3. Group work
Show that the Sx, Sy and Sz matrices can be written as a linear combination of projection operators. (Projection operators are outer products of the eigenvectors with themselves.) The coefficient of each term is the eigenvalue associated with the eigenvector used to make the projection operator.

4. Postulates of Quantum Mechanics
1. Normalized ket vector contains all the information about the state of
a quantum mechanical system.
2. Operator A describes a physical observable and acts on kets.
3. One of the eigenvalues an of A is the only possible result of a measurement.
4. The probability of obtaining the eigenvalue an :
5. State vector collapse :
6. Schrödinger Equation :
Time evolution of a quantum system

5. What do we know so far?
Atoms (and fundamental particles, i.e., electrons) can have “intrinsic spin” – can be a 2-level spin system, where all measurements of the intrinsic angular momentum yield + or – hbar/2
Can measure this with S-G device
Find that by looking at the % of atoms found along different spin projections (i.e. Sz) we can infer the initial state
Find that making a measurement to determine the state of an atom changes the state -> “collapses” the atom into the new state – this is related to “projection”

6. Group work
Show that the spin operator matrices: Sx and Sy can be written as a linear combination of projection operators, where the projection operators are outer products of the eigenvectors with themselves and the coefficient of each term is the eigenvalue associated with the eigenvector used to make the projection operator
If a spin operator acts on a vector, what transformation does it correspond to?

7. Hermetian Operators: At = A
We normalize the eigenkets!
ALWAYS have real eigenvalues and orthogonal eigenfunctions that form a complete basis set
Linear transformations
What kind of transformations have you observed with matrices acting on vectors – what do they do?
Come up with at least one “physical” example of a linear transformation (operator) (you don’t need to write the matrix, just say what it does)
What did the projection operators do to the vectors?
What can you say about the operators Sz, Sx, and Sy?

8. Projection operator Operators “embed” the kets and eigenvalues
The projector operator MODELS measurements – it tells us what state (ket) the atom is in after the measurement:
It tells us about the probability of finding a particular eigenvalue from a measurement
P+|ψ> = |+><+| ψ> = ψ+|+> = coefficient of Psi along +z spin, in the +z spin direction
(this new ket is NOT normalized!)

9. “Fixing our equation” to make the new ket always normalized