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

1. Apr 17, 2010

### raisin_raisin

Hello,
If the dynamics of the system are descibed by a Hamiltonian, H please could someone explain when should I be using
$$|\right \psi(t) \rangle=H\left |\right \psi(0) \rangle$$
and when to use
$$|\right \psi(t) \rangle=U\left |\right \psi(0) \rangle$$
where
$$U=e^{-iHt/\hbar}$$

Thank you

2. Apr 17, 2010

### haael

$$H$$ is the operator of energy. You use it, when you want to know the energy of a state from the eigenstate equation:
$$H \phi(0) = e \phi(0)$$

$$U(t)$$ 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 $$t$$, provided you know it at time 0.
$$\phi(t) = U(t) \phi(0)$$

You must first know energy from the first equation before you check time evolution from second equation.

3. Apr 18, 2010

### tom.stoer

No.

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.

4. Apr 18, 2010

Thank you