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?
1 Answer
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
Δp⋅Δx≥h4π , where
Now, the uncertainty in momentum can be thought of as the uncertainty in velocity multiplied, in your case, by the mass of the mosquito.
Δp=m⋅Δv
You know that the mosquito has a mass of
Δv=0.01 m/s=10−2m 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 =10−3g =10−6kg
So, rearrange the equation to solve for
Δx≥h4π⋅1Δp=h4π⋅1m⋅Δv
Δx≥6.626⋅10−34m2kgs−14π⋅11.60⋅10−6kg⋅10−2ms−1
Δx≥0.32955⋅10−26m=3.30⋅10−27m
The answer is rounded to three sig figs.
