QUANTUM ANHARMONIC OSCILLATOR

Quartic anharmonic oscillator with potential V(x)= x² + g²x⁴ was the first non-exactly-solvable problem tackled by the newly-written Schrödinger equation in 1926. Since that time thousands of articles have been published on the subject, mostly about the domain of small g² (weak coupling regime), although physics corresponds to g² ~ 1, and they were mostly about energies.

This book is focused on studying eigenfunctions as a primary object for any g². Perturbation theory in g² for the logarithm of the wavefunction is matched to the true semiclassical expansion in powers of ℏ: it leads to locally-highly-accurate, uniform approximation valid for any g²∈[0,∞) for eigenfunctions and even more accurate results for eigenvalues. This method of matching can be easily extended to the general anharmonic oscillator as well as to the radial oscillators. Quartic, sextic and cubic (for radial case) oscillators are considered in detail as well as quartic double-well potential.

Contents:

• Preface
• Introduction
• Personal Recollections, History, Acknowledgement (A V Turbiner)
• A Brief Recapitulation (J C del Valle)
• The One-Dimensional Anharmonic Oscillator:
  • Generalities
  • Riccati-Bloch Equation, Weak/Strong Coupling Regime
  • Generalized Bloch Equation, Semiclassical Expansion
  • Matching Perturbation Theory and Semiclassical Expansion
  • Quartic Anharmonic Oscillator
  • Sextic Anharmonic Oscillator
• The Radial Anharmonic Oscillator:
  • Spherical Symmetrical Potentials: Generalities
  • Radial AHO: Matching Perturbation Theory and Semiclassical Expansion
  • Radial Cubic Anharmonic Oscillator
  • Radial Quartic Anharmonic Oscillator
  • Radial Sextic Anharmonic Oscillator
• Appendices of Part I:
  • Classical Quartic Anharmonic Oscillator
  • Computational Realization of PT on the Riccati–Bloch Equation
  • One-Dimensional Quartic AHO: The First 6 Eigenstates: Interpolating Parameters, Nodes, Energies
  • One-Dimensional Sextic AHO: The First 6 Eigenstates: Interpolating Parameters, Nodes, Energies
  • Numerical Evaluation of nth Correction ε_n and γ_n(ν): Ground State
  • The Lagrange Mesh Method
• Appendices of Part II:
  • d-Dimensional Radial Oscillator: Lagrange Mesh Method
  • First PT Corrections and Generating Functions G_3,4 for the Cubic Anharmonic Oscillator
  • First PT Corrections and Generating Functions G_4,6 for the Quartic Anharmonic Oscillator
  • First PT Corrections and Generating Functions G_8,12 for the Sextic Anharmonic Oscillator
• Bibliography

Readership: Graduate, postgraduates, and academicians in Physics or Mathematics.

Key Features:

• Among compact wave functions in quantum mechanics, one can find those which are close to the exact wave functions, thus obtaining an approximate solution of the original quantum problem. Such approximations are valuable to calculate the energies and expectation values of the studied system. Where there is no single textbook that presents a comprehensive and detailed guide about the construction of such compact approximations, this book attempts to fill the gap, showing even zero approximation in developing convergent iteration procedure — the perturbation theory
• The book develops general techniques to study a wide variety of systems in the framework of nonrelativistic quantum mechanics, and is not limited to anharmonic quantum systems
• A new approach to semiclassical considerations is presented, based on the development of perturbation theory for the logarithm of the wave function
• The book is supplemented with computational codes written in MATHEMATICA, showing how to numerically realize the techniques discussed in the book