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\def\fb{}

%% \subhead   Notes from Baranger's talk.   \endsubhead

From Schr\"{o}dinger equation,
  $i\hbar{\partial\psi\over\partial t}={\fb H}\psi$,
we can get

$$\psi(t+dt)= \psi(t) + {\partial\psi\over\partial t}dt
	=\psi(t) - {i\over\hbar}{\fb H}\psi(t) dt
	= e^{-{i\over\hbar}{\fb H} dt}\psi(t)$$

This last form is unitary.

$$ e^{-{i\over\hbar}{\fb H} dt} \approx e^{-{i\over\hbar}
                                                       {{\fb V}\over2} dt}
	e^{-{i\over\hbar}{\fb H} dt} e^{-{i\over\hbar}
                                                   {{\fb V}\over2} dt} $$

$$ \left| \psi(t) \right>
	= e^{-{i\over\hbar}{\fb H}T} \left| \psi(0) \right> $$

$$ \left< xy | \psi(t) \right>
      =\left< xy \right| e^{-{i\over\hbar}{\fb H}T} \left| \psi(0) \right>
      =\left< xy \right| e^{-{i\over\hbar}{\fb H}T} \left| \psi(0) \right>
$$
