Q23P In Problem聽4.3 you showed that ... [FREE SOLUTION]

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Introduction to Quantum Mechanics

David J. Griffiths

Physics

2nd Edition 路 469 pages

ISBN: 9780131118928

Chapter 4: Q23P (page 170)

In Problem4.3 you showed that $Y_{2}^{1} (胃, 蠒) = - \sqrt{15 / 8 蟺} s i n 胃 c o s 胃 e^{i 蠒}$ . Apply the raising operator to find localid="1656065252558" $Y_{2}^{2} (胃, 蠒)$ . Use Equation 4.121to get the normalization.

Short Answer

Expert verified

The value of $Y_{2}^{2} (胃, 蠒) is \frac{1}{4} \sqrt{\frac{15}{2 蟺}} h {(e^{i 蠒} \sin 胃)}^{2}$ .

Step by step solution

Define Raising Operator

An operator that raises or lowers the eigenvalue of another operator (collectively called as ladder operators) is referred to as a raising or lowering operator.

Find the value of Y22(θ,ϕ)

In this step evaluate first the spherical harmonic is $Y_{2}^{1} (胃, 蠒)$ .

From problem 4.3, the value of $Y_{2}^{1}$

$Y_{2}^{1} = - \sqrt{\frac{15}{8 蟿蟿}} \sin 胃 \cos 胃 e^{i o} L_{+} = h e^{i 蠒} [\frac{鈭�}{鈭� 胃} + i c o t 胃 \frac{鈭�}{鈭� 蠒}] L_{+} Y_{2}^{1} = Y_{2}^{2}$

Now, determine the spherical harmonic $Y_{2}^{2} (胃, 蠒)$ as follows,

localid="1658464225727" $Y_{2}^{1} = - \sqrt{\frac{15}{8 蟺}} h e^{i 蠒} [e^{i 蠒} \frac{鈭�) (\sin 胃 \cos 胃)}{鈭� 胃} + i \cot 胃 \sin 胃 \cos 胃 \frac{鈭� e^{i 蠒}}{鈭� 蠒}] = - \sqrt{\frac{15}{8 蟺}} h e^{i 蠒} [e^{i 蠒} (\cos^{2} 胃 - \sin^{2} 胃) + i \cot 胃 \sin 胃 \cos 胃 (i e^{i 蠒})] = - \sqrt{\frac{15}{8 蟺}} h [e^{i 蠒} \cos 2 胃 - \cos^{2} 胃 e^{i 蠒}] = \sqrt{\frac{15}{2 蟺}} h e^{2 i 蠒} \sin^{2} 胃 = \frac{1}{2} \sqrt{\frac{15}{2 蟺}} h {(e^{2 i 蠒} \sin 胃)}^{2}$

Thus, the value of $Y_{2}^{2} (胃, 蠒)$ islocalid="1658464244244" $\frac{1}{2} \sqrt{\frac{15}{2 蟺}} h {(e^{i 蠒} \sin^{2} 胃)}^{2}$ .

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91影视

In Problem4.3 you showed that $Y_{2}^{1} (胃, 蠒) = - \sqrt{15 / 8 蟺} s i n 胃 c o s 胃 e^{i 蠒}$ . Apply the raising operator to find localid="1656065252558" $Y_{2}^{2} (胃, 蠒)$ . Use Equation 4.121to get the normalization.

Short Answer

Step by step solution

Define Raising Operator

Find the value of Y22(θ,ϕ)

One App. One Place for Learning.

Most popular questions from this chapter

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91影视

In Problem4.3 you showed that Y21(胃,蠒)=-15/8蟺sin胃cos胃ei蠒. Apply the raising operator to find localid="1656065252558" Y22(胃,蠒). Use Equation 4.121to get the normalization.

Short Answer

Step by step solution

Define Raising Operator

Find the value of Y22(θ,ϕ)

One App. One Place for Learning.

Most popular questions from this chapter

Recommended explanations on Physics Textbooks

Radiation

Famous Physicists

Particle Model of Matter

Atoms and Radioactivity

Quantum Physics

Fields in Physics

Study anywhere. Anytime. Across all devices.

In Problem4.3 you showed that $Y_{2}^{1} (胃, 蠒) = - \sqrt{15 / 8 蟺} s i n 胃 c o s 胃 e^{i 蠒}$ . Apply the raising operator to find localid="1656065252558" $Y_{2}^{2} (胃, 蠒)$ . Use Equation 4.121to get the normalization.