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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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Most popular questions from this chapter

The proton and electron had been identified by 1920, but the neutron wasn't found until 1932. Meanwhile, the atom was a mystery. Helium, for example, has a mass about four times the proton mass but a charge only twice that of the proton. Of course, we now know that its nucleus is two protons and two neutrons of about the same mass. But before the neutron's discovery, it was suggested that the nucleus contained four protons plus two electrons, accounting for the mass (electrons are "light") and the total charge. Quantum mechanics makes this hypothesis untenable. A confined electron is a standing wave. The fundamental standing wave on a string satisfies L=12λ, and the "length of the string" in the nucleus is its diameter.2Rnuc, so, the electron's wavelength could be no longer than about4RmucAssuming a typical nuclear radius of 4×10-15m determine the kinetic energy of an electron standing wave continued in the nucleus. (Is it moving "slow" or "fast"?) The charge of a typical nucleus is +20e , so the electrostatic potential energy of an electron at its edge would be-(1/2πε0)20e2/Rnuc(it would be slightly lower at the center). To escape. the electron needs enough energy to get far away, where the potential energy is 0. Show that it definitely would escape.

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