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Chapter 40: Q. 37 (page 1176)

A typical electron in a piece of metallic sodium has energy-E0compared to a free electron, where E0is the2.7eV work function of sodium.

a. At what distance beyond the surface of the metal is the electron’s probability density 10%of its value at the surface?

b. How does this distance compare to the size of an atom?

Short Answer

Expert verified

a.1.4A0

Step by step solution

01

Part (a) Step 1: Given information

We have given,

E0=2.7eV

P=10%

we have to find the distance.

02

Simplify

We know that,

Ptunnel=e-2w/η

and,

η=h2π2m(U0-E)η=(6.62×10-34J.s)2π2×9.1×10-31kg(2.7×1.6×10-19J)η=0.12nm

Put this value in probability equation

P=e-2w/(0.12)=0.1-2w0.12=-2.3w=0.14nmw=1.4A0

03

Part (b) Step 1: Given information

We have to find its distance compare with the atom size.

04

Simplify 

Since the found distance is comparable to the size of the atom because the most atoms have to size inA0.

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Most popular questions from this chapter

Show that the penetration distance ηhas units of m.

a. What is the probability that an electron will tunnel through a0.50nmair gap from a metal to a STM probe if the work function is 4.0eV?

b. The probe passes over an atom that is0.050nm“tall.” By what factor does the tunneling current increase?

c. If a 10%current change is reliably detectable, what is the smallest height change the STM can detect?

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

Even the smoothest mirror finishes are “rough” when viewed at a scale of 100nm. When two very smooth metals are placed in contact with each other, the actual distance between the surfaces varies from0nmat a few points of real contact to ≈100nm. The average distance between the surfaces is≈50nm. The work function of aluminum is 4.3eV. What is the probability that an electron will tunnel between two pieces of aluminum that are 50nmapart? Give your answer as a power of10rather than a power ofe.

Figure 40.27a modeled a hydrogen atom as a finite potential well with rectangular edges. A more realistic model of a hydrogen atom, although still a one-dimensional model, would be the electron + proton electrostatic potential energy in one dimension:

U(x)=-e24πε0x

a. Draw a graph of U(x) versus x. Center your graph at x=0.

b. Despite the divergence at x=0, the Schrödinger equation can be solved to find energy levels and wave functions for the electron in this potential. Draw a horizontal line across your graph of part a about one-third of the way from the bottom to the top. Label this line E2, then, on this line, sketch a plausible graph of the n=2wave function.

c. Redraw your graph of part a and add a horizontal line about two-thirds of the way from the bottom to the top. Label this line E3, then, on this line, sketch a plausible graph of the n=3 wave function.
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