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Chapter 40: Q. 45 (page 1177)

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?

Short Answer

Expert verified

The probability that the proton will tunnel through the barrier is1.22%.

Step by step solution

01

Given Information 

We need to find the probability that the proton will tunnel through the barrier.

02

Simplify

Finding the penetration distance of the proton into the forbidden region:

η=ħ2mU0−E

but we given that the proton's energy is 1.0MeVbelow the top of the well, which represents the value of U0−E. Then, U0−E=1MeV.

Since:

η=1.05×10−34J⋅s21.67×10−27kg1×106eV×1.6×10−19J/eVη=4.54×10−15m=4.54fm

the probability that the proton will tunnel through the barrier is given as:

Ptunnel=e−2w/η

There, wthe width of the barrier.

Therefore:

Ptunnel=exp−20fm4.54fm=0.0122=1.22%

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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?

Sketch the n=4 wave function for the potential energy shown in FIGURE EX40.13.

An electron in a finite potential well has a 1.0nm penetration distance into the classically forbidden region. How far below U0 is the electron’s energy?

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

An electron in a rigid box absorbs light. The longest wavelength in the absorption spectrum is600nm. How long is the box?
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