GerritJan Linker

Linear operators  operator representation in a basis A linear vector or operator Ê is defined by the way that for each vector a of V a vector Êa is added. This has to happen in a linear way: Ê ( ßb + µu ) = ßÊb + µÊu where b and v are vectors and ß and µ are scalars. Because the vector Êe_{i} is also in V we can write Êe_{i} = ∑_{j} E_{ji}e_{j} ,in Dirac notation: Êi> = ∑_{j}E_{ji}j> This means that operator Ê is completely defined by the matrix E_{ij} and the basis e_{1}, e_{2}, ... In other words, matrix E is the operator representation for Ê in that basis. All the matrix elements E_{ij} can be found by left multiplying by the ket vector <k: <kÊi> = ∑_{j}E_{ji}<kj> = ∑_{j}E_{ji}delta_{kj} = E_{ki}
