Chapter 7: Q49E (page 282)

Question: Verify the correctness of the normalization constant of the radial wave function given in Table 7.4 as
$\frac{1}{{(2 a_{0})}^{3 / 2} \sqrt{3} a_{0}}$

Short Answer

Expert verified

Answer

It has been proved that the normalization for the case 2pstateis correct.

Step by step solution

Given data

The radial wave function corresponding to the state is,

$R_{n / l} (r) = \frac{r e^{- \frac{r}{2 a_{0}}}}{{(2 a_{0})}^{3 / 2} \sqrt{3} a_{0}}$

Normalization condition

The radial part of the Hydrogen atom wave function should satisfy the condition as,

$\int_{0}^{\infty} {R_{n / l} (r)}^{2} r^{2} d r = 1$

Determining whether the given normalization constant is correct

Check equation (I) with the given wave function as:

$= \frac{1}{{(2 a_{0})}^{3} 3 a_{0}} \int_{0}^{\infty} r^{4} e^{- \frac{r}{a_{0}}} d r = \frac{1}{24 a_{0}^{3}} \int_{0}^{\infty} r^{4} e^{- \frac{r}{a_{0}}} d r$

Let us assume,

$r / a_{0} = z r = a_{0} z d r = a_{0} d z$

Then the integral becomes

$= \frac{1}{24 a_{0}^{5}} a_{0}^{5} \int_{0}^{\infty} z^{4} e^{- z} d z = \frac{Γ (5)}{24} = \frac{4!}{24} = \frac{24}{24} = 1$

Here $Γ (5)$ is the gamma function.

Thus the normalization is correct.

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Question: Verify the correctness of the normalization constant of the radial wave function given in Table 7.4 as
$\frac{1}{{(2 a_{0})}^{3 / 2} \sqrt{3} a_{0}}$

Short Answer

Step by step solution

Given data

Normalization condition

Determining whether the given normalization constant is correct

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Short Answer

Step by step solution

Given data

Normalization condition

Determining whether the given normalization constant is correct

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Question: Verify the correctness of the normalization constant of the radial wave function given in Table 7.4 as
$\frac{1}{{(2 a_{0})}^{3 / 2} \sqrt{3} a_{0}}$