Supersymmetry of 𝒫𝒯-symmetric tridiagonal Hamiltonians

We extend the study of supersymmetric tridiagonal Hamiltonians to the case of non-Hermitian Hamiltonians with real or complex conjugate eigenvalues. We find the relation between matrix elements of the non-Hermitian Hamiltonian [Formula: see text] and its supersymmetric partner [Formula: see text] in a given basis. Moreover, the orthogonal polynomials in the eigenstate expansion problem attached to [Formula: see text] can be recovered from those polynomials arising from the same problem for [Formula: see text] with the help of kernel polynomials. Besides its generality, the developed formalism in this work is a natural home for using the numerically powerful Gauss quadrature techniques in probing the nature of some physical quantities such as the energy spectrum of [Formula: see text]-symmetric complex potentials. Finally, we solve the shifted [Formula: see text]-symmetric Morse oscillator exactly in the tridiagonal representation.

Quantized Vibrational Motion

The harmonic oscillator of chapter 2 is visited again, now in its quantum theoretical version. The solution of the Schrodinger equation (SE) is shown step-by step, as it features steps that are very similar to those used in solving the equations for the angular momentum and hydrogen-like orbitals in later chapters. The Morse oscillator has a potential function that is much more representative of the vibrations of atoms in molecules as the harmonic potential. The solutions of the harmonic and Morse oscillator are compared. It is then shown how nuclear vibrations in poly-atomic molecules are treated at the harmonic level. This requires the separation of internal degrees of freedom from the overall translation and rotation of a molecule, leading to the normal modes. The chapter also discusses basic aspects of vibrational spectroscopy and the selection rules of infrared and Raman vibrational spectroscopy.

On the Husimi Version of the Classical Limit of Quantum Correlation Functions

We extend the Husimi (coherent state) based version of linearized semiclassical theories for the calculation of correlation functions to the case of survival probabilities. This is a case that could be dealt with before only by use of the Wigner version of linearized semiclassical theory. Numerical comparisons of the Husimi and the Wigner case with full quantum results as well as with full semiclassical ones will be given for the revival dynamics in a Morse oscillator with and without coupling to an additional harmonic degree of freedom.

On the method of eigenvalues determination of the “truncated” matrix with morse hamiltonian as an example

A method of precise eigenvalues determination is developed on the basis of high order perturbation theory and applied to di-atomic molecule, as an example. The proposed method makes it possible not only to obtain energy values, but also to estimate a prediction accuracy and limits of its applicability for a specific implemented model. Numerical calculations are performed with the use of the extended Morse oscillator functions. The sixth power of the Morse coordinate is included into the potential function expansion. Analysis of possibility to make calculation in the model of the "truncated" matrix of the Hamiltonian is performed. The comparative possibilities of the method are analyzed with respect to other approaches of the potential functions determination for polyatomic molecules.

Dynamics of the Morse Oscillator: Analytical Expressions for Trajectories, Action-Angle Variables, and Chaotic Dynamics

We consider the one degree-of-freedom Hamiltonian system defined by the Morse potential energy function (the “Morse oscillator”). We use the geometry of the level sets to construct explicit expressions for the trajectories as a function of time, their period for the bounded trajectories, and action-angle variables. We use these trajectories to prove sufficient conditions for chaotic dynamics, in the sense of Smale horseshoes, for the time-periodically perturbed Morse oscillator using a Melnikov type approach.