Postulates of Quantum Mechanics

Quantum Mechanics divides the world into two parts, commonly called the system and the observer.  Except at specified times the system and the observer do not interact.  An interaction at those specified times is called a measurement.  Quantum Mechanics predicts all the information that the observer can possibly obtain about the system.  This information can be represented in different ways.  It is often represented in terms of a wavefunction.  A measurement changes the information an observer has about the system and therefore changes the wavefunction of the system.

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 |y(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.

How can we predict the results of a measurement?

2. Every measurable physical quantity is described by a Hermitian operator acting in E.  (The eigenvectors of the operator form a basis for the vector space and its eigenvalues are 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 |y>, the probability of measuring the eigenvalue a_n (a_a) is given by
   ,
   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_a> is the eigenvector corresponding to the eigenvalue a_a; we assume a_a is a non degenerate continuous eigenvalue of A.)  This may be written in terms of the projector:
   .
5. If a measurement on a system in the state |y> gives the result a_n, then the state of the system immediately after the measurement is the normalized projection of |y> onto the eigensubspace associated with a_n;
6. The Cartesian components of the observables R and P satisfy   .  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 , 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.
7. The time evolution of the state vector is governed by the Schroedinger equation,  where H(t) is the observable associated with the total energy of the system.