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Chapter 40: Q. 9 (page 1174)

An electron has a 0.0100 probability (a 1.00% chance) of tunneling through a potential barrier. If the width of the barrier is doubled, will the tunneling probability be0.0050,0.0025or0.0001? Explain.

Short Answer

Expert verified

The width of the barrier is doubled , will the tunneling probability be0.0050,0.0025or0.0001isPtunnel,2=0.0001.

Step by step solution

01

Given Information

We have to given that an electron has a 0.0100, the width of the barrier is doubled, will the tunneling probability be 0.0050,0.0025or0.0001.

02

Simplify

Ptunnel,2=The probability that the electron is the tunnel through the barrier is given as the equation:

Ptunnel=e-2ωη

and we are given that the initial probability is 0.01, so let the probability after the width of the barrier is doubled be Ptunnel,2since:

ω1is the initial width after doubling. Notice that eab=eab. Then:

role="math" localid="1650132194389" Ptunnel,2=exp-4ω1η=exp-2ω1η2=Ptunnel,12Ptunnel,2=0.0001

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Most popular questions from this chapter

Model an atom as an electron in a rigid box of length 0.100nm, roughly twice the Bohr radius.

a. What are the four lowest energy levels of the electron?

b. Calculate all the wavelengths that would be seen in the emission spectrum of this atom due to quantum jumps between these four energy levels. Give each wavelength a label λn→mto indicate the transition.

c. Are these wavelengths in the infrared, visible, or ultraviolet portion of the spectrum?

d. The stationary states of the Bohr hydrogen atom have negative energies. The stationary states of this model of the atom have positive energies. Is this a physically significant difference? Explain.

e. Compare this model of an atom to the Bohr hydrogen atom. In what ways are the two models similar? Other than the signs of the energy levels, in what ways are they different?

A proton’s energy is 1.0MeV below the top of a 10fm-wide energy barrier. What is the probability that the proton will tunnel through the barrier?

The correspondence principle says that the average behavior of a quantum system should begin to look like the Newtonian solution in the limit that the quantum number becomes very large. What is meant by “the average behavior” of a quantum system?

Consider a particle in a rigid box of length L. For each of the states n=1,n=2,and n=3:

a. Sketch graphs of ψ(x)2. Label the points x=0and x=L.

b. Where, in terms of L, are the positions at which the particle is most likely to be found?

c. Where, in terms of L, are the positions at which the particle is least likely to be found?

d. Determine, by examining your ψ(x)2graphs, if the probability of finding the particle in the left one-third of the box is less than, equal to, or greater than 13. Explain your reasoning.

e. Calculate the probability that the particle will be found in the left one-third of the box

What is the quantum number of the particle in FIGURE Q40.4? How can you tell?
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