 
Summary: 1 Summary of things you should already know
"I think I can safely say that nobody understands quantum mechanics" Richard Feynman
1.1 Operators and Observables
It is a premise of QM that any measurable quantity is associated with a Hermitian operator.
So far we are used to analytic expressions for wavefunctions and operators. But if a ket is full
description of the state of a system, it must also contain some implicit information. The abstract
braket notation includes this.
Consider the electric charge. Obviously this is measurable, so it should be associated with an
operator ^Q, such that e.g.
^Q = e
where is the wavefunction of an electron. e meets all the criteria for a quantum number,
and the above equation is obviously a true representation of reality, but it cannot be proved
algebraically. Thus the meaning of the ket  is broader than a simple spatial function, and
operators can also be nonalgebraic. This is especially important in particle physics where all
manner of quantum numbers appear (isospin, strangeness, baryon number etc. etc.)
1.2 Hamiltonians and eigenstates
Schroedinger's equation ^H = i¯h/t shows us that the Hamiltonian (energy operator) is related
to the change in wavefunction in time. A system prepared in an eigenstate of the Hamiltonian
has time invariant probability density. A system prepared in an eigenstate of a noncommuting
operator has a probability density which varies in time. It is this time independence (conservation
