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When to use H when to use U(H)?

  1. Apr 17, 2010 #1
    If the dynamics of the system are descibed by a Hamiltonian, H please could someone explain when should I be using
    [tex]|\right \psi(t) \rangle=H\left |\right \psi(0) \rangle[/tex]
    and when to use
    [tex]|\right \psi(t) \rangle=U\left |\right \psi(0) \rangle[/tex]

    Thank you
  2. jcsd
  3. Apr 17, 2010 #2
    [tex]H[/tex] is the operator of energy. You use it, when you want to know the energy of a state from the eigenstate equation:
    [tex]H \phi(0) = e \phi(0)[/tex]

    [tex]U(t)[/tex] as you defined it, is a time shift operator. You use it when you want to know what will happen with your state after time [tex]t[/tex], provided you know it at time 0.
    [tex]\phi(t) = U(t) \phi(0)[/tex]

    You must first know energy from the first equation before you check time evolution from second equation.
  4. Apr 18, 2010 #3


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    You can apply the time development operator to systems w/o knowing their energy; this works even for systems (wavefunctions) that are not solutions to the Schrödinger equation specified by H. This is used both in scattering and in time-dependent perturbation theory: you can e.g. look at the scattering of plane waves in a given potential V (a certain H=T+V); it is clear that the plane waves do not solve the Schrödinger equation, therefore they are not eigenstates of H, but nevertheless you can use U (or some scattering operator derived from U) to evolve the plane waves in time and study the scattering matrix.
  5. Apr 18, 2010 #4
    Thank you
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