Physics For Scientists And Engineers With Modern Physics, 9th Edition, The Ohio State University
Physics For Scientists And Engineers With Modern Physics, 9th Edition, The Ohio State University
9th Edition
ISBN: 9781305372337
Author: Raymond A. Serway | John W. Jewett
Publisher: Cengage Learning
Question
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Chapter 41, Problem 62CP

(a)

To determine

Plot of wave function as a function of x.

(a)

Expert Solution
Check Mark

Answer to Problem 62CP

The wave function was plotted as a function of x.

Explanation of Solution

Write the given wave function of the electron.

    ψ(x)={Aeαxforx>0Ae+αxforx<0                                                                                    (I)

Here, ψ(x) is the wave function, A is the constant, α is a constant.

Write the formula to calculate the probability of finding a particle in a certain range.

    P=abψ*ψdx:                                                                                                           (II)

Here, P is the probability, ψ* is the complex conjugate of the wave function, [a,b] is the range in which the probability is found out.

Refer equation (I) and plot the wave function as a function of x.

Figure 1 below shows the plot of wave function.

Physics For Scientists And Engineers With Modern Physics, 9th Edition, The Ohio State University, Chapter 41, Problem 62CP , additional homework tip  1

Conclusion:

The wave function was plotted as a function of x.

(b)

To determine

Plot of probability density as a function of x.

(b)

Expert Solution
Check Mark

Answer to Problem 62CP

The probability density was plotted as a function of x.

Explanation of Solution

Write the given wave function of the electron.

    ψ(x)={Aeαxforx>0Ae+αxforx<0                                                                                    (I)

Here, ψ(x) is the wave function, A is the constant, α is a constant.

Write the formula to calculate the probability of finding a particle in a certain range.

    P=abψ*ψdx:                                                                                                          (II)

Here, P is the probability, ψ* is the complex conjugate of the wave function, [a,b] is the range in which the probability is found out.

Refer equation (II) and plot the probability density as a function of x.

Figure 2 below shows the plot of wave function.

Physics For Scientists And Engineers With Modern Physics, 9th Edition, The Ohio State University, Chapter 41, Problem 62CP , additional homework tip  2

Conclusion:

The probability density was plotted as a function of x.

(c)

To determine

To show that ψ(x) is physically reasonable wave function.

(c)

Expert Solution
Check Mark

Answer to Problem 62CP

The ψ(x) satisfy all the conditions to be a reasonable wave function. Thus ψ(x) is a physically reasonable wave function.

Explanation of Solution

Write the given wave function of the electron.

    ψ(x)={Aeαxforx>0Ae+αxforx<0                                                                                     (I)

Here, ψ(x) is the wave function, A is the constant, α is a constant.

For the wave function to be a reasonable wave function, there are set of condition.

The ψ(x) has to be continuous to be a reasonable wave function. The given wave function is continuous everywhere except at infinity.

As x± the ψ(x) must go to zero to be a reasonable wave function. The given wave function satisfy this condition.

The ψ(x) can also be normalized which is an essential requirement to be a wave function.

The ψ(x) satisfy all the conditions to be a reasonable wave function. Thus ψ(x) is a physically reasonable wave function.

Conclusion:

The ψ(x) satisfy all the conditions to be a reasonable wave function. Thus ψ(x) is a physically reasonable wave function.

(d)

To determine

To normalize the wave function.

(d)

Expert Solution
Check Mark

Answer to Problem 62CP

The normalization constant of the given wave function is α.

Explanation of Solution

Write the given wave function of the electron.

    ψ(x)={Aeαxforx>0Ae+αxforx<0                                                                                     (I)

Here, ψ(x) is the wave function, A is the constant, α is a constant.

Write the condition for normalized wave function.

    |ψ|2dx=1

The wave function is symmetric. Thus re-write the above condition.

    20|ψ|2dx=1

Substitute equation (I) in the above equation.

    2A20e2αxdx=1(2A22α)(ee0)=12A22α=1A=α

Conclusion:

The normalization constant of the given wave function is α.

(e)

To determine

The probability of finding the electron in the range 12αx12α.

(e)

Expert Solution
Check Mark

Answer to Problem 62CP

The probability of finding the particle in the range 12αx12α is 0.632.

Explanation of Solution

Refer section (d) and write the given normalized wave function of the electron.

    ψ(x)={αeαxforx>0αe+αxforx<0                                                                                  (II)

Here, ψ(x) is the wave function, and α is a constant.

Write the formula to calculate the probability of finding a particle in a certain range.

    P=ab|ψ|2dx:                                                                                                      (III)

Here, P is the probability, and [a,b] is the range in which the probability is found out.

Refer equation (II) in equation (III) to determine probability in range 12αx12α.

    P=(a)21/2α1/2αe2αxdx=2(a)2x=01/2αe2αxdx=(2α2α)(e2α/2α1)=(1e1)=0.632

Conclusion:

The probability of finding the particle in the range 12αx12α is 0.632.

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From number 2 and 3 I just want to show all problems step by step please do not short cut look for formula
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Chapter 41 Solutions

Physics For Scientists And Engineers With Modern Physics, 9th Edition, The Ohio State University

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