to obtain an expression for the standard deviation in the time variable for the

pulse. This may be labeled ${}_t($or $t,$ to be in conformity with the energy$-$time

uncertainty principle, $Et\frac{}{2}.)$

$($c$).$ Your expression for $P_f()$ provides the probability of exciting the molecular

transition as a function of the difference between the frequency of the radiation,

$,$ and the frequency of the transition, ${}_{fi}.$ It therefore gives the line shape that

would be observed in a spectrum, given the fact that you are using a pulsed source

of radiation.

Now, compare your expression for $P_f()$ to the standard form of a

Gaussian probability distribution to obtain the standard deviation of the

distribution of frequencies that can excite the transition. This may be designated

as either ${}_{}$ or $.$ Convert your expression for $$ to an energy uncertainty, $E$

using the relation $E=.$

$($d$).$ Combine your expressions for $E$ and $t$ to obtain an expression for the uncertainty

product $Et$ for the case of a Gaussian laser pulse. You should find that the

energy$-$time uncertainty principle, $Et\frac{}{2}$ is obeyed.