(a)
The radial probability density for the
(a)
Answer to Problem 81AP
The radial probability density for the
Explanation of Solution
Write the expression for the wave function of electron in the
Here,
Write the expression for the probability density for
Here,
Write the complex conjugate of
Conclusion:
Use expressions (III), and (I) in expression (II) to find
Therefore, the radial probability density for the
(b)
The derivative of the radial probability density with respect to
(b)
Answer to Problem 81AP
The derivative of the radial probability density with respect to
Explanation of Solution
Write the expression for
Take the derivative of expression (IV) with respect to
Simplify expression (V).
Conclusion:
Therefore, the derivative of the radial probability density with respect to
(c)
Three values of
(c)
Answer to Problem 81AP
The three values possible for
Explanation of Solution
Minimum points of the function can be found by equating the derivative to zero.
The derivative equals to zero when any of the product terms equals to zero. Thus
First term of expression
Equate the second term to zero.
Solve expression (VII) to find
Equate third of expression (VI) to zero.
Expression (VII) is true if and only if
Conclusion:
Therefore, the three values possible for
(d)
The two values of
(d)
Answer to Problem 81AP
The two values of
Explanation of Solution
The maximum points of the function can be found by equating the quadratic equation of the derivative to zero.
Write the general expression for a quadratic expression.
Write the expression to find the solution of expression (X).
Conclusion:
Compare expressions (IX) and (X).
Substitute values of
Therefore, the two values of
(e)
The value of
(e)
Answer to Problem 81AP
The value of
Explanation of Solution
Write the expression for probability as found in subpart (a).
Conclusion:
Substitute
Substitute
Thus the value of probability is greater when
Therefore, the value of
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Chapter 42 Solutions
Physics For Scientists And Engineers With Modern Physics, 9th Edition, The Ohio State University
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