Chapter 4: Q23P (page 170)

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

Short Answer

Expert verified

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

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 $Y_{2}^{1} (θ, ϕ)$ .

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

$Y_{2}^{1} = - \sqrt{\frac{15}{8 τ τ}} \sin θ \cos θ e^{i o} L_{+} = h e^{i ϕ} [\frac{\partial}{\partial θ} + i c o t θ \frac{\partial}{\partial ϕ}] L_{+} Y_{2}^{1} = Y_{2}^{2}$

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

localid="1658464225727" $Y_{2}^{1} = - \sqrt{\frac{15}{8 π}} h e^{i ϕ} [e^{i ϕ} \frac{\partial) (\sin θ \cos θ)}{\partial θ} + i \cot θ \sin θ \cos θ \frac{\partial e^{i ϕ}}{\partial ϕ}] = - \sqrt{\frac{15}{8 π}} h e^{i ϕ} [e^{i ϕ} (\cos^{2} θ - \sin^{2} θ) + i \cot θ \sin θ \cos θ (i e^{i ϕ})] = - \sqrt{\frac{15}{8 π}} h [e^{i ϕ} \cos 2 θ - \cos^{2} θ e^{i ϕ}] = \sqrt{\frac{15}{2 π}} h e^{2 i ϕ} \sin^{2} θ = \frac{1}{2} \sqrt{\frac{15}{2 π}} h {(e^{2 i ϕ} \sin θ)}^{2}$

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

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

Short Answer

Step by step solution

Define Raising Operator

Find the value of Y22(θ,ϕ)

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In Problem4.3 you showed that Y21(θ,ϕ)=-15/8πsinθcosθeiϕ. Apply the raising operator to find localid="1656065252558" Y22(θ,ϕ). 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

Fields in Physics

Electric Charge Field and Potential

Force

Space Physics

Medical Physics

Electrostatics

Study anywhere. Anytime. Across all devices.

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