Electron-Beam Interactions with Solids pp 91-94 | Cite as

# Matrices and Operators

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## Abstract

It is always possible to represent a linear operator
. (A.1) If
. (A.2) If {
. (A.5)

*L*as a matrix. The representation of the linear operator*L*on the basis represented by a complete set of eigenfunctions {*u*_{n}(*)} is given by the matrix constituted by the following matrix elements:***r**$$
L_{nm} = \left( {u_n ,Lu_m } \right) = \int {u_n^* Lu_m } d^3 x
$$

(4)

*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)

*u*_{n}(*)} is an orthonormal set of eigenfunctions of the Hilbert space, then (***r***u*_{n},*u*_{m}) =δ_{nm}. (A.3) If {*u*_{n}(*)} is a set of orthonormal eigenfunctions of the operator***r***L*, then the representation of*L*on the basis {*u*_{n}(*)} is a diagonal matrix. This can be written as***r***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)

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© Springer-Verlag Berlin Heidelberg 2003