Operator methods in Quantum Mechanics

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 Pn 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. 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 .y a-y a-(a-y) .a+(a+y) a+ y y P460 - operators and H.O.

8. H.O.- algebraic/group theory • Can prove • this uses • asimilar 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 <x>, <p>, <x2>, <p2> 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 <x2>, <p2> 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 <x2>, <p2> and uncertainty relationship for ground state. • using “mixed” terms P460 - operators and H.O.