engleski

On the interaction of two finite dimensional quantum systems

Interaction of quantum system A described by the generalised eigenvalue equation ( ) with quantum system B described by the generalised eigenvalue equation ( ) is considered. With the system A is associated -dimensional space and with the system B is associated an n-dimensional space that is orthogonal to . Combined system is described by the generalised eigenvalue equation ( ) where operators and represent interaction between those two systems. All operators are Hermitian, while operators , and are in addition positive definite. It is shown that each eigenvalue of the combined system is the eigenvalue of the eigenvalue equation . Operator in this equation is expressed in terms of the eigenvalues of the system and in terms of matrix elements and where vectors form a base in . Eigenstate of this equation is the projection of the eigenstate of the combined system on the space . Projection of on the space is given by where is inverse of in . Hence, if the solution to the system is known, one can obtain all eigenvalues and all the corresponding eigenstates of the combined system as a solution of the above eigenvalue equation that refers to the system alone. Slightly more complicated expressions are obtained for the eigenvalues and the corresponding eigenstates, provided such eigenvalues and eigenstates exist.

interaction of quantum systems; perturbation; diagonalisation; generalised eigenvalue equation; eigenvalues; eigenstates

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