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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 ∂|ψ(t)> /∂t We can write this equation as: (- i/)H ∂t = ∂|ψ(t)> / |ψ(t)> Its integration yields log |ψ(t)> = (- i/)H t + constant1. Therefore: |ψ(t)> = constant2 exp {(- i/)H t} At t = 0 the system was in the state |ψ(0)> Hence constant2 = |ψ(0)>. Then: |ψ(t)> = exp {(- i/)H t} |ψ(0)> The state of the system evolves as: |ψ(t)> = exp {- iHt/ } |ψ(0)> U = exp {- iHt/} is the time evolution operator or the propagator.
  


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