How is the state of a quantum mechanical system described?
(1)  At a fixed time t₀ the state of a physical system is defined by specifying a ket |Ψ(t₀)> belonging to the space ℰ.  ℰ is a complex, separable Hilbert space, a complex linear vector space in which an inner product is defined and which possesses a countable orthonormal basis.  The vectors in such a space have the properties mathematical objects must have in order to be capable of describing a quantum system.
Consider two kets, |Ψ(t₀)> and ae^iθ|Ψ(t₀)>.  They represent the same physical state.
But |Φ> = a₁e^iθ1|Ψ₁> + a₂e^iθ2|Ψ₂> ≠  a₁|Ψ₁> + a₂|Ψ₂>  if θ₂ ≠ θ₂.
Global phase factors and multipliers do not affect the physical predictions, but relative phases and multipliers are significant.

How can we predict the results of a measurement?
(2)  Every measurable physical quantity is described by a Hermitian operator acting in ℰ.  (We want the eigenvectors of the operator to form a basis for the vector space and its eigenvalues to be real.)
(3)  The only possible results of a measurement are the eigenvalues of the Hermitian operator describing this measurement.
(4)  When a physical quantity described by the operator A is measured on a system in a normalized state |Ψ>, the probability of measuring the eigenvalue a_n (a_α) is given by
P(a_n) = ∑_i=1^gn|<u_nⁱ|Ψ> |²,    (dP(a_α) = |<v_α|Ψ> |²dα),
where {|u_nⁱ>} (i = 1, 2, ..., g_n) is an orthonormal basis in the eigensubspace ℰ_n associated with the eigenvalue a_n; (where |v_α> is the eigenvector corresponding to the eigenvalue a_α; we assume a_α is a non degenerate continuous eigenvalue of A.)
This may be written in terms of the projector
P(a_n) = <Ψ|P_n|Ψ>,  P_n = ∑_i=1^gn|u_nⁱ><u_nⁱ|,    (dP(a_α) = <Ψ|P_α|Ψ>,  P_α = |v_α><v_α|dα).  
(5)  If a measurement of a system in the state |Ψ> gives the result a_n, then the state of the system immediately after the measurement is the normalized projection of |Ψ> onto the eigensubspace associated with a_n;
|Ψ> --> P_n|Ψ>/(<Ψ|P_n|Ψ>)^1/2 after measuring a_n.
(6)  The Cartesian components of the observables R and P satisfy   [R_i,R_j] = [P_i,P_j] = 0,   [R_i,P_j] = ihδ_ij.  These are called the canonical commutation relations.  R is the position operator and P is the operator corresponding to the conjugate momentum or canonical momentum.

How do we find the operator corresponding to a physical quantity that is classically defined?
(a)  Express the physical quantity in terms of the fundamental dynamical variables r and the conjugate momenta p.  We define the conjugate momentum through p_x = ∂L/∂x, ... , where L = Lagrangian.
(b)  Symmetrize the expression with respect to r and p, then replace the variables r and p with the operators R and P. 
Example: 

• r∙p --> ½(r∙p + p∙r) --> ½(R∙P + P∙R)  

How is the state of a quantum mechanical system evolve?
(7)  The time evolution of the state vector is governed by the Schroedinger equation,
(iħ∂/∂t)|Ψ(t)> = H(t)|Ψ(t)>, 
where H(t) is the observable associated with the total energy of the system.
For a conservative system, where all the forces can be derived by taking the gradient of a scalar potential, the classical Hamiltonian of a particle is written as 
H = T + U = p²/(2m) + U(r,t).
This is an expression for the total energy of the system.
The quantum mechanical operator is found by replacing p² with P² = P_x² + P_y² + P_z² and U(r,t) with U(R,t).  No product of non-commuting operators is involved, so symmetrization is not required.  The Schroedinger equation becomes
iħ∂/∂t)|Ψ(t)> = [P²/(2m) + U(R,t)]|Ψ(t)>.
The Schroedinger equation is first order in t.  Given |Ψ(t₀)>, |Ψ(t)> is uniquely determined. The quantum state evolves in a perfectly deterministic way between measurements.

How can we know |Ψ(t₀)> ?
If, at t = 0, we measure a complete set of commuting observables, then |Ψ(t₀)> is uniquely defined.  The state is an eigenvector of the unique common eigenbasis.
Two commuting observables can be measured simultaneously, i.e. the measurement of one does not cause loss of information obtained in the measurement of the other.  If we measure a C.S.C.O., then the state of the system after the measurement is one element of an unique eigenbasis. The results of the measurement specify the state completely.

Example:

• Take two commuting observables, A and B, [A,B] = 0.  There exists a common eigenbasis
  {|a_n,b_p,i(n,p)>}.
  If both operators have degenerate eigenvalues, then |a_n,b_p,i> denotes the ith eigenvector with eigenvalues a_n and b_p. 
  We have
  A|a_n,b_p,i(n,p)> = a_n|a_n,b_p,i(n,p)>, 
  B|a_n,b_p,i(n,p)> = b_p|a_n,b_p,i(n,p)>.
  
  Assume that the system is in the state
  Ψ(t₀)> =  ∑_n,p,iC_n,p,i(t₀)|a_n,b_p,i> with  <Ψ(t₀)|Ψ(t₀)> = 1.
  At t = 0 you measure A.  
  The result will be one of the eigenvalues a_n and after the measurement the system will be in the normalized eigenstate 
  P_n|Ψ(t₀)>/(<Ψ(t₀)|P_n|Ψ(t₀)>)^1/2  (postulate 5).
  P_n|Ψ(t₀)> = ∑_p,i|a_n,b_p,i><a_n,b_p,i|∑_n',p',i'C_n',p',i'(t₀)|a_n',b_p',i'> =  ∑_p,iC_n,p,i(t₀)a_n,b_p,i>.
  The probability of obtaining the eigenvalue a_n is  P(a_n) =  ∑_p,i|C_n,p,i(t₀)|²   (postulate 4).
  An infinitesimal time later you measure B.
  The result of this measurement will be one of the eigenvalues b_p, and after the measurement the system will be in the corresponding normalized eigenstate
  P_pP_n|Ψ(t₀)>/(<Ψ(t₀)|P_pP_n|Ψ(t₀)>)^1/2.
  P_pP_n is the projector onto the subspace of eigenvectors with eigenvalues a_n and b_p  
  P_pP_n|Ψ(t₀)> = ∑_n',i'|a_n',b_p,i'><a_n',b_p,i'| ∑_p'',i''C_n,p'',i''(t₀)|a_n,b_p'',i''> = ∑_iC_n,p,i(t₀)|a_n,b_p,i>.
  [P_p,P_n] = 0.  This only holds because [A,B] = 0 and a common eigenbasis exist.  If [A,B] ≠ 0 then P_pP_n is not a projector and [P_p,P_n] ≠ 0.
  P_pP_n|Ψ(t₀)> = ∑_iC_n,p,i(t₀)|a_n,b_p,i>.
  P_an(b_p) = probability of obtaining b_p after obtaining a_n  = ∑_i|C_n,p,i(t₀)|²/(∑_pi|C_n,p,i(t₀)|²).
  P(a_n,b_p) =  probability of the composite event  =  ∑_i|C_n,p,i(t₀)|² = <Ψ(t₀)|P_pP_n|Ψ(t₀)>.
  
  The measurement of B does not cause loss of information obtained in the measurement of A (or vice versa), on the contrary, it adds to it.  After the measurement of A we know the system is in an eigenstate of A, after the measurement of B we know the system is in an eigenstate of A and in an eigenstate of B.

In the Schroedinger picture of quantum mechanics the state vector |Ψ> of the system evolves in time.  Its evolution is described by the Schroedinger equation (iħ∂/∂t)|Ψ(t)> = H(t)|Ψ(t)>.   Results of measurements are predicted by letting operators, which do not evolve in time (which are time independent unless they contain time explicitly), operate on the evolving state vector.  The predictions of quantum mechanics (the probabilities of obtaining given results if measurements are made) are expressed in terms of scalar products and matrix elements of operators.

Let {|a_i>} be a unique eigenbasis of the operator A, A|a_i> = a_i|a_i>, and let |Ψ> be a normalized state vector.  The probability of obtaining a_i as a result of a measurement of A is P(a_i) = |<a_i|Ψ>|² and the average value of the outcome of a measurement of A is
<A> = ∑_ia_iP(a_n) = ∑_ia_i<Ψ|a_i><a_i|Ψ> = ∑_i<Ψ|A|a_i><a_i||Ψ> = <Ψ|A|Ψ>.
 
These predictions of quantum mechanics do not depend on the representation, i.e. which basis we choose to express our state vector.