Complex density probability in non-Hermitian quantum mechanics: Interpretation and a formula for resonant tunneling probability amplitude

Abstract

Non-Hermitian quantum mechanics has been developed to study the dynamics of nuclear, atomic, and molecular sys- tems that can be prepared in metastable finite-lifetime states ~so-called resonance states !@ 1,2# for the study of delocaliza- tion phenomenon which is relevant in different fields, such as bacteria populations, vortex spinning in superconductors, and for the study of the stability conditions of the solutions of hydrodynamical problems @3,4#. In cases where resonance phenomena are studied, the Hamiltonians are non-Hermitian due to the specific boundary conditions that are imposed on the solutions of the Schrodinger equation. The asymptotic solutions should be exponential divergent wave functions ~known as Siegert functions!. In other cases @3,4# the Hamil- tonian is non-Hermitian due to the inclusion of a non- Hermitian operator such as a vii, while the boundary con- ditions of ''conventional'' Hermitian quantum mechanics are kept. There is a way to unify the two types of non-Hermitian quantum problems. Upon complex scaling, i.e., x !x exp(iu), the exponentially divergent metastable reso- nance eigenfunctions become square integrable and thereby become part of the generalized Hilbert space @2#. Therefore, the resonances are the eigenfunctions of a complex scaled non-Hermitian Hamiltonian with the same boundary condi- tions as in the conventional Hermitian quantum mechanics. Let us denote the complex non-Hermitian Hamiltonian by H ˆ . A matrix representation of H ˆ ~denoted by H) is obtained by using a finite number of orthogonal functions as a basis set. Since the usual boundary conditions are applied the basis functions can be square integrable or periodic functions. The right and left eigenfunctions of H ˆ, which are defined asC j R and C j , are associated with the right and left eigenvectors of H: