Matrices and Operators

Part of the Springer Tracts in Modern Physics book series (STMP, volume 186)


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:
$$ L_{nm} = \left( {u_n ,Lu_m } \right) = \int {u_n^* Lu_m } d^3 x $$
. (A.1) If L = L°, then L mn * = L mn. In detail,
$$ \left( {L^\dag } \right)_{nm} = \left( {u_n ,L^\dag u_m } \right) = \left( {Lu_n ,u_m } \right) = \left( {u_m ,Lu_n } \right)^* = L_{mn}^* $$
. (A.2) If {u n(r)} is an orthonormal set of eigenfunctions of the Hilbert space, then (u n, u m) =δnm. (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 = λn u n, (A.4) and, as a consequence,
$$ L_{nm} = (u_n ,Lu_m ) = \lambda _n (u_n ,u_m ) = \lambda _n \delta _{nm} {\mathbf{ }}. $$
. (A.5)


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© Springer-Verlag Berlin Heidelberg 2003

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