The Quadratic Zeeman Effect in Hydrogen: an Example of Semi-classical Quantization of a Strongly Non-separable but Almost Integrable System
نویسندگان
چکیده
Semi-classical quantization of multidimensional systems is discussed both in terms of the Einstein-Brillouin-Keller quantization on invariant tori, and in terms of infinite families of periodic orbits. The notions of separability, integrability, and non-integrability of classical systems are introduced. An approximate integrability is used to quantize the quadratic Zeeman problem, via analytic calculation of the Birkhoff-Gustavson normal form.
منابع مشابه
Parabolic maps with spin: Generic spectral statistics with non-mixing classical limit
We investigate quantised maps of the torus whose classical analogues are ergodic but not mixing. Their quantum spectral statistics shows non-generic behaviour, i. e. it does not follow random matrix theory (RMT). By coupling the map to a spin 1/2, which corresponds to changing the quantisation without altering the classical limit of the dynamics on the torus, we numerically observe a transition...
متن کاملTHE REVIEW OF ALMOST PERIODIC SOLUTIONS TO A STOCHASTIC DIERENTIAL EQUATION
This paper proves the existence and uniqueness of quadratic mean almost periodic mild so-lutions for a class of stochastic dierential equations in a real separable Hilbert space. Themain technique is based upon an appropriate composition theorem combined with the Banachcontraction mapping principle and an analytic semigroup of linear operators.
متن کاملIntegrable systems, symmetries, and quantization
These notes correspond to a mini-course given at the Poisson 2016 conference in Geneva. Starting from classical integrable systems in the sense of Liouville, we explore the notion of non-degenerate singularity and expose recent research in connection with semi-toric systems. The quantum and semiclassical counterpart will also be presented, in the viewpoint of the inverse question: from the quan...
متن کاملA Super-Integrable Two-Dimensional Non-Linear Oscillator with an Exactly Solvable Quantum Analog⋆
Two super-integrable and super-separable classical systems which can be considered as deformations of the harmonic oscillator and the Smorodinsky–Winternitz in two dimensions are studied and identified with motions in spaces of constant curvature, the deformation parameter being related with the curvature. In this sense these systems are to be considered as a harmonic oscillator and a Smorodins...
متن کاملHydrogen Abstraction Reaction of Hydroxyl Radical with 1,1-Dibromoethane and 1,2-Dibromoethane Studied by Using Semi-Classical Transition State Theory
The hydrogen abstraction reaction by OH radical from CH2BrCH2Br (R1) and CH₃CHBr2 (R2) is investigated theoretically by semi-classical transition state theory. The stationary points for both reactions are located by using ωB97X-D and KMLYP density functional methods along with cc-pVTZ basis. Single-point energy calculations are performed at the QCISD(T) and CCSD(T) levels of theory with differe...
متن کامل