Postulates of Quantum Mechanics
                       (from “quantum mechanics” by Claude Cohen-Tannoudji)
      nd                                               Q
     2  postulate: Every measurable physical quantity
       is described by an operator ˆ  This operator is an observable. 
                                   Q. 
             3rd postulate: The only possible result of the 
             measurement of a physical quantity Q is one of the eigenvalues
                                           €  ˆ 
             of the corresponding             Q .
                      € 
             observable
      4th postulate (non-degenerate): When the physical quantity Q
                                     €            ψ
 is measured on a system in the normalized state      the probability of
                           a      €                             ˆ 
 obtaining the eigenvalue   n of the corresponding             Q is
                      2        observable                              ˆ 
    P a = u ψ            where u      is the normalized eigenvector of Q 
      ( n)      n                 n                   € 
                         associated with the eigenvalue a .
                                      €                  n
               €                                   € 
€                   €                                     € 
                                            € 
    Physical interpretation of ψ
         2    *
       ψ =ψψ is a probability density. The probability of 
  finding the particle in the volume element dxdydz at time t is
              € 2
       ψ(x,y,z,t) dxdydz.
€ 
                       €            € 
   General solution for ψ(x,y,z,t)
€                              ψ(x,y,z,t) =ψ (x,y,z)θ (t)
            Try separation of variables:   n       n
   Plug into TDSE to arrive at the pair of linked equations:
          €    −iE t/h
      θ t =e n         €     ˆ 
       n( )            and  H ψ = E ψ
                                n     n n
€                € 
      Orthogonality:
         ψ ,ψ                                         Hψ =Eψ
    For    a   b  which are different eigenvectors of     n      n n
                                   *
     we have orthogonality:      ψψ =0
                               ∫
                                   a  b
€                                        € 
     Let us prove this to introduce the bra/ket 
     notation used in the textbook
                  €