Corrected bohr-sommerfeld quantum conditions for nonseparable systems

A S Tarnovskiĭ

A. C. de la Torre, Pedro Jose Catuogno, Sebastián Ferrando

Joseph Bishop Keller

Classical mechanics and the quantum conditions of Planck, Bohr, Sommerfeld, Wilson and Einstein are presented. The virtues and defects of this “old quantum theory” are pointed out. Its replacement by quantum mechanics is described, leading to the Schrödinger equation for the wave function and the corresponding energy eigenvalues. For separable systems, the reduction of this equation to ordinary differential equations and their asymptotic solution by the WKB method are described, as well as the resulting corrected quantum conditions with integer or half-integer quantum numbers. For nonseparable systems, the analogous asymptotic solution constructed by the author is described, together with the corrected quantum conditions to which it leads. Examples of the use of these conditions in the solution of eigenvalue problems are presented. It is explained that difficulties arise in using this method when the classical motion is stochastic or chaotic. Suggestions for overcoming these difficulties are mentioned.

G. C. Ghirardi

Richard H. Cushman, Jędrzej Śniatycki

The seminal paper by Niels Bohr followed by a paper by Arnold Sommerfeld led to a revolutionary Bohr–Sommerfeld theory of atomic spectra. We are interested in the information about the structure of quantum mechanics encoded in this theory. In particular, we want to extend Bohr–Sommerfeld theory to a full quantum theory of completely integrable Hamiltonian systems, which is compatible with geometric quantization. In the general case, we use geometric quantization to prove analogues of the Bohr–Sommerfeld quantization conditions for the prequantum operators Pf. If a prequantum operator Pf satisfies the Bohr–Sommerfeld conditions and if it restricts to a directly quantized operator Qf in the representation corresponding to the polarization F, then Qf also satisfies the Bohr–Sommerfeld conditions. The proof that the quantum spherical pendulum is a quantum system of the type we are looking for requires a new treatment of the classical action functions and their properties. For the sake of completeness we have provided an extensive presentation of the classical spherical pendulum. In our approach to Bohr–Sommerfeld theory, which we call Bohr–Sommerfeld–Heisenberg quantization, we define shifting operators that provide transitions between different quantum states. Moreover, we relate these shifting operators to quantization of functions on the phase space of the theory. We use Bohr–Sommerfeld–Heisenberg theory to study the properties of the quantum spherical pendulum, in particular, the boundary conditions for the shifting operators and quantum monodromy.

Nanny Fröman, Per Olof Fröman

This investigation concerns some model potentials with very steep walls, for which the quantal Bohr–Sommerfeld half-integer quantization condition, if necessary generalized to correspond to modified phase-integral approximations of arbitrary order, and used without or with higher-order corrections included, can be used for obtaining, very accurately, the energy eigenvalues of the bound states, apart possibly from the lowest ones. One of the cases treated is the potential proportional to cot2x, for which a modified Bohr–Sommerfeld half-integer quantization condition yields the energy eigenvalues exactly. That the Bohr–Sommerfeld half-integer quantization condition is applicable to potentials with very steep walls may at first sight seem surprising in view of the well-known fact that the energy eigenvalues of a square-well potential with infinitely high walls are obtained exactly from the Bohr–Sommerfeld integer quantization condition, i.e., the quantization condition obtained by replacing (s+1/2) π by (s+1) π in the right-hand member of the Bohr–Sommerfeld half-integer quantization condition. From the study of a potential with horizontal bottom and linearly rising walls, which goes over into a square well when the inclination of the walls tends to zero, it can easily be understood, why the half-integer quantization condition is appropriate when the steep walls have a finite slope.