On the interaction of two finite-dimensional quantum systems
The author considers the following formal model of a composite system. Suppose that the quantum system \(\mathcal S\), whose associated Hilbert space is the finite dimensional space \(X\), consists of two subsystems \({\mathcal S}_a\) and \({\mathcal S}_b\). One associates with \({\mathcal S}_a\) a subspace \(X^a\) of \(X\), and with \({\mathcal S}_b\) a subspace \(X^b\) of \(X\), and \(X^a\oplus X^b= X\). The interaction of \(S^a\) is modelled by the generalized eigenvalue equation \[ A\Theta_s=E_sS^a\Theta_s,\quad s= 1, \dots, \text{dim} X^a \] the interaction of \(S^b\) is described by the generalized eigenvalue equation \[ B\Phi_i=\lambda_iS^b\Phi_i,\quad i=1, \dots, \text{dim} X^b, \] whereas the combined system \({\mathcal S}\) is described by \[ (A+B+V)\Psi_k=\varepsilon_i[S^a+S^b+P]\Psi_k. \] Here \(A\), \(S^a\) are self-adjoint operators on \(X^a\), \(B\), \(S^b\) are self-adjoint operators on \(X^b\), and \(S^a\) and \(S^b\) are positive definite. The interaction between \({\mathcal S}_a\) and \({\mathcal S}_b\) is described by the self-adjoint operators \(V\) and \(P\) on \(X\), where \(S^a+S^b+P\) is supposed to be positive definite. The aim of the paper is to show that each eigenvalue \(\varepsilon_k\) and eigenstate \(\Psi_k\) of \({\mathcal S}\) can be found in a somewhat ``decoupled'' way. More precisely, once the solutions to the system \({\mathcal S}_b\) are known, the eigenvalues \(\varepsilon_k\) and eigenstates \(\Psi_k\) can be expressed in terms of \(A\), \(S^a\), \(V\), \(P\), \(\{\lambda_i\}_i\) and \(\{\Psi_i\}_i\). The paper contains numerical considerations concerning the above method of computation of \(\varepsilon_k\) and \(\Psi_k\), and also some numerical examples.
scientific article; zbMATH DE number 1878957