Chapter 7: Q47E (page 282)

Question: Show that the angular normalization constant in Table 7.3 for the case $(l, m_{l}) = (1, 0)$ is correct.

Short Answer

Expert verified

Answer

It has been proved that the normalization for the case $(l, m_{l}) = (1, 0)$ is correct.

Step by step solution

Given data

The wave function corresponding to $(l, m_{l}) = (1, 0)$ is,

$Θ_{l, m_{l}} (θ) Φ_{m_{l}} (ϕ) = \sqrt{\frac{3}{4 π}} \cos θ$

Normalization

$TheangularpartoftheHydrogenatomwavefunctionshouldsatisfytheflowingcondition, \int_{0}^{π} {\{Θ_{l, m_{l}} (θ)\}}^{2} 2 π s i n θ d θ = 1 . . . . . (I)$

Determining whether the given normalization constant is correct

In the given wave function,

$Θ_{l, m_{l}} (θ) = \sqrt{\frac{3}{4 π}} \cos θ$

Check equation (I) as,

$= \frac{3}{4 π} \times 2 π \int_{0}^{π} \cos^{2} θ \sin θ d θ = \frac{3}{2} \int_{0}^{π} \cos^{2} θ \sin θ d θ$

Let us assume,

$\cos θ = z - \sin θ d θ = d z$

Then the integral becomes,

$= \frac{3}{2} \int_{1}^{- 1} z^{2} (- d z) = \frac{3}{2} \int_{- 1}^{1} z^{2} d z = \frac{3}{2} {[\frac{z^{3}}{3}]}_{- 1}^{1} = \frac{3}{2} \times \frac{2}{3} = 1$

Thus the normalization is correct.

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Question: Show that the angular normalization constant in Table 7.3 for the case $(l, m_{l}) = (1, 0)$ is correct.

Short Answer

Step by step solution

Given data

Normalization

Determining whether the given normalization constant is correct

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Question: Show that the angular normalization constant in Table 7.3 for the case (l,ml)=(1,0) is correct.

Short Answer

Step by step solution

Given data

Normalization

Determining whether the given normalization constant is correct

One App. One Place for Learning.

Most popular questions from this chapter

Recommended explanations on Physics Textbooks

Electricity and Magnetism

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Question: Show that the angular normalization constant in Table 7.3 for the case $(l, m_{l}) = (1, 0)$ is correct.