Precise determination of the energy levels of the anharmonic oscillator from the quantization of the angle variable
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In this article, an ansatz motivated by the classical form of el phi, where phi is the angle variable, was used to construct operators which satisfy the commutation relations of the creation-annihilation operators for the anharmonic oscillator.Abstract:
Using an ansatz motivated by the classical form of el phi , where phi is the angle variable, we construct operators which satisfy the commutation relations of the creation-annihilation operators for the anharmonic oscillator. The matrix elements of these operators can be expressed in terms of entire functions in the position complex plane. These functions provide solutions of the Ricatti equation associated with the time-independent Schrodinger equation. We relate the normalizability of the eigenstates to the global properties of the flows of this equation. These exact results yield approximations which complement the WKB approximation and allow an arbitrarily precise determination of the energy levels. We give numerical results for the first 10 levels with 30 digits. We address the question of the quantum integrability of the system.read more
Citations
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References
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Book
Methods of Mathematical Physics
Richard Courant,David Hilbert +1 more
TL;DR: In this paper, the authors present an algebraic extension of LINEAR TRANSFORMATIONS and QUADRATIC FORMS, and apply it to EIGEN-VARIATIONS.
Journal ArticleDOI
Proof of a theorem of a.?n.?kolmogorov on the invariance of quasi-periodic motions under small perturbations of the hamiltonian
TL;DR: In this paper, the rotatory motion of a heavy asymmetric rigid body is studied and the theorems of the rotational motion of such a rigid body are formulated and proved.
Journal ArticleDOI
Eigenvalues of λx2m anharmonic oscillators
TL;DR: In this paper, the ground state and excited energy levels of the generalized anharmonic oscillator defined by the Hamiltonian Hm = − d2/dx2+x2+ λx2m, m = 2,3, …, have been calculated using the Hill determinants.
Journal ArticleDOI
Numerological analysis of the WKB approximation in large order
TL;DR: In this paper, the one-dimensional two-turning-point eigenvalue problem for analytic potentials to all orders in the WKB approximation has been studied, and it is shown how to compute the eigenvalues of the potential to twelfth order.
Journal ArticleDOI
Statistical Properties of Energy Levels of Chaotic Systems: Wigner or Non-Wigner?
TL;DR: Two counterexamples - the hydrogen atom in a magnetic field and the quartic oscillator - which display nearest neighbor statistics strongly different from the usual Wigner distribution are presented.