Perturbation Expansions of First Order Eigenvalue Equations in Quantum Mechanics

T Koide

Abstract We discuss the systematic expansion of the solution of the Fokker–Planck equation with the help of the eigenfunctions of the time-dependent Fokker–Planck operator. The expansion parameter is the time derivative of the external parameter which controls the form of an external potential. Our expansion corresponds to the perturbative calculation of the adiabatic motion in quantum mechanics. With this method, we derive a new formula to calculate the irreversible work order by order, which is expressed as the expectation value with a pseudo density matrix. Applying this method to the case of the harmonic potential, we show that the first order term of the expansion gives the exact result. Because we do not need to solve the coupled differential equations of moments, our method simplifies the calculations of various functions such as the fluctuation of the irreversible work per unit time. We further investigate the exact optimized protocol to minimize the irreversible work by calculating its variation with respect to the control parameter itself.

B. S. Lee

Michel A. Vandyck

An expansion of Schrodinger's theory in powers of h is presented. It has the property that the standard WKB results, together with the Bohr-Sommerfeld quantization condition are obtainable by a zeroth-order perturbation in h, rather than by a first-order one (in contrast with the usual expansion methods). It also makes the connection between the (classical) Hamilton-Jacobi equation and the WKB approximation more transparent.

Wei-Ming Chen, Chia-Yi Ju

We develop a Rayleigh--Schrödinger-like perturbation theory for non-Hermitian quantum systems at an exceptional point of order $N$. Working in the Jordan basis of the unperturbed Hamiltonian and employing a Puiseux expansion of the perturbed eigenvalues and eigenstates, we derive explicit recursion relations for the expansion coefficients. The corrections to the unperturbed eigenvalue in the Puiseux expansion govern the splitting near the exceptional point; the first two are obtained iteratively in two equivalent forms. One is given in terms of the perturbation Hamiltonian in the Jordan basis, and the other in terms of the generator that drives the eigenvalue evolution with respect to the perturbation. The latter constitutes the exceptional-point counterpart of a geometric perturbation method recently developed for the non-exceptional-point regime. Both representations are verified explicitly for the $N = 2$ and $N = 3$ cases.

Luigi E. Picasso, L. Bracci, Emilio d’Emilio