Chapter 1: Q48E (page 1)

Question: Show that the normalization constant $\sqrt{15 / 32 π}$ given in Table 7.3 for the angular parts of the $l = 2, m_{l} = \pm 2$ wave function is correct.

Short Answer

Expert verified

Answer

It has been proved that the normalization for the case $l = 2, m_{l} = \pm 2$ is correct.

Step by step solution

Given data

$Thewavefunctioncorrespondingto (l, m_{l}) = (2, \pm 2) is, Θ_{l, m_{l}} (θ) Φ_{m_{l}} (ϕ) = \sqrt{\frac{15}{32 π}} \sin^{2} θ e^{\pm 2 i ϕ}$

Normalization

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

Determining whether the given normalization constant is correct

In the given wave function,

$Θ_{l, m_{l}} (θ) = \sqrt{\frac{15}{32 π}} \sin^{2} θ$

Check equation (I) as,

$= \frac{15}{32 π} \times 2 π \int_{0}^{π} \sin^{5} θ d θ = \frac{15}{16} \int_{0}^{π} \sin^{4} θ \sin θ d θ = \frac{15}{16} \int_{0}^{π} {(\sin^{2} θ)}^{2} \sin θ d θ = \frac{15}{16} \int_{0}^{π} {(1 - \cos^{2} θ)}^{2} \sin θ d θ$

Let us assume,

$\cos θ = z - \sin θ d θ = d z$

Then the integral becomes,

$= \frac{15}{16} \int_{1}^{- 1} {(1 - z^{2})}^{2} (- d z) = \frac{15}{16} \int_{- 1}^{1} (1 + z^{4} - 2 z^{2}) d z = \frac{15}{16} {[z + \frac{z^{5}}{5} - 2 \frac{z^{3}}{3}]}_{- 1}^{1} = \frac{15}{16} (2 + \frac{2}{5} - \frac{4}{3}) = 1$

Thus the normalization is correct.

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Question: Show that the normalization constant $\sqrt{15 / 32 π}$ given in Table 7.3 for the angular parts of the $l = 2, m_{l} = \pm 2$ wave function 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.

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Question: Show that the normalization constant 15/32π given in Table 7.3 for the angular parts of the l=2,ml=±2 wave function 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

Linear Momentum

Geometrical and Physical Optics

Oscillations

Fluids

Solid State Physics

Electricity and Magnetism

Study anywhere. Anytime. Across all devices.

Question: Show that the normalization constant $\sqrt{15 / 32 π}$ given in Table 7.3 for the angular parts of the $l = 2, m_{l} = \pm 2$ wave function is correct.