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Exact quantization method for the polynomial 1D Schrödinger equation | Méthode de quantification exacte pour l'équation de Schrödinger polynômiale 1D

André Voros

Published Jan 1, 2000• 0 citations
OpenAlex

Abstract

The stationary 1D Schrödinger equation with a polynomial potential $V(q)$ of degree $N$ is reduced analytically to a system of (complex) exact quantization conditions of Bohr--Sommerfeld form. These reduced equations can be probed numerically for effective solvability: in test cases, they appear to generate discrete dynamical systems with contractive behavior, and converging towards the exact quantum data as their fixed point. (To date, this statement is verified only empirically and for potentials close to purely quartic or sextic ones.)

Related papers

Exercises in exact quantization | Exercices de quantification exacte

André Voros

The formalism of exact 1D quantization is reviewed in detail and applied to the spectral study of three concrete Schrödinger Hamiltonians $[-{\rm d}^2/{\rm d} q^2 + V(q)]^\pm$ on the half-line $\{q>0\}$, with a Dirichlet ($-$) or Neumann ($+$) condition at $q=0$. Emphasis is put on the analytical investigation of the spectral determinants and spectral zeta functions with respect to singular perturbation parameters. We first discuss the homogeneous potential $V(q)=q^N$ as $N \to +\infty$ vs its (solvable) $N=\infty$ limit (an infinite square well): useful distinctions are established between regular and singular behaviours of spectral quantities; various identities among the square-well spectral functions are unraveled as limits of finite-$N$ properties. The second model is the quartic anharmonic oscillator: its zero-energy spectral determinants $\det( -{\rm d}^2/{\rm d} q^2 + q^4 + v q^2)^\pm$ are explicitly analyzed in detail, revealing many special values, algebraic identities between Taylor coefficients, and functional equations of a quartic type coupled to asymptotic $v \to +\infty$ properties of Airy type. The third study addresses the potentials $V(q)=q^N+v q^{N/2-1}$ of even degree: their zero-energy spectral determinants prove computable in closed form, and the generalized eigenvalue problems with $v$ as spectral variable admit exact quantization formulae which are perfect extensions of the harmonic oscillator case (corresponding to $N=2$); these results probably reflect the presence of supersymmetric potentials in the family above.

•Jan 1, 2000•0 citations

Exact Quantization Rules for the One-Dimensional Schrödinger Equation with Turning Points

Paul B. Bailey

It is pointed out that, for a number of problems, exact quantization rules exist which closely resemble that of Bohr-Wilson-Sommerfeld. In some cases it is shown how these rules may be derived mathematically from the Schrödinger equation.

Journal of Mathematical Physics•Sep 1, 1964•13 citations

Analytical Solutions of Two-Dimensional Schrödinger Equation

Vladimír Tichý

The goal of the dissertation is to find new method of solving two-dimensional Schrödinger equation in such cases, when the separation of the variables is not applicable. The results are applied to the two-dimensional Schrödinger equation with the potentials of the form of the quartic polynomial, of the sextic polynomial and of the quartic Morse potential. For these cases, the analytical formulae for the ground state wave functions and the corresponding energies have been found. For the specific class of the potentials of the form of the quartic polynomial, analytical formula for one of the excited states and for the corresponding energy have been found.

•Jan 1, 2012•0 citations

Quantum Correction in Exact Quantization Rules

Zhong-Qi Ma, Bo-Wei Xu

Abstract. – An exact quantization rule for the Schrödinger equation is presented. In the exact quantization rule, in addition to Nπ, there is an integral term, called the quantum correction. For the exactly solvable systems we find that the quantum correction is an invariant, independent of the number of nodes in the wave function. In those systems, the energy levels of all the bound states can be easily calculated from the exact quantization rule and the solution for the ground state, which can be obtained by solving the Riccati equation. With this new method, we re-calculate the energy levels for the one-dimensional systems with a finite square well, with the Morse potential, with the symmetric and asymmetric Rosen-Morse potentials, and with the first and the second Pöschl-Teller potentials, for the harmonic oscillators both in one dimension and in three dimensions, and for the hydrogen atom. In the development of quantum mechanics, the Bohr-Sommerfeld quantization rules of the old quantum theory [1] occupy a position intermediate between classical and quantum mechanics. The WKB approximation [2–4] is a method for the approximate treatment of the Schrödinger wave function with another quantization rule [1]. Both quantization rules are approximate. We are going to derive an exact quantization rule for the one-dimensional Schrödinger equation: d2 ψ(x) = −2M dx2 2 [E − V (x)] ψ(x), (1) ¯h where M is the mass of the particle, and the potential V (x) is a piecewise continuous real function of x satisfying for definiteness V (x)> E, − ∞ < x < xA or xB < x < ∞, V (x) = E, x = xA or x = xB, V (x) < E, xA < x < xB, (2) where xA and xB are two turning points. Between two turning points, the momentum is k(x) = √ 2M [E − V (x)]/¯h.

•Jan 1, 2005•122 citations

Development of the perturbation theory using polynomial solutions

Fethi Mohamed Maiz

The number of quantum systems for which the stationary Schrodinger equation is exactly solvable is very limited. These systems constitute the basic elements of the quantum theory of perturbation. The exact polynomial solutions for real quantum potential systems provided by the use of Lagrange interpolation allows further development of the quantum perturbation theory. In fact, the first order of correction for the value of the energy appears to be sufficient since the chosen perturbation Hamiltonian is very small or even negligible compared to the main Hamiltonian. Here, we use the perturbation theory to derive polynomial solutions, and we then find that our approximated results agree very well with previous published or numerically achieved ones. We believe that this study is an operational tool for the verification and improvement of numerical and approximate methods.

Journal of Mathematical Physics•Jan 1, 2019•2 citations

Exact Quantization Conditions

C. Rosenzweig, Joseph B. Krieger

The method of Froman and Froman for proving exact quantization conditions is reviewed. This formalism, unlike the usual WKB approximation to which it bears a close resemblance, requires consideration of the behavior of the potential everywhere it is defined. This approach leads to proofs that certain quantization conditions are exact without having to compare the results to solutions of the Schrödinger equation obtained by other means. Using the formalism, we prove the correctness of all previously known exact quantization rules for the one-dimensional and radial cases. Furthermore, it is shown that exact quantization rules can be proved for two other potentials. For one of these, no analytic solutions to the Schrödinger equation are known. For the latter case, the proof is checked by numerical integration of the Schrödinger equation for a special case.

Journal of Mathematical Physics•Jun 1, 1968•64 citations
The stationary 1D Schrödinger equation with a polynomial potential V(q)V(q)V(q) of degree NNN is reduced analytically to a system of (complex) exact quantization conditions of Bohr--Sommerfeld form. These reduced equations can be probed numerically for effective solvability: in test cases, they appear to generate discrete dynamical systems with contractive behavior, and converging towards the exact quantum data as their fixed point. (To date, this statement is verified only empirically and for potentials close to purely quartic or sextic ones.)
The formalism of exact 1D quantization is reviewed in detail and applied to the spectral study of three concrete Schrödinger Hamiltonians [−d2/dq2+V(q)]±[-{\rm d}^2/{\rm d} q^2 + V(q)]^\pm[−d2/dq2+V(q)]± on the half-line {q>0}\{q>0\}{q>0}, with a Dirichlet (−-−) or Neumann (+++) condition at q=0q=0q=0. Emphasis is put on the analytical investigation of the spectral determinants and spectral zeta functions with respect to singular perturbation parameters. We first discuss the homogeneous potential V(q)=qNV(q)=q^NV(q)=qN as N→+∞N \to +\inftyN→+∞ vs its (solvable) N=∞N=\inftyN=∞ limit (an infinite square well): useful distinctions are established between regular and singular behaviours of spectral quantities; various identities among the square-well spectral functions are unraveled as limits of finite-NNN properties. The second model is the quartic anharmonic oscillator: its zero-energy spectral determinants det⁡(−d2/dq2+q4+vq2)±\det( -{\rm d}^2/{\rm d} q^2 + q^4 + v q^2)^\pmdet(−d2/dq2+q4+vq2)± are explicitly analyzed in detail, revealing many special values, algebraic identities between Taylor coefficients, and functional equations of a quartic type coupled to asymptotic v→+∞v \to +\inftyv→+∞ properties of Airy type. The third study addresses the potentials V(q)=qN+vqN/2−1V(q)=q^N+v q^{N/2-1}V(q)=qN+vqN/2−1 of even degree: their zero-energy spectral determinants prove computable in closed form, and the generalized eigenvalue problems with vvv as spectral variable admit exact quantization formulae which are perfect extensions of the harmonic oscillator case (corresponding to N=2N=2N=2); these results probably reflect the presence of supersymmetric potentials in the family above.
The goal of the dissertation is to find new method of solving two-dimensional Schrödinger equation in such cases, when the separation of the variables is not applicable. The results are applied to the two-dimensional Schrödinger equation with the potentials of the form of the quartic polynomial, of the sextic polynomial and of the quartic Morse potential. For these cases, the analytical formulae for the ground state wave functions and the corresponding energies have been found. For the specific class of the potentials of the form of the quartic polynomial, analytical formula for one of the excited states and for the corresponding energy have been found.
Abstract. – An exact quantization rule for the Schrödinger equation is presented. In the exact quantization rule, in addition to Nπ, there is an integral term, called the quantum correction. For the exactly solvable systems we find that the quantum correction is an invariant, independent of the number of nodes in the wave function. In those systems, the energy levels of all the bound states can be easily calculated from the exact quantization rule and the solution for the ground state, which can be obtained by solving the Riccati equation. With this new method, we re-calculate the energy levels for the one-dimensional systems with a finite square well, with the Morse potential, with the symmetric and asymmetric Rosen-Morse potentials, and with the first and the second Pöschl-Teller potentials, for the harmonic oscillators both in one dimension and in three dimensions, and for the hydrogen atom. In the development of quantum mechanics, the Bohr-Sommerfeld quantization rules of the old quantum theory [1] occupy a position intermediate between classical and quantum mechanics. The WKB approximation [2–4] is a method for the approximate treatment of the Schrödinger wave function with another quantization rule [1]. Both quantization rules are approximate. We are going to derive an exact quantization rule for the one-dimensional Schrödinger equation: d2 ψ(x) = −2M dx2 2 [E − V (x)] ψ(x), (1) ¯h where M is the mass of the particle, and the potential V (x) is a piecewise continuous real function of x satisfying for definiteness V (x)> E, − ∞ < x < xA or xB < x < ∞, V (x) = E, x = xA or x = xB, V (x) < E, xA < x < xB, (2) where xA and xB are two turning points. Between two turning points, the momentum is k(x) = √ 2M [E − V (x)]/¯h.
The number of quantum systems for which the stationary Schrodinger equation is exactly solvable is very limited. These systems constitute the basic elements of the quantum theory of perturbation. The exact polynomial solutions for real quantum potential systems provided by the use of Lagrange interpolation allows further development of the quantum perturbation theory. In fact, the first order of correction for the value of the energy appears to be sufficient since the chosen perturbation Hamiltonian is very small or even negligible compared to the main Hamiltonian. Here, we use the perturbation theory to derive polynomial solutions, and we then find that our approximated results agree very well with previous published or numerically achieved ones. We believe that this study is an operational tool for the verification and improvement of numerical and approximate methods.
The method of Froman and Froman for proving exact quantization conditions is reviewed. This formalism, unlike the usual WKB approximation to which it bears a close resemblance, requires consideration of the behavior of the potential everywhere it is defined. This approach leads to proofs that certain quantization conditions are exact without having to compare the results to solutions of the Schrödinger equation obtained by other means. Using the formalism, we prove the correctness of all previously known exact quantization rules for the one-dimensional and radial cases. Furthermore, it is shown that exact quantization rules can be proved for two other potentials. For one of these, no analytic solutions to the Schrödinger equation are known. For the latter case, the proof is checked by numerical integration of the Schrödinger equation for a special case.