- #1

alexgmcm

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## Homework Statement

Following a question that asks for the Energy-Time uncertainty principle, [tex]\Delta E \Delta T \geq \frac{\hbar}{2}[/tex]

Show that for any wave, the fractional uncertainty in wavelength, [tex]\frac{\Delta \lambda}{\lambda}[/tex] is the same in magnitude as the fractional uncertainty in frequency [tex]\frac{\Delta f}{f}[/tex]

## Homework Equations

Energy-Time uncertainty principle as stated above:

[tex]\Delta E \Delta T \geq \frac{\hbar}{2}[/tex]

The relation between Energy and Frequency of a wave:

[tex] E = hf [/tex]

## The Attempt at a Solution

Using the above equations we can deduce that:

[tex]\Delta E = h \Delta f [/tex]

but from E = hf we can say that [tex] h = \frac{E}{f}[/tex]

[tex] \therefore \Delta E = \frac{E}{f} \Delta f [/tex]

[tex] \therefore \frac{\Delta E}{E} = \frac{\Delta f}{f} [/tex]

Now I thought I might be able to do a similar thing for wavelength using [tex]E = \frac{hv}{\lambda}[/tex] but this runs into problems stemming from the fact that lambda is the denominator of that equation not the numerator. I have tried several rearrangements and even considered using Position-Momentum Uncertainty with De Broglie, but I still can't get see how I can equate the fractional uncertainties. I'm not even sure if my rearrangement for the frequency is on the right track or not.

Any help would be greatly appreciated.