GerritJan Linker

Transformation of operator in another basis set If we have an operator Ê in the basis {α>}_{N}, E_{αβ}= <αÊβ> E_{αβ} is a matrix element of the matrix E. Matrix E is said to be the matrix representation of the operator Ê in the basis {α>}_{N}. If we would like to get the representation of Ê in another basis set, say {i>}_{N} we can find a matrix U that brings E into the new basis set: A=U^{†}EU where U^{†} is the adjoint of U. Derivation: E_{αβ}= <αÊβ> = <α 1 Ê 1 β> We will use the closure relation: Σ_{i}i><i = 1 => <α 1 Ê 1 β> = <α Σ_{i}i><i Ê Σ_{j}j><jβ> = Σ_{i}<αi>Σ_{j}<iÊj><jβ> = Σ_{i}Σ_{j}<αi><iÊj><jβ> = Σ_{i}Σ_{j}U_{αi}E_{ij}U_{jβ} = Σ_{i}Σ_{j}U^{*}_{iα}E_{ij}U_{jβ} = U^{†}EU = A So, we have transformed the matrix representation E of Ê in the basis {α>}_{N} into another matrix representation A in the basis {i>}_{N} using the transformation matrix U. The last formula Σ_{i}Σ_{j}U^{*}_{iα}E_{ij}U_{jβ} = U^{†}EU=A should be read as follows. Use Σ_{i}Σ_{j}U^{*}_{iα}E_{ij}U_{jβ} to compute the matrix element A_{αβ}
