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The frequency spread of a wavepacket from a femtosecond laser pulse

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!

Question marking scheme for the femtosecond laser task

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