173k views
2 votes
. Position-Momentum Uncertainty Principle (2 pt). An entomologist is studying a small beetle. She records the beetle's weight to be 1.0mg and its diameter to be 3 mm. At one point she is using a magnifier and observes the beetle to be stationary with an uncertainty of 10 −2

mm. How fast might the beetle actually be moving?

User Mrxra
by
7.8k points

1 Answer

1 vote

Final answer:

The question involves the Heisenberg Uncertainty Principle to estimate the minimum uncertainty in the velocity of a beetle. By applying the principle's formula and the given uncertainties in position, we can calculate the beetle's potential velocity despite it appearing stationary.

Step-by-step explanation:

The question deals with the Heisenberg Uncertainty Principle in physics, which states that there is a fundamental limit to the precision with which certain pairs of physical properties of a particle, such as position (Δx) and momentum (Δp), can be known simultaneously. For a beetle observed to be stationary with an uncertainty in position of 10-2 mm, we will estimate the uncertainty in its velocity (Δv).

The uncertainty principle is represented by the inequality ΔxΔp ≥ ħ/2, where ħ is the reduced Planck's constant (1.055 × 10-34 kg·m2/s). First, convert the uncertainty in position to meters (Δx = 10-5 m). Next, we calculate Δp using the relation Δp ≥ ħ/(2Δx). Finally, we can compute the uncertainty in velocity by dividing the uncertainty in momentum by the mass of the beetle (m = 1.0 × 10-6 kg).

For our calculations: Δp ≥ (1.055 × 10-34 kg·m2/s) / (2 × 10-5 m) gives the minimum Δp. Then, Δv ≥ Δp / m gives the minimum uncertainty in the beetle's velocity, which is how fast it might actually be moving in grounding theoretical terms.

User Jkindwall
by
6.6k points