- #1

- 288

- 2

## Homework Statement

"Show that if the Hamiltonian depends on time and [itex][H(t_1),H(t_2)]=0[/itex], the time development operator is given by

[tex]U(t)=\mathrm{exp}\left[-\frac{i}{\hbar}\int_0^t H(t')dt'\right]."[/tex]

## Homework Equations

[itex]i\hbar\frac{d}{dt}U=HU[/itex]

[itex]U(dt)=I-\frac{i}{\hbar}H(t)dt[/itex]

## The Attempt at a Solution

The first thing I tried was to rearrange the first of the relevant equations:

[tex]\left(\frac{d}{dt}U\right)U^{-1}=-\frac{i}{\hbar}H(t).[/tex]

I can then integrate both sides; if the LHS could turn into an expression like [itex]\ln{U}[/itex] I'd be done, but that didn't work out. Any hints?