Operations with observables

1. Time-evolution operator
Let's H be a time-independant hamiltonian  of a quantum system. 
At the time "t", the state of the system is described by the wave 
function ψ(x,t) or |ψ(t)>. The Schrodinger time-dependent 
equation is: 
H|ψ(t)>  = i ℏ ∂|ψ(t)> /∂t 
We can write this equation as:
(- i/ℏ)H ∂t   =  ∂|ψ(t)> / |ψ(t)> 
Its integration yields
log |ψ(t)>  = (- i/ℏ)H t + constant1. Therefore:
|ψ(t)> = constant2 exp {(- i/ℏ)H t}
At t = 0 the system was in the state |ψ(0)> 
Hence constant2 = |ψ(0)>. Then:
|ψ(t)> = exp {(- i/ℏ)H t} |ψ(0)>

The state of the system evolves as:
|ψ(t)> = exp {- iHt/ℏ } |ψ(0)>

U = exp {- iHt/ℏ} is the time  evolution operator or the propagator.