Book edition 10th Edition

Author(s) David Halliday

Pages 1328 pages

ISBN 9781118230718

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Short Answer

The wave function for the hydrogen-atom quantum state represented by the dot plot shown in Fig. 39-21, which has n = 2 and $l = m_{l} = 0$ , is
$Ψ_{200} (r) = \frac{1}{4 \sqrt{2 π}} a^{- 3 / 2} (2 - \frac{r}{a}) e^{- r / 2 a}$
in which a is the Bohr radius and the subscript on $Ψ (r)$ gives the values of the quantum numbers $n, l, m_{l}$ . (a) Plot $Ψ {}_{200}^{2}(r)$ and show that your plot is consistent with the dot plot of Fig. 39-21. (b) Show analytically that $Ψ {}_{200}^{2}(r)$ has a maximum at $r = 4 a$ . (c) Find the radial probability density $P_{200} (r)$ for this state. (d) Show that
$\int_{0}^{\infty} P_{200} (r) d r = 1$
and thus that the expression above for the wave function $Ψ_{200} (r)$ has been properly normalized.

The plot is shown in the below figure.

b. It is proved that $Ψ_{200}^{2} (r)$ has a maximum at r = 4a.

c. The radial probability is $P_{200} (r) = \frac{r^{2}}{8 a^{3}} (2 - \frac{r}{a}) e^{- r / a}$ .

d. It is proved that $\int_{0}^{\infty} P_{200} (r) d r = 1$ .

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Step by Step Solution

TABLE OF CONTENTS :

TABLE OF CONTENTS

Step 1: Describe the given information

The wave function is given by,

$Ψ_{200} (r) = \frac{1}{4 \sqrt{2} π} a^{- 3 / 2} (2 - \frac{r}{a}) e^{- r / 2 a} . . . . . . . . . . (1)$

Step 2: Plot Ψ2002(r) and show that the plot is consistent with the dot plot of the given figure(a)

Take the values of r on the horizontal axis, but actually, the values of r/a must be taken on the horizontal axis. Here, a is the Bohr radius, and it is a constant value.

The plot is shown in the below figure.

From the above figure, it can be observed that there is a high central peak between r = 0 and r = 2A. At r = 2a, the value of the wave function $Ψ_{200} (r)$ must be equal to zero . Also, the low peak value is reached its maximum value at r = 4a. The graph is not consistent with the dot plot of the given figure.

Step 3: Show that Ψ2002(r) has a maximum at r = 4a(b)

Differentiate the given wave function and equate to zero to find the maximum.

$\frac{\partial}{\partial r} [Ψ_{200} (r)] = 0 \frac{\partial}{\partial r} [\frac{1}{4 \sqrt{2 π}} a^{- 3 / 2} (2 - \frac{r}{a}) e^{- r / 2 a}] = 0 \frac{\partial}{\partial r} (2 - \frac{r}{a}) e^{- r / 2 a} = 0 (2 - \frac{r}{a}) e^{- r / 2 a} (- \frac{1}{2 a}) + (- \frac{r}{a}) e^{- r / 2 a} = 0$

Simplify further.

$(2 - \frac{r}{a}) (- \frac{1}{2}) - 1 = 0 - 1 + \frac{r}{2 a} - 1 = 0 \frac{r}{2 a} = 2 r = 4 a$

Therefore, it is proved that $Ψ_{200}^{2} (r)$ has a maximum at r = 4a.

Step 4: Find the radial probability density P200(r) for this state(c)

The expression for the radial probability is as follows.

$P_{200} (r) = 4 {πr}^{2} Ψ_{200}^{2} (r) = \frac{r^{2}}{8 a^{3}} (2 - \frac{r}{a}) e^{- r / a}$

Therefore, the radial probability for the given state is $P_{200} (r) = \frac{r^{2}}{8 a^{3}} (2 - \frac{r}{a}) e^{- r / a}$ .

Step 5: Show that ∫0∞P200(r)dr=1 (d)

Show that the value of $\int_{0}^{\infty} P_{200} (r) d r = 1$ as follows.

$\int_{0}^{\infty} P_{200} (r) d r = \int_{0}^{\infty} [\frac{r^{2}}{8 a^{3}} {(2 - \frac{r}{a})}^{2} e^{- r / a}] d r = \frac{1}{8} \int_{0}^{\infty} x^{2} {(2 - x)}^{2} e^{- x} d x = \frac{1}{8} \int_{0}^{\infty} [x^{4} - 4 x^{3} + 4 x^{2}] e^{- x} d x = \frac{1}{8} [4! - 4 (3!) + 4 (2!)]$

Simplify further.

$\int_{0}^{\infty} P_{200} (r) d r = \frac{1}{8} [24 - 24 + 8] = 1$

Therefore, it is proved that $\int_{0}^{\infty} P_{200} (r) d r = 1$ .

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Fundamentals Of Physics

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Short Answer

Step by Step Solution

Step 1: Describe the given information

Step 2: Plot Ψ2002(r) and show that the plot is consistent with the dot plot of the given figure(a)

Step 3: Show that Ψ2002(r) has a maximum at r = 4a(b)

Step 4: Find the radial probability density P200(r) for this state(c)

Step 5: Show that ∫0∞P200(r)dr=1 (d)

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Fundamentals Of Physics

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Short Answer

Step by Step Solution

Step 1: Describe the given information

Step 2: Plot Ψ2002(r) and show that the plot is consistent with the dot plot of the given figure(a)

Step 3: Show that Ψ2002(r) has a maximum at r = 4a(b)

Step 4: Find the radial probability density P200(r) for this state(c)

Step 5: Show that ∫0∞P200(r)dr=1 (d)

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