Chapter 4: Q21P (page 170)

(a)Derive Equation 4.131 from Equation 4.130. Hint: Use a test function; otherwise you're likely to drop some terms.
(b)Derive Equation 4.132 from Equations 4.129 and 4.131 .Hint : Use Equation 4.112.

Short Answer

Expert verified

(a)Equation 4.131 is derived.

(b) Equation 4.132 is derived.

Step by step solution

Representation of some equations

Write equation 4.130.

$L_{\pm} = \pm h e^{\pm iϕ} (\frac{\partial}{\partial θ} \pm i \cot θ \frac{\partial}{\partial ϕ})$

Write equation 4.129.

$L_{z} = \frac{h}{i} \frac{\partial}{\partial ϕ}$

(a) Derivation of equation 4.131

Solve to derive equation 4.131

$L_{+} L_{-} f = - h^{2} e^{iφ} \{(\frac{\partial}{\partial θ} + i \cot θ \frac{\partial}{\partial ϕ}) [e^{- iφ} (\frac{\partial f}{\partial θ} - i \cot θ \frac{\partial f}{\partial ϕ})]\} = - h^{2} e^{iφ} \{\begin{matrix} e^{iφ} (\frac{\partial^{2} f}{\partial θ^{2}} + i \csc^{2} θ \frac{\partial f}{\partial ϕ} - i \cot θ \frac{\partial^{2}}{∆ ϕ \partial θ}) \\ + i \cot θ [- e^{- iφ (\frac{\partial f}{\partial θ}) + e^{- iφ}} (\frac{\partial^{2} f}{\partial θ \partial ϕ} - i \cot θ \frac{\partial^{2} f}{\partial ϕ^{2}})] \end{matrix}\} = - h^{2} e^{iφ} \{\begin{matrix} e^{iφ} (\frac{\partial^{2} f}{\partial θ^{2}} + {ie}^{- iφ} \csc^{2} θ \frac{\partial f}{\partial ϕ} - {ie}^{- iφ} \cot θ \frac{\partial^{2}}{\partial \partial \partial ϕ} + e^{iφ} cotθ \frac{\partial f}{\partial θ} - {ie}^{- iφ} \cot^{2} θ \frac{\partial f}{\partial ϕ}) \\ + {ie}^{- iφ} \cot θ \frac{\partial^{2} f}{\partial ϕ \partial θ} + e^{iφ} \cot^{2} θ \frac{\partial^{2} f}{\partial ϕ^{2}} \end{matrix}\} = - h^{2} \{\frac{\partial^{2}}{\partial θ^{2}} + i (\csc^{2} θ - \cot^{2} θ) \frac{\partial}{\partial ϕ} + cotθ \frac{\partial}{\partial θ} + \cot^{2} θ \frac{\partial^{2}}{\partial ϕ^{2}}\} f L_{+} L_{-} = - h^{2} \{\frac{\partial^{2}}{\partial θ^{2}} + i \frac{\partial}{\partial ϕ} + cotθ \frac{\partial}{\partial θ} + \cot^{2} θ \frac{\partial^{2}}{\partial ϕ^{2}}\}$

Thus, equation 4.131 is derived.

(b) Derivation of equation 4.132

Solve to derive equation 4.132.

$L^{2} = - h^{2} \frac{\partial^{2}}{\partial θ^{2}} - i h^{2} \frac{\partial}{\partial ϕ} - h^{2} \cot^{2} θ \frac{\partial^{2}}{\partial ϕ^{2}} - h^{2} \frac{\partial^{2}}{\partial ϕ^{2}} + i h^{2} \frac{\partial}{\partial ϕ} = - h^{2} (\frac{\partial^{2}}{\partial θ^{2}} + \cot θ \frac{\partial}{\partial θ} + \frac{1}{\sin^{2} θ} \frac{\partial^{2}}{\partial ϕ^{2}}) = - h^{2} (\frac{1}{\sin θ} \frac{\partial}{\partial θ} (\sin θ \frac{\partial}{\partial θ}) + \frac{1}{\sin^{2} θ} \frac{\partial^{2}}{\partial ϕ^{2}}) = - h^{2} (\frac{1}{\sin θ} \frac{\partial}{\partial θ} (\sin θ \frac{\partial}{\partial θ}) + \frac{1}{\sin^{2} θ} \frac{\partial^{2}}{\partial ϕ^{2}})$

Thus, equation 4.132 is derived.

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(a)Derive Equation 4.131 from Equation 4.130. Hint: Use a test function; otherwise you're likely to drop some terms.
(b)Derive Equation 4.132 from Equations 4.129 and 4.131 .Hint : Use Equation 4.112.

Short Answer

Step by step solution

Representation of some equations

(a) Derivation of equation 4.131

(b) Derivation of equation 4.132

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(a)Derive Equation 4.131 from Equation 4.130. Hint: Use a test function; otherwise you're likely to drop some terms.(b)Derive Equation 4.132 from Equations 4.129 and 4.131 .Hint : Use Equation 4.112.

Short Answer

Step by step solution

Representation of some equations

(a) Derivation of equation 4.131

(b) Derivation of equation 4.132

One App. One Place for Learning.

Most popular questions from this chapter

Recommended explanations on Physics Textbooks

Torque and Rotational Motion

Rotational Dynamics

Kinematics Physics

Radiation

Thermodynamics

Mechanics and Materials

Study anywhere. Anytime. Across all devices.

(a)Derive Equation 4.131 from Equation 4.130. Hint: Use a test function; otherwise you're likely to drop some terms.
(b)Derive Equation 4.132 from Equations 4.129 and 4.131 .Hint : Use Equation 4.112.