On the Interaction of Two Finite-Dimensional Quantum Systems

Interaction of quantum system Sa described by the generalised ρ×ρ eigenvalue equation A|Θs〉=EsSa|Θs〉 (s=1,...,ρ) with quantum system Sb described by the generalised n×n eigenvalue equation B|Φi〉=λiSb|Φi〉 (i=1,...,n) is considered. With the system Sa is associated ρ-dimensional space Xρa and with the system Sb is associated an n-dimensional space Xnb that is orthogonal to Xρa. Combined system S is described by the generalised (ρ+n)×(ρ+n) eigenvalue equation [A+B+V]|Ψk〉=εk[Sa+Sb+P]|Ψk〉 (k=1,...,n+ρ) where operators V and P represent interaction between those two systems. All operators are Hermitian, while operators Sa,Sb and S=Sa+Sb+P are, in addition, positive definite. It is shown that each eigenvalue εk∉λi of the combined system is the eigenvalue of the ρ×ρ eigenvalue equation \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document} $$[\Omega (\varepsilon _k ) + A]|\Psi _k^a \rangle = \varepsilon _k S^a |\Psi _k^a \rangle $$ \end{document}. Operator \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document} $$\Omega (\varepsilon )$$ \end{document} in this equation is expressed in terms of the eigenvalues λi of the system Sb and in terms of matrix elements 〈χs|V|Φi〉 and 〈χs|P|Φi〉 where vectors |χs〉 form a base in Xρa. Eigenstate |Ψka〉 of this equation is the projection of the eigenstate |Ψk〉 of the combined system on the space Xρa. Projection |Ψkb〉 of |Ψk〉 on the space Xnb is given by |Ψkb〉=(εkSb−B)−1(V−εkP})|Ψka〉 where (εkSb−B)−1 is inverse of (εkSb−B) in Xnb. Hence, if the solution to the system Sb is known, one can obtain all eigenvalues εk∉λi} and all the corresponding eigenstates |Ψk〉 of the combined system as a solution of the above ρ×ρ eigenvalue equation that refers to the system Sa alone. Slightly more complicated expressions are obtained for the eigenvalues εk∈λi} and the corresponding eigenstates, provided such eigenvalues and eigenstates exist.