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.
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 : P  an 
5. State vector collapse :  ' 
Pn 
 Pn 
6. Schrödinger Equation : i d  (t )  H (t )  (t )
dt
Time evolution of a quantum system
2
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”
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?
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?
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!)
“Fixing our equation” to make the new ket always normalized