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Question: What is uncertainty in the location of a photon of wavelength \(5000 \overset{{}^\circ }{\mathop{A}}...

What is uncertainty in the location of a photon of wavelength 5000A∘ 5000 \overset{{}^\circ }{\mathop{A}}\,, if wavelength is known to an accuracy of 1 pm?
A. 7.96×10−4m7.96\times {{10}^{-4}}m
B. 0.02 m
C. 3.9×10−8m3.9\times {{10}^{-8}}m
D. None of these

Explanation

Solution

Heisenberg’s uncertainty principle tells us that it is impossible to find the position and momentum of any electron or sub-atomic species, with full accuracy. This principle explains the dual nature of matter, that is particle and wave nature.
Formula used:
Heisenberg’s uncertainty principle Δx×Δp=h4π\Delta x\times \Delta p=\dfrac{h}{4\pi } , where h is planck’s constant, x is position, p is momentum.

Complete answer:
We have been given a wavelength of a photon, λ=5000A∘ \lambda =5000\overset{{}^\circ }{\mathop{A}}\, for which we have to find the uncertainty in position. So from Heisenberg’s uncertainty formula we will first take out change in momentum to determine uncertainty in position.
As we know from de-broglie equation, λ=hp\lambda =\dfrac{h}{p} ,
We have p=hλ=Δpp=Δλλp=\dfrac{h}{\lambda }=\dfrac{\Delta p}{p}=\dfrac{\Delta \lambda }{\lambda }
Therefore, we will have change in momentum, Δp=Δλ(pλ)=Δλhλ2\Delta p=\Delta \lambda \left( \dfrac{p}{\lambda } \right)=\dfrac{\Delta \lambda h}{{{\lambda }^{2}}} ,
Δp=(10−12m)×6.6×10−34Js(5000s)2\Delta p=({{10}^{-12}}m)\times \dfrac{6.6\times {{10}^{-34}}Js}{{{(5000s)}^{2}}}
Δp=2.6×10−33\Delta p=2.6\times {{10}^{-33}}
So, now uncertainty in position will be calculated as, Δx=hΔp×4π\Delta x=\dfrac{h}{\Delta p\times 4\pi }
Δx=6.6×10−344π×2.6×10−33\Delta x=\dfrac{6.6\times {{10}^{-34}}}{4\pi \times 2.6\times {{10}^{-33}}}
Δx\Delta x= 0.02 m
Hence, the uncertainty in position is calculated to be 0.02 m. So, option B is correct.
Additional information: the significance of Heisenberg’s uncertainty principle is applicable for the sub atomic particles which are microscopic. It is insignificant for macroscopic species.

Note:
The value of wavelength 5000A∘ \overset{{}^\circ }{\mathop{A}}\,is converted to picometer and then to meter per second to obtain velocity, with the conversion factor of 1A∘ =10−2pm=10−12m1\overset{{}^\circ }{\mathop{A}}\,={{10}^{-2}}pm={{10}^{-12}}m, as the accuracy is given in pico meter. h is planck’s constant whose value is 6.6×10−34Js6.6\times {{10}^{-34}}Js.