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Chapter 40: Q. 5 (page 1175)

FIGURE EX 40.5is the probability density for an electron in a rigid box. What is the electron’s energy, in eV?

Short Answer

Expert verified

The electron’s energy, in eV is16.9eV.

Step by step solution

01

Given Information

We need to find the electron’s energy, ineV.

02

Simplify

The given graph of the probability density has three antinodes, which means that the wave function also has three antinodes. And we know that the graph of ψnxhas nantinodes .). So, the wave function shown in the graph corresponds to the n=3state. But the energy of an electron in a rigid box is given by:

En=n2h28mL2

Notice that L is given to be0.45nm from the graph, which is the maximum value of x

substitute the numerical values for the quantities in the equation above to find E3

E3=9×6.626×10−34J⋅s28×9.11×10−31kg×0.45×10−9m2=2.7×10−18J

or in electronvolt

E3=2.7×10−18J×1eV1.6×10−19JE3=16.9eV

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Most popular questions from this chapter

Showed that a typical nuclear radius is 4fm As you’ll learn in Chapter 42, a typical energy of a neutron bound inside the nuclear potential well isEn=-20MeV. To find out how “fuzzy” the edge of the nucleus is, what is the neutron’s penetration distance into the classically forbidden region as a fraction of the nuclear radius?

For the quantum-well laser of Figure 40.16, estimate the probability that an electron will be found within one of the GaAlAs layers rather than in the GA As layer. Explain your reasoning.

A neutron is confined in a 10fm-diameter nucleus. If the nucleus is modeled as a one-dimensional rigid box, what is the probability that a neutron in the ground state is less than 2.0fm from the edge of the nucleus?

In most metals, the atomic ions form a regular arrangement called a crystal lattice. The conduction electrons in the sea of electrons move through this lattice. FIGURE CP40.47is a one-dimensional model of a crystal lattice. The ions have mass m, charge eand an equilibrium separation b.

a. Suppose the middle charge is displaced a very small distance x≪bfrom its equilibrium position while the outer charges remain fixed. Show that the net electric force on the middle charge is given approximately by

F=−e2b3πε0x

In other words, the charge experiences a linear restoring force.

b. Suppose this crystal consists of aluminum ions with an equilibrium spacing of 0.30nm. What are the energies of the four lowest vibrational states of these ions?

c. What wavelength photons are emitted during quantum jumps between adjacent energy levels? Is this wavelength in the infrared, visible, or ultraviolet portion of the spectrum?

For a particle in a finite potential well of width L and depthU0 , what is the ratio of the probability Prob ( in δxat x=L+η) to the probability Prob ( in δx atx=L)?
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