Question

Aman walks at$\mathbf{1}\mathit{m}\mathbf{/}\mathit{s}$, known to within an uncertainty (unrealistically small) of$\mathbf{\text{1}}\mathbf{}\mathit{\mu }\mathbf{\text{m/h}}$.

(a) Compare the minimum uncertainty in his position to his actual physical dimension in his direction of motion $\mathbf{25}\mathit{c}\mathit{m}$,from front to back.

(b) Is it sensible to apply the uncertainty principle to the man?

Short answer

1. $\Delta \text{x}~{\text{10}}^{-26}\times 25\text{cm}$
2. No

Step-by-step solution

 Uncertainty Principle

The position and speed of particles (like electron and photon) is not known with perfect accuracy. This is the principle of Uncertainty.

Given data

Man weight$65kg$

Speed$1m/s$

Uncertainty$\text{1}\mu \text{m/h}$

Step 3(a): Compare the minimum uncertainty in his direction of motion

Given the uncertainty in the velocityv, we need to figure out the position measurement$\text{(}\Delta \text{x)}$constraint.

$$\begin{gathered} \Delta \text{p}\Delta \text{x}⩾\frac{h}{2} \\ \text{m}\Delta \text{v}\Delta \text{x}⩾\frac{h}{2} \\ 65kg\times \frac{{\displaystyle {\text{10}}^{\text{- 6}}}}{{\displaystyle \text{3600}}}m/s\times \Delta \text{x≥}\frac{{\displaystyle {\text{1.054×10}}^{\text{- 34}}\frac{kg\times {\displaystyle {{\displaystyle \text{m}}}^{\text{2}}}}{s}}}{{\displaystyle \text{2}}} \\ \Delta \text{x≥}\frac{{\displaystyle {\text{1.054×10}}^{\text{-34}}\text{m}}}{2\times {\displaystyle {{\displaystyle \text{1.81×10}}}^{\text{- 8}}}} \\ \Delta {\text{x≥2.91×10}}^{\text{- 27}}\text{m} \\ \Delta \text{x}~{\text{10}}^{-26}\text{×}25\text{cm} \end{gathered}$$

Hence the solution is $\Delta \text{x}~{\text{10}}^{-26}\times 25\text{cm}$

Step 4(b):Find whether uncertainty principle is applicable to man

The ambiguity in position is around ${\text{10}}^{\text{- 26}}$ orders of magnitude smaller than the man's width. Beyond the experimental observations, this is a very small value; therefore applying the uncertainty relation is pointless. The answer is No,it is not sensible to apply the uncertainty principle to the man.