Course calendar, week 1

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Wed, 9/1:
Course information.
Superposition.
Example of spin-1/2 states.
Two state "qubits", N-qubit states, e.g.  |0110100011>, spanning 2^N dimensions.
Wavefunction Psi(x) of a particle.
Wavefunctional Psi[B(x)] of a field, e.g. the (electro)magnetic field.
Fock (number state) of photons |{n_(k,e)}>.

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Fri, 9/3:
Defintion of a vector space.
Linear operators.
Matrix representation of a linear operator with respect to a basis {v_i}: Lv_i = L_ji v_j (sum on j).
Composition of linear operators corresponds to matrix multiplication.
Trace and determinant as invariants of a linear operator.
Other invariants from other combinations of the eigenvalues
(e.g., those appearing in the characteristic polynomial det(L - aI)).
Definition of inner product.
Unitary operators: (Uv,Uw)=(v,w) for all vectors v,w.
Adjoint or Hermitian conjugate A* (or A^{\dagger}) defined by (v,Aw)=(A*v,w) for all vectors v,w.
Hermitian or self-adjoint operator: A=A*.
Eigenvalues and eigenvectors.
Eigenvalues of unitary operators have unit modulus, e-values of hermitian operator are real.
Matrices of unitary and hermitian operators in an orthonormal basis.
Dirac notation: A vector v is denoted by a "ket" |v>.
(v, ) is a linear map from vectors to complex numbers, also denoted by a "bra" <v|.
The inner product looks like a "bracket" : (v,w)=<v|w> (not <v||w>).
The matrix elements of an operator A in an orthonormal (ON) basis {|i>} are A_ij = <i|A|j>.
Resolution of the identity: I = Sum_i |i><i|, where {|i>} is an ON basis.
Example: <i|AB|j> = <i|AIB|j> = <i|A|k><kB|j> (sum on k). This is useful if we know something
about the matrix elements of A and/or B in this basis, for example if it is an eigenbasis of A or B.