On the Interaction of Two Finite-Dimensional Quantum Systems

Interaction of quantum system S^a described by the generalised × eigenvalue equation A|_s=E_sS^a|_s (s=1,...,) with quantum system S^b described by the generalised n×n eigenvalue equation B|_i=_iS^b|_i (i=1,...,n) is considered. With the system S^a is associated -dimensional space X^a and with the system S^b is associated an n-dimensional space X_n^b that is orthogonal to X^a. Combined system S is described by the generalised (+n)×(+n) eigenvalue equation [A+B+V]|_k=_k[S^a+S^b+P]|_k (k=1,...,n+) where operators V and P represent interaction between those two systems. All operators are Hermitian, while operators S^a,S^b and S=S^a+S^b+P are, in addition, positive definite. It is shown that each eigenvalue _ki of the combined system is the eigenvalue of the × eigenvalue equation . Operator in this equation is expressed in terms of the eigenvalues _i of the system S^b and in terms of matrix elements _s|V|_i and _s|P|_i where vectors |_s form a base in X^a. Eigenstate |_k^a of this equation is the projection of the eigenstate |_k of the combined system on the space X^a. Projection |_k^b of |_k on the space X_n^b is given by |_k^b=(_kS^b–B)^–1(V–_kP})|_k^a where (_kS^b–B)^–1 is inverse of (_kS^b–B) in X_n^b. Hence, if the solution to the system S^b is known, one can obtain all eigenvalues _ki} and all the corresponding eigenstates |_k of the combined system as a solution of the above × eigenvalue equation that refers to the system S^a alone. Slightly more complicated expressions are obtained for the eigenvalues _ki} and the corresponding eigenstates, provided such eigenvalues and eigenstates exist.