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
bra-ket 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 non-algebraic. 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 non-commuting
operator has a probability density which varies in time. It is this time independence (conservation