1. 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

2. 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.