Class minutes---week 2

Mon, 9/6: Labor day, no class.

Wed, 9/8:
Components of bra = (components of ket)*
<v|w> = <v|i><i|w> = v_i* w_i (sum on i)
matrix notation
projection operator: |v><v| (if <v|v>=1) projects orthogonally onto 1-dimensional subspace
general projection operators: PP=P, P*=P, eigenvalues=0,1, projects onto some subspace.

Each quantum mechanical system has an associated a Hilbert space, and a maximal specification of a state of the system is a normalized vector |v> in the Hilbert space. The probability of finding the system in the state |w> if it is actually in the state |v> is |<w|v>|^2. More generally, the probability of finding it in the subspace defined by a projector P is ||P |v>||^2 = <v|P||v>.

Example of spin-1/2: |+x> = (|+z> + |-z>)/\sqrt{2}.
Probability of |+x> in state |+z> is |<+x|+z>|^2 = 1/2.
Example of higher dimensional projection: particle with spin and spatial degrees of freedom.
Specify location but not spin: two-dimensional subspace. Specify spin but not location:
infinite dimensional subspace.

Fri, 9/10:
# Other examples of projections: for a particle, specify location in a region but not at a point: infinite dimensional subspace. For a composite system: specify the state of some part of the system. E.g. , specify the state of the first qubit of a two-qubit system: P=|00><00| + |01><01|.
# "Observable"<->hermitian operator.
Spectral representation of hermitian operator: A=a_i |i><i|, sum on i, where {|i>} is an ON basis
of eigenvectors of A with eigenvalues a_i. Clump together terms with same value of a_i: A=aP_a, sum on a, where P_a=|i><i| summed over the values of i for which a_i=a, i.e. the projection onto the subspace with e-value a. The eigenvectors of a hermitian operator span the Hilbert space.
# Average value or expectation value of A in state |v> is <A>_v=Sum_i prob_i a_i=Sum_i <v|i><i|v>a_i=<v|A|v> (using the spectral rep. of A).
# Time evolution preserves inner product and superposition, so it is given by a unitary operator, |v(t2)>=U(t2,t1)|v(t1>, where U(t2,t1) is unitary.
(Amazingly one can get the same conclusion just from the assumption that the probabilities |<v|w>|^2 are preserved by time evolution, without assuming anything about superpositions (linearity). Wigner showed (see Weinberg, The Quantum Theory of Fields, Vol. 1, Ch.2, Appendix A) that this implies the operator is either unitary or antiunitary, and only the former is continuously connected to the identity.)
# Infinitesimal time evolution: U(t+dt,t)=1+Adt, U*U=1 => A+A*=0 => A anti-hermitian, A=(-i/hbar)H for some hermitian H, called the Hamiltonian. dU(t,0)/dt=[U(t+dt,0)-U(t,0)]/dt=[U(t+dt,t)U(t,0)-U(t,0)]/dt=A(t)U(t,0), so (d/dt)|v>=A(t)|v>,
or i hbar (d/dt)|v>=H(t)|v>, the Schrodinger eqn.
# If H(t)=H is time independent, the solution to the Schrodinger eqn is |v(t)>=exp(-iHt/hbar)|v(0)>.
If |v(0)> is an eigenvector |E> of H with eigenvalue E, then |E,t>=exp(-iEt/hbar)|E,0>=exp(-iwt)|E,0>,
where E=hbar w, the same as Einstein's relation between energy and frequency of a photon.
# Planck's constant: hbar=6.58E-16 eV-s = 0.658 ev-fs = 2/3 eV-fs (1 fs = E-15 s = one femtosecond)
# composite systems: e.g. two qubits: ON basis {|0>|0>, |0>|1>,|1>|0>,|1>|1>}, or in simpler notation, {|00>,|01>,|10>,|11>}.
Generally, system A&B with Hilbert spaces H_A & H_B combine by tensor product: H_AB = H_A * H_B where "*" stands for tensor product. That is, a basis for the composite system is the set of outer products of basis vectors, one from each of the two Hilbert spaces. The dimension of the tensor product is thus the
product of the dimensions.
The inner product on H_AB is such that outer products of ON basis vectors are normalized and
orthogonal to each other unless both basis vectors are the same. This is the same as saying that (v1w1,v2w2)=(v1,v2)(w1,w2)
(plus extensions by linearity to non-product tensors).
PHYSICALLY, this corresponds to the fact that probabilities for the composite system are joint probabilities, so the probabilities should combine by multiplication.
# Entangled states: e.g. |w>=a |00> + b |11>.
What is the state of the first qubit? It is |0> with probability |a|^2 and |1> with probability |b|^2.
It cannot be assigned a definite state on its own. Entangled states of two systems are states that cannot
be written as a product of two states, one from each component.