Physics Asked on April 4, 2021
I want to ask a question about the frequency spread of a wavepacket from a femtosecond laser pulse.
I was presented with the following question today:
Compute the frequency spread of a wavepacket created by a femtosecond laser
with a pulse of $Delta t=85$ fs. Report your answers in wavenumbers $left(cm^{-1}right)$ and comment on your answer.
I have attempted my working out below.
From the uncertainty principle, we have:
$$Delta t Delta E = frac{hbar}{2pi}$$
and using Planck’s equation:
$$E = frac{hc}{lambda}$$
I first decided to convert Planck’s equation into wavenumbers $left(cm^{-1}right)$ as shown below:
$$1m = 100 cm$$
thus
$$frac{1}{m} = frac{1}{100cm}$$
$$E = frac{hc}{lambda_{m}} = frac{hc}{100 lambda_{cm}}$$
I first calculated $Delta E$ explicitly:
$$Delta E = frac{frac{hbar}{2pi}}{85times 10^{-15}} = 1.240 times 10^{-21} $$
followed by the use of Planck’s equation:
$$1.240 times 10^{-21} = frac{6.626 times 10^{-34} times 2.997 times 10^{8}}{100 lambda_{cm}}$$
such that
$$frac{1.240 times 10^{-21} times 100}{6.626 times 10^{-34} times 2.997 times 10^{8}} = lambda_{cm} = 6.25 times 10^{5} cm^{-1}$$
However, I’m expecting an answer (according to the mark scheme) of $400$ cm$^{-1}$, which is far more appropriate, and justifies the laser’s behaviour to creating a wavepacket that excites a series of states in a coherent fashion.
What am I doing wrong here? Have I miscalculated something or is the answer scheme incorrect with regards to the final expected answer?
I am a penultimate-year chemist studying QM, so I may not be too well versed in the mathematics of QM entirely!
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