Question

Using Heisenberg's uncertainty principle, how would you calculate the uncertainty in the position of a $1.60mg$ mosquito moving at a speed of $1.50\text{ }m/s$ if the speed is known to within $0.0100m/s$ ?

Answer 1

Hint : The uncertainty principle, also known as the Heisenberg uncertainty principle or the indeterminacy principle, is a statement made (1927) by the German physicist Werner Heisenberg that the position and velocity of an object cannot be measured precisely at the same time, even in theory.

Complete Step By Step Answer:
According to the Heisenberg uncertainty principle, it is impossible to accurately determine both the position and velocity of particles that have both particle and wave natures at the same time.
According to the Heisenberg Uncertainty Principle, you cannot measure a particle's momentum and position with arbitrarily high precision at the same time.
Simply put, the uncertainty for each of those two measurements must satisfy the inequality.
$\Delta p\cdot \Delta x\ge \dfrac{h}{4\pi },\text{ }$ where
$\Delta p$ =the uncertainty in momentum;
$\Delta x$ =the uncertainty in position;
$h$ = planck’s constant= $6.626\cdot {{10}^{-34}}{{m}^{2}}kg\text{ }{{s}^{-1}}$
Now, the uncertainty in momentum can be thought of as the uncertainty in velocity multiplied by the mass of the mosquito in your case.
$\Delta p=m\cdot \Delta v$
You are aware that the mosquito weighs $1.60mg$ and that the uncertainty in its velocity is
$\Delta v=0.01\text{ }m/s={{10}^{-2}}m\text{ }{{s}^{-1}}$
Before entering your values into the equation, keep in mind that Planck's constant is expressed in kilograms.
This means you'll have to convert the mosquito's mass from milligrams to kilograms using the conversion factor.
$1\text{ }mg={{10}^{-3}}g={{10}^{-6}}kg$
So, rearrange the equation to solve for $\Delta$ and plug in your values,
$\begin{aligned} & \Delta x\ge \dfrac{h}{4\pi }\cdot \dfrac{1}{\Delta p}=\dfrac{h}{4\pi }\cdot \dfrac{1}{m.\Delta \upsilon } \\\ & \Delta x\ge \dfrac{{{6.626.10}^{-34}}{{m}^{2}}kg{{s}^{-1}}}{4\pi }.\dfrac{1}{{{1.60.10}^{-6}}kg{{.10}^{-2}}m{{s}^{-1}}} \\\ & \Delta x\ge {{0.32955.10}^{-26}}m={{3.30.10}^{-27}}m \\\ \end{aligned}$

Note :
Since this uncertainty principle is such a fundamental result in quantum mechanics, typical quantum mechanics experiments routinely observe aspects of it. Certain experiments, on the other hand, may purposefully test a specific form of the uncertainty principle as part of their main research program. Tests of number–phase uncertainty relations in superconducting or quantum optics systems. Extremely low-noise technology, such as that required in gravitational wave interferometers, is one application that relies on the uncertainty principle for operation.