Semi-classical calculations of energy levels and wave functions of hamiltonian systems with one and several degrees of freedom based on the method of classical and quantum normal forms

Abstract

The paper presents two schemes for the sequential construction of the classical normal form and its quantum analogue for some classes of classical Hamiltonian systems. For quantum normal forms, a method for solving their eigenvalue problem is indicated. Based on these normal forms, a semi-classical method for solving Schrodinger equations for classical Hamiltonian systems under their quantum consideration is proposed. With this proposed method, some quantum problems were solved and it was found that this method gives a very accurate prediction for energy levels. However, this accuracy in the field of the existence of classical chaos is deteriorating. The same semiclassical method solved the quantum problem for a flat hydrogen atom in a homogeneous magnetic field. The proposed method allows carrying out all calculations using modern computer systems of analytical calculations.

Keywords

• classical normal form
• quantum analog of normal form
• weyl-mccoy rule
• energy levels
• eigenfunctions
• mathematical modeling