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 m/s if the speed is known to within 0.0100m/s?

Answer 1

`3.30 * 10^(-27)"m"`

Explanation:

The Heisenberg Uncertainty Principle states that you cannot simultaneously measure both the momentum of a particle and its position with arbitrarily high precision.

Simply put, the uncertainty you get for each of those two measurements must always satisfy the inequality

`color(blue)(Deltap * Deltax>= h/(4pi))" "`, where

`Deltap` - the uncertainty in momentum;
`Deltax` - the uncertainty in position;
`h` - Planck's constant - `6.626 * 10^(-34)"m"^2"kg s"^(-1)`

Now, the uncertainty in momentum can be thought of as the uncertainty in velocity multiplied, in your case, by the mass of the mosquito.

`color(blue)(Deltap = m * Deltav)`

You know that the mosquito has a mass of `"1.60 mg"` and that the uncertainty in its velocity is

`Deltav = "0.01 m/s" = 10^(-2)"m s"^(-1)`

Before plugging your values into the equation, notice that Planck's constant uses kilograms as the unit of mass.

This means that you will have to convert the mass of the mosquito from miligrams to kilograms by using the conversion factor

`"1 mg " = 10^(-3)"g " = 10^(-6)"kg"`

So, rearrange the equation to solve for `Delta` and plug in your values

`Deltax >= h/(4pi) * 1/(Deltap) = h/(4pi) * 1/(m * Deltav)`

`Deltax >= (6.626 * 10^(-34)"m"^color(red)(cancel(color(black)(2)))color(red)(cancel(color(black)("kg"))) color(red)(cancel(color(black)("s"^(-1)))))/(4pi) * 1/(1.60 * 10^(-6)color(red)(cancel(color(black)("kg"))) * 10^(-2)color(red)(cancel(color(black)("m")))color(red)(cancel(color(black)("s"^(-1)))))`

`Deltax >= 0.32955 * 10^(-26)"m" = color(green)(3.30 * 10^(-27)"m")`

The answer is rounded to three sig figs.