P460 - operators and H.O.1 Operator methods in Quantum Mechanics Section 6-1 outlines some formalism – don’t get lost; much you understand define ket and bra vectors and dot product add in operators to this formalism. Let A be an operator

P460 - operators and H.O.2 Orthonormal States can usually define a set of orthonormal states |n> (eigenfunctions). Can rotate to this basis (diagonalize operater) any other function can be made from these identical to 2D vectors

P460 - operators and H.O.3 Projection operator this defines the projection operator P n which when it acts on an arbitrary state projects it into the state |n> so “projection” along “vector” n. Again for 2D vectors

P460 - operators and H.O.4 Heisenberg Picture Section 6-4 discusses the difference between the Heisenberg and Schrodinger picture. Don’t worry about it – 460 mostly uses “Schrodinger” one determines expectation values of operators the two ways vary in whether the operator or the wavefunction changes with time. In some sense, different base vectors are being used

P460 - operators and H.O.5 H.O. - algebraic/group theory write down the Hamiltonian in terms of p,x operators try to factor but p and x do not commute. Explore some relationships, Define step-up and step-down operators

P460 - operators and H.O.6 H.O.- algebraic/group theory Reminder: look at [x,p] with this by substitution get and Sch. Eq. Can be rewritten in one of two ways Look for E eigenvalues and wave functions which satisfy S.E. Note books uses just “A” for step down operator

P460 - operators and H.O.7 H.O.- algebraic/group theory Start with eigenfunction with eigenvalue E so these two new functions are also eigenfunctions of H with different energy eigenvalues a + is step-up operator: moves up to next level a - is step-down operator: goes to lower energy.a + (a +  a +    a -  a - (a - 

P460 - operators and H.O.8 H.O.- algebraic/group theory Can prove this uses a similar proof can be done for step-down can raise and lower wave functions. But there is a lowest energy level…..

P460 - operators and H.O.9 H.O. eigenfunctions For lowest energy level can’t “step-down” easy differential equation to solve can determine energy eigenvalue step-up operator then gives energy and wave functions for states n=1,2,3…..

P460 - operators and H.O.10 H.O. eigenfunctions can also use operators to determine normalization (see book) giving relatively easy to prove/see that

P460 - operators and H.O.11 H.O. Example Compute,,, and uncertainty relationship for ground state. Could do just by integrating. Instead using step-up and step-down operator. With this

P460 - operators and H.O.12 H.O. Example Compute, and uncertainty relationship for ground state. consider when you step-up and step-down (or vice-versa) you get back to the same state modulo a normalization term

P460 - operators and H.O.13 H.O. Example Compute, and uncertainty relationship for ground state. using “mixed” terms