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Chapter 4: Q40E (page 136)

A visual inspection of an ant of mass 0.5 mg verifies that it to within an uncertainty of 0.7μmof a given point, apparently stationary. How fast might the ant actually be moving?

Short Answer

Expert verified

The ant moves at a speed of 1.5x10-22m/s.

Step by step solution

01

Given data.

Mass is 0.5mg10-6kg1mg=5×10-7kg.

Uncertainty in position ∆x=7×10-7m.

h=1.0545×10-34J·s.

02

Heisenberg uncertainty principle.

The following equation describes the Heisenberg uncertainty principle for momentum:

∆p∆x≥h2 ……………….. (1)

03

Momentum.

Knowing that momentum may be stated as p=mv we can derive the following relationship using the equation (1):

∆p∆x≥h2m∆v∆x≥h2

04

Uncertainty in velocity.

The formula for velocity uncertainty is then derived:

∆p∆x≥h2∆v≥h2m∆x

05

Velocity of ant.

We can calculate the ant's velocity by solving for∆v:

∆v≥h2m∆x∆v≥1.0545×10-34J·s2×5×10-7kg×7×10-7m∆v≥1.5×10-22m/s

The ant moves at a speed of ∆v≥1.5×10-22m/s.

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