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Chapter 40: Q. 45 (page 1177)

A proton鈥檚 energy is $1.0 M e V$ below the top of a $10 f m$ -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 is $1.22 %$ .

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:

$畏 = \frac{魔}{\sqrt{2 m (U_{0} 鈭� E)}}$

but we given that the proton's energy is $1.0 M e V$ below the top of the well, which represents the value of $(U_{0} 鈭� E)$ . Then, $(U_{0} 鈭� E = 1 MeV)$ .

Since:

$\begin{matrix} 畏 = \frac{1.05 脳 10^{鈭� 34} J 鈰� s}{\sqrt{2 (1.67 脳 10^{鈭� 27} kg) (1 脳 10^{6} eV 脳 1.6 脳 10^{鈭� 19} J / eV)}} \\ 畏 = 4.54 脳 10^{鈭� 15} m = 4.54 fm \end{matrix}$

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

$P_{t u n n e l} = e^{鈭� 2 w / 畏}$

There, $(w)$ the width of the barrier.

Therefore:

$P_{t u n n e l} = \exp (鈭� \frac{20 fm}{4.54 fm}) = 0.0122 = 1.22 %$

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Most popular questions from this chapter

A particle confined in a rigid one-dimensional box of length $10 fm$ has an energy level $E_{n} = 32.9 MeV$ and an adjacent energy level $E_{n + 1} = 51.4 MeV$ .

a. Determine the values of n and n + 1.

b. Draw an energy-level diagram showing all energy levels from 1 through n + 1. Label each level and write the energy beside it.

c. Sketch the n + 1 wave function on the n + 1 energy level.

d. What is the wavelength of a photon emitted in the $n + 1 鈫� n$ transition? Compare this to a typical visible-light wavelength.

e. What is the mass of the particle? Can you identify it?

The graph in FIGURE EX40.16 shows the potential-energy function U(x of a particle. Solution of the Schr枚dinger equation finds that the n=3 level has $E_{3} = 0.5 e V$ and that the n=6 level has $E_{6} = 2.0 e V$ .

a. Redraw this figure and add to it the energy lines for the n=3 and n=6 states.

b. Sketch the n=3 and n=6 wave functions. Show them as oscillating about the appropriate energy line.

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 ${|蠄 (x)|}^{2}$ . Label the points $x = 0$ and $x = L$ .

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 ${|蠄 (x)|}^{2}$ graphs, if the probability of finding the particle in the left one-third of the box is less than, equal to, or greater than $\frac{1}{3}$ . Explain your reasoning.

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

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

An electron confined in a harmonic potential well emits a 1200nm photon as it undergoes a $3 鈫� 2$ quantum jump. What is the spring constant of the potential well?

See all solutions

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