Last hour: If every element of state space can be expanded in one (and only one) way in terms of a set of countable, orthonormal functions uj , we call the {uj} a discrete basis: If every element of state space can be written in terms of functions with a continuously varying parameter p as we call the p a continuous basis. Adjoint operator Â+: <f|Âg> = <Â+f|g> Hermitian operator: <f|Âg> = <Âf|g> All operators representing observables in QM are Hermitian! Hermitian Operators are self-adjoint: Â+ = Â. All eigenvalues of Hermitian operators are real. All eigenfunctions of a Hermitian operator with different eigenvalues are orthogonal . The eigenfunctions of a Hermitian operator form a basis

Learning Goals for Chapter 6 – Average values After this chapter, the related homework problems, and reading the relevant parts of the textbook, you should be able to: calculate average values of observables for given wave functions; explain how obtaining a QM average as <Ô> = <|Ô|> relates to the expansion of a wave function; calculate the time derivative of a QM average; determine if an observable is a constant of motion in a QM system.