansions.

Main Methods:

  • Developed a zeroth-order solution including nonperturbative quantum corrections.
  • Applied corrections to WKB and Thomas-Fermi solutions.
  • Constructed uniformly converging perturbative expansions.

Main Results:

  • Introduced a novel quantization condition.
  • Achieved exact eigenenergies for harmonic-oscillator potentials.
  • Achieved exact eigenenergies for Morse potentials.

Conclusions:

  • The new semiclassical method offers improved accuracy.
  • The approach is effective for both linear and nonlinear Schrödinger equations.
  • Provides exact solutions for important physical potentials.