Quantum Anharmonic Oscillator

The following sections are included:

• Preface
• Contents
• Introduction
• Personal Recollections, History, Acknowledgement (A.V. Turbiner)
• A Brief Recapitulation (J.C. del Valle)

The following sections are included:

• The Potentials of One-Dimensional Anharmonic Oscillators
• From the Schrödinger Equation to the Riccati Equation

The following sections are included:

• Riccati-Bloch Equation
• Weak Coupling Expansion
• Non-Linearization Procedure and Perturbation Theory
  • Ground State (No Nodes)
  • Excited States (The Case of K Nodes)
  • Anharmonic Oscillator (Excited States)
• Strong Coupling Expansion

In this Chapter it will be derived the so-called Generalized Bloch equation, which has no dependence on the Planck constant ħ similarly to the Riccati-Bloch equation, derived in Chapter 2. The generalized Bloch equation will allow us to construct the asymptotic expansion at large distances and the semiclassical expansion of the solution. By matching the expansion at small distances (or perturbation theory) with the expansion at large distances (or semiclassical expansion) leads to the remarkable simple, approximate expression for the eigenfunction, which holds for any anharmonic oscillator.

In this short Chapter we will elaborate the main result of Part I: the uniform approximation of the eigenfunctions of the one-dimensional AHO with a polynomial potential of even degree (2p). First, we consider the ground state function and construct its approximation in the form of a prefactor multiplied by an exponential both with several free parameters, which are fixed by using the standard variational procedure. Second, we take the ground state function approximation with arbitrary parameters, then (i) we modify their prefactor accordingly, and then (ii) we multiply it by a polynomial of degree K with real coefficients, whose all roots are real and simple (no roots coincide), which plays the role of extra prefactor. The coefficients of the polynomial are fixed by imposing the orthogonality to the approximations of lower energy eigenstates with lower degree polynomials, which are assumed to be known already. In turn, the free parameters of the modified ground state type approximation are fixed by making the minimization of the variational energy…

In this Chapter the knowledge presented in Chapters 2 and 3 for the general polynomial anharmonic oscillator will be applied to the one-dimensional quantum single-well symmetric anharmonic oscillator with potential V (x) = x² + g²x⁴. In this case the Perturbation Theory (PT) in powers of g² (weak coupling regime) and the semiclassical expansion in powers of ħ for the energies (eigenvalues) coincide. This is related to the fact that the dynamics in x-space and in (gx)-space corresponds to the same energy spectrum with effective coupling constant ħg². The two equations which govern the dynamics in those two spaces, the Riccati-Bloch (RB) and the Generalized Bloch (GB) equations, respectively, are derived. The PT in g² for the logarithmic derivative of the wave function leads to the PT (with polynomial in x coefficients) of the RB equation and to the true semi-classical expansion in powers of ħ of the GB equation, which corresponds to a loop expansion of the density matrix in the path integral formalism. A 2-parametric interpolation of these two expansions leads to a uniform approximation of the wavefunction in x-space with an unprecedented accuracy of ∼ 10⁻⁶ locally and an unprecedented accuracy ∼ 10⁻⁹ − 10⁻¹⁰ for the energy associated with it for any g² > 0. This Chapter is written in maximally self-contained way.

In this Chapter the knowledge presented in Chapters 2 and 3 for the general polynomial anharmonic oscillator and applied in Chapter 5 for the quartic anharmonic oscillator will be employed for the one-dimensional quantum single-well symmetric anharmonic oscillator with potential V (x) = x² + g⁴x⁶. In this case the Perturbation Theory (PT) in powers of g⁴ (weak coupling regime) and the Semiclassical Expansion (SE) in powers of ħ for the energies (eigenvalues) coincide. Again this is related to the fact that the dynamics in x-space and in (gx)-space correspond to the same energy spectrum with effective coupling constant ħ²g⁴. The two equations which govern the dynamics in those two spaces, the Riccati-Bloch (RB) and the Generalized Bloch (GB) equations, respectively, will be presented. The PT in g⁴ for the logarithmic derivative of the wave function leads to the PT (with polynomial in x coefficients) of the RB equation and to the true SE in powers of ħ of the GB equation, which corresponds to a loop expansion of the density matrix in the path integral formalism. A 4-parametric interpolation of these two expansions leads to a uniform approximation of the wavefunction in x-space with an unprecedented accuracy of ∼10⁻⁶ (six significant figures) locally and an unprecedented accuracy of ∼10⁻¹⁰ − 10⁻¹² (ten-twelve significant figures) for the associated energy eigenvalue, it remains true for any g² > 0, which will be studied. This Chapter is written in a brief manner and maximally self-contained manner.

The following sections are included:

• Radial Schrödinger Equation
• Radial Anharmonic Oscillators: Potentials
• Radial Riccati-Bloch Equation
  • Radial Generalized Bloch Equation
• The Weak Coupling Regime
  • The Non-Linearization Procedure
  • Radial Anharmonic Oscillator: The Weak Coupling Expansion from Radial Riccati-Bloch Equation
  • Radial Anharmonic Oscillator: The Weak Coupling Expansion from Radial Generalized Bloch Equation
  • Connection between Y and Ƶ Expansions

In this short Chapter we will present the main result of the Part II: the uniform approximation of the eigenfunctions of the radial AHO with a polynomial potential of degree p. As the first step we consider the ground state function and construct its approximation in the form of prefactor multiplied by exponential which depend on with several free parameters, then they will be fixed in variational procedure. Then, as the second step, we take the ground state function approximation which again depends on free parameters, modify the degrees of the prefactor accordingly, multiply it by r^ℓ and by the polynomial of degree n_r in r² with real coefficients, for which also all roots in r² are assumed positive and simple (no roots coincide). The coefficients of the polynomial are fixed by imposing the condition of the orthogonality to the approximations with lower-then-n_r-degree polynomials. In turn, the free parameters of this approximation are fixed by making the minimization of the variational energy…

The simplest radial anharmonic oscillator is characterized by a cubic anharmonicity (1.20),

In this Chapter we consider the radial anharmonic oscillator (rAHO) with quartic anharmonicity (1.21),

In this Chapter we consider radial anharmonic oscillator (rAHO) with sextic anharmonicity (1.22),

The following sections are included:

• Conclusions
• Appendices of Part I
  • Appendix A. Classical Quartic Anharmonic Oscillator
  • Appendix B. Computational Realization of PT on the Riccati-Bloch Equation
    • Ground State of the One-Dimensional Quartic Anharmonic Oscillator
    • Ground State of the One-Dimensional Sextic Anharmonic Oscillator
  • Appendix C. One-dimensional Quartic AHO; The First 6 Eigenstates: Interpolating Parameters, Nodes, Energies
    • Interpolations for Parameters A and B in (5.58)
      • Ground State (0, 0)
      • First Excited State (0, 1)
      • Second Excited State (1, 0)
      • Third Excited State (1, 1)
      • Fourth Excited State (2, 0)
      • Fifth Excited State (2, 1)
    • Numerical Results (Energies)
      • Energies of the First Six States (Tables)
      • Energies of the First Six States (Fits)
  • Appendix D. One-Dimensional Sextic AHO; The First 6 Eigenstates: Interpolating Parameters, Nodes, Energies
    • Numerical Results (Energies)
      • Energies of the First Six States (Tables)
    • Interpolations for Parameters A,B,C,D and Energies E
      • Ground State Energy (0, 0)
      • First Excited State Energy (0, 1)
      • Second Excited State Energy (1, 0)
      • Third Excited State Energy (1, 1)
      • Fourth Excited State Energy (2, 0)
      • Fifth Excited State Energy (2, 1)
  • Appendix E. Numerical Evaluation of nth Correction ε_n and Y_n(v): Ground State
    • Generalities
    • Code
      • Output
  • Appendix F. The Lagrange Mesh Method
    • Gauss Quadrature
    • Lagrange Functions
    • The Secular Equation and Lagrange Mesh Equations
    • Optimization of the Mesh: Rescaling of Mesh Points
    • Hermite and Laguerre Meshes
      • One-dimensional Case
    • Jacobi Mesh
    • Minimal Code for the LMM
• Appendices of Part II
  • Appendix G. d-dimensional Radial Oscillator: Lagrange Mesh Method
    • Cubic Radial Oscillator r³
  • Appendix H. First PT Corrections and Generating Functions G_3,4 for the Cubic Anharmonic Oscillator
  • Appendix I. First PT Corrections and Generating Functions G_4,6 for the Quartic Anharmonic Oscillator
  • Appendix J. First PT Corrections and Generating Functions G_8,12 for the Sextic Anharmonic Oscillator
• Bibliography