Matrices and Operators

Abstract

It is always possible to represent a linear operator L as a matrix. The representation of the linear operator L on the basis represented by a complete set of eigenfunctions {u _n(r)} is given by the matrix constituted by the following matrix elements:

$$ L_{nm} = \left( {u_n ,Lu_m } \right) = \int {u_n^* Lu_m } d^3 x $$

(4)

. (A.1) If L = L°, then L ^*_mn = L _mn. In detail,

$$ \left( {L^\dag } \right)_{nm} = \left( {u_n ,L^\dag u_m } \right) = \left( {Lu_n ,u_m } \right) = \left( {u_m ,Lu_n } \right)^* = L_{mn}^* $$

(5)

. (A.2) If {u _n(r)} is an orthonormal set of eigenfunctions of the Hilbert space, then (u _n, u _m) =δ_nm. (A.3) If {u _n(r)} is a set of orthonormal eigenfunctions of the operator L, then the representation of L on the basis {u _n(r)} is a diagonal matrix. This can be written as Lu _n = λ_n u _n, (A.4) and, as a consequence,

$$ L_{nm} = (u_n ,Lu_m ) = \lambda _n (u_n ,u_m ) = \lambda _n \delta _{nm} {\mathbf{ }}. $$

(6)

. (A.5)