Q42E In Appendix G. the operator for ... [FREE SOLUTION]

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Chapter 7: Q42E (page 281)

In Appendix G. the operator for the square of the angular momentum is shown to be
${\hat{L}}^{2} = - h^{2} [c s c 胃 \frac{鈭�}{鈭� 胃} (s i n 胃 \frac{鈭�}{鈭� 胃}) + c s c^{2} 胃 \frac{{鈭�}^{2}}{鈭� 蠒^{2}}]$
Use this to rewrite equation (7-19) as ${\hat{L}}^{2} 鈿� 桅 = - C h^{2} 鈿� 桅$

Short Answer

Expert verified

${\hat{L}}^{2} 胃桅 = - C h^{2} 胃桅$

Step by step solution

Given data

The angular momentum operator is given as:

${\hat{L}}^{2} = - h^{2} [c s c 胃 \frac{鈭�}{鈭� 胃} (s i n 胃 \frac{鈭�}{鈭� 胃}) + c s c^{2} 胃 \frac{{鈭�}^{2}}{鈭� 蠒^{2}}]$

Where, $h$ is the reduced Planck鈥檚 constant.

Calculation

The separation of the variable of the above equation can be written as:

$(c s c 胃 \frac{鈭�}{鈭� 胃} (s i n 胃 \frac{鈭�}{鈭� 胃}) + c s c^{2} 胃 \frac{{鈭�}^{2}}{鈭� 蠒^{2}}) 螛桅 = C 螛桅 (c s c 胃 \frac{鈭�}{鈭� 胃} (s i n 胃 \frac{鈭� 螛桅}{鈭� 胃}) + c s c^{2} 胃 \frac{{鈭�}^{2} 螛桅}{鈭� 蠒^{2}}) = C 螛桅 \frac{1}{螛} c s c 胃 \frac{鈭�}{鈭� 胃} (s i n 胃 \frac{鈭� 螛}{鈭� 胃}) + c s c^{2} 胃 \frac{1}{桅} \frac{{鈭�}^{2} 桅}{鈭� 蠒^{2}} = C$

Now we can write the operator as:

${\hat{L}}^{2} = - h^{2} [c s c 胃 \frac{鈭�}{鈭� 胃} (\sin 胃 \frac{鈭�}{鈭� 胃}) c s c^{2} 胃 \frac{{鈭�}^{2}}{鈭� 蠒^{2}}] {\hat{L}}^{2} 鈯� 桅 = - h^{2} [c s c 胃 \frac{鈭�}{鈭� 胃} (\sin 胃 \frac{鈭�}{鈭� 胃}) c s c^{2} 胃 \frac{{鈭�}^{2}}{鈭� 蠒^{2}}] 鈯� 桅 = - h^{2} 桅 c s c 胃 \frac{鈭�}{鈭� 胃} (\sin 胃 \frac{鈭�}{鈭� 胃}) - h^{2} 桅 c s c^{2} 胃 \frac{{鈭�}^{2} 桅}{鈭� 蠒^{2}} = - h^{2} 鈯� 桅 \frac{1}{鈯�} c s c 胃 \frac{鈭�}{鈭� 胃} (\sin 胃 \frac{鈭� 桅}{鈭� 胃}) c s c^{2} 胃 \frac{1}{桅} \frac{{鈭�}^{2} 桅}{鈭� 蠒^{2}} {\hat{L}}^{2} 鈯� 桅 = - C h^{2} 鈯� 桅$

Conclusion

Thus, it can be written as ${\hat{L}}^{2} 鈯� 桅 = - C h^{2} 鈯� 桅$ .

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91影视

In Appendix G. the operator for the square of the angular momentum is shown to be L^2=-h2[csc胃鈭�鈭�胃sin胃鈭�鈭�胃+csc2胃鈭�2鈭�蠒2]Use this to rewrite equation (7-19) asL^2鈿�桅=-Ch2鈿�桅

Short Answer

Step by step solution

Given data

Calculation

Conclusion

One App. One Place for Learning.

Most popular questions from this chapter

Recommended explanations on Physics Textbooks

Work Energy and Power

Famous Physicists

Linear Momentum

Circular Motion and Gravitation

Further Mechanics and Thermal Physics

Engineering Physics

Study anywhere. Anytime. Across all devices.