Chapter 4.1, Problem 4.7P

To determine

The spherical harmonics $Y_{\mathcal{l}}^{\mathcal{l}}\left(\theta ,\varphi \right)$ and $Y_3^2\left(\theta ,\varphi \right)$ and whether they satisfy the angular equation for appropriate value of $m$ and $\mathcal{l}$.

Answer to Problem 4.7P

The spherical harmonics $Y_{\mathcal{l}}^{\mathcal{l}}\left(\theta ,\varphi \right)$ is $\underset{\_}{\frac{1}{\mathcal{l}!}\sqrt{\frac{\left(2\mathcal{l}+1\right)!}{4\pi }}{\left(-\frac{1}{2}{e}^{i\varphi }\sin \theta \right)}^{\mathcal{l}}}$ and $Y_3^2\left(\theta ,\varphi \right)$ is $\underset{\_}{\frac{1}{4}\sqrt{\frac{105}{2\pi }}{e}^{2i\varphi }{\sin }^2\theta \cos \theta }$ and it satisfies with the angular equation for appropriate value of $m$ and $\mathcal{l}$.

Explanation of Solution

Write the general expression for the spherical harmonics $Y_{\mathcal{l}}^{\mathcal{l}}\left(\theta ,\varphi \right)$.

    $Y_{\mathcal{l}}^{\mathcal{l}}\left(\theta ,\varphi \right)=\sqrt{\frac{\left(2\mathcal{l}+1\right)}{4\pi }\frac{1}{\left(2\mathcal{l}\right)!}}{e}^{i\mathcal{l}\varphi }{P}_{\mathcal{l}}^{\mathcal{l}}\left(\cos \theta \right)$        (I)

Here, $P_{\mathcal{l}}^{\mathcal{l}}\left(\cos \theta \right)$ is the associated Legendre function.

Write the expression for the associated Legendre function.

    $P_{\mathcal{l}}^{\mathcal{l}}\left(x\right)={\left(-1\right)}^{\mathcal{l}}{\left(1-x^2\right)}^{\mathcal{l}/2}{\left(\frac{d}{dx}\right)}^{\mathcal{l}}{P}_{\mathcal{l}}\left(x\right)$        (II)

Here, $P_{\mathcal{l}}\left(x\right)$ is the Legendre function.

Write the expression for the Legendre function.

    $P_{\mathcal{l}}\left(x\right)=\frac{1}{2^{\mathcal{l}}\mathcal{l}!}{\left(\frac{d}{dx}\right)}^{\mathcal{l}}{\left(x^2-1\right)}^{\mathcal{l}}$        (III)

Use equation (III) in (II) to solve for $P_{\mathcal{l}}^{\mathcal{l}}\left(x\right)$.

    $P_{\mathcal{l}}^{\mathcal{l}}\left(x\right)=\frac{{\left(-1\right)}^{\mathcal{l}}}{2^{\mathcal{l}}\mathcal{l}!}{\left(1-x^2\right)}^{\mathcal{l}/2}{\left(\frac{d}{dx}\right)}^{2\mathcal{l}}{\left(x^2-1\right)}^{\mathcal{l}}$        (IV)

The ${\left(x^2-1\right)}^{\mathcal{l}}=x^{2\mathcal{l}}+\cdot \cdot \cdot ,$ where all other terms involve powers of $x$ less than $2\mathcal{l}$, and hence give zero when differentiated $2\mathcal{l}$ times.

Then the equation (IV) becomes,

    $P_{\mathcal{l}}^{\mathcal{l}}\left(x\right)=\frac{{\left(-1\right)}^{\mathcal{l}}}{2^{\mathcal{l}}\mathcal{l}!}{\left(1-x^2\right)}^{\mathcal{l}/2}{\left(\frac{d}{dx}\right)}^{2\mathcal{l}}{x}^{2\mathcal{l}}$        (V)

The expansion of ${\left(\frac{d}{dx}\right)}^n{x}^n$ can be written as

    ${\left(\frac{d}{dx}\right)}^n{x}^n=n!$        (VI)

Use equation (VI) in (V) and it becomes,

    $P_{\mathcal{l}}^{\mathcal{l}}\left(x\right)=\frac{{\left(-1\right)}^{\mathcal{l}}\left(2\mathcal{l}\right)!}{2^{\mathcal{l}}\mathcal{l}!}{\left(1-x^2\right)}^{\mathcal{l}/2}$        (VII)

Use equation (VII) in (I) to solve for the spherical harmonics $Y_{\mathcal{l}}^{\mathcal{l}}\left(\theta ,\varphi \right)$.

    $\begin{array}{c}{Y}_{\mathcal{l}}^{\mathcal{l}}\left(\theta ,\varphi \right)={\left(-1\right)}^{\mathcal{l}}\sqrt{\frac{\left(2\mathcal{l}+1\right)!}{4\pi }}{e}^{i\mathcal{l}\varphi }\frac{\left(2\mathcal{l}\right)!}{2^{\mathcal{l}}\mathcal{l}!}{\left(\sin \theta \right)}^{\mathcal{l}}\\ =\frac{1}{\mathcal{l}!}\sqrt{\frac{\left(2\mathcal{l}+1\right)!}{4\pi }}{\left(-\frac{1}{2}{e}^{i\varphi }\sin \theta \right)}^{\mathcal{l}}\end{array}$        (VIII)

Use equation (VIII) to find $Y_3^2\left(\theta ,\varphi \right)$.

    $\begin{array}{c}{Y}_3^2\left(\theta ,\varphi \right)=\sqrt{\frac{7}{4\pi }\cdot \frac{1}{5!}}{e}^{2i\varphi }{P}_3^2\left(\cos \theta \right)\\ =\sqrt{\frac{7}{4\pi }\cdot \frac{1}{5!}}15e^{2i\varphi }\cos \theta {\sin }^2\theta \\ =\frac{1}{4}\sqrt{\frac{105}{2\pi }}{e}^{2i\varphi }{\sin }^2\theta \cos \theta \end{array}$        (IX)

Consider $\frac{1}{\mathcal{l}!}\sqrt{\frac{\left(2\mathcal{l}+1\right)!}{4\pi }}{\left(-\frac{1}{2}\right)}^{\mathcal{l}}=A$ and the spherical harmonics $Y_{\mathcal{l}}^{\mathcal{l}}\left(\theta ,\varphi \right)$ can be written as

    $Y_{\mathcal{l}}^{\mathcal{l}}\left(\theta ,\varphi \right)=A{\left(e^{i\varphi }\sin \theta \right)}^{\mathcal{l}}$        (X)

Differentiate equation (X) with respect to $\theta$ and it becomes,

    $\begin{array}{c}\frac{\partial Y_{\mathcal{l}}^{\mathcal{l}}}{\partial \theta }=A{e}^{i\mathcal{l}\varphi }l{\left(\sin \theta \right)}^{\mathcal{l}-1}\cos \theta \\ \sin \theta \frac{\partial Y_{\mathcal{l}}^{\mathcal{l}}}{\partial \theta }=A{e}^{i\mathcal{l}\varphi }l\cos \theta Y_{\mathcal{l}}^{\mathcal{l}}\\ \sin \theta \frac{\partial }{\partial \theta }\left(\sin \theta \frac{\partial Y_{\mathcal{l}}^{\mathcal{l}}}{\partial \theta }\right)=\mathcal{l}\cos \left(\sin \theta \frac{\partial Y_{\mathcal{l}}^{\mathcal{l}}}{\partial \theta }\right)-\mathcal{l}{\sin }^2\theta Y_{\mathcal{l}}^{\mathcal{l}}\\ =\left({\mathcal{l}}^2{\cos }^2\theta -\mathcal{l}{\sin }^2\theta \right)Y_{\mathcal{l}}^{\mathcal{l}}\end{array}$        (XI)

Take the second derivative of $Y_{\mathcal{l}}^{\mathcal{l}}\left(\theta ,\varphi \right)$ with respect to $\varphi$ and it becomes,

    $\frac{{\partial }^2{Y}_{\mathcal{l}}^{\mathcal{l}}}{\partial {\varphi }^2}=-{\mathcal{l}}^2{Y}_{\mathcal{l}}^{\mathcal{l}}$        (XII)

The sum of the equations (XII) and (XI) can be written as

    $\begin{array}{c}\sin \theta \frac{\partial }{\partial \theta }\left(\sin \theta \frac{\partial Y_{\mathcal{l}}^{\mathcal{l}}}{\partial \theta }\right)+\frac{{\partial }^2{Y}_{\mathcal{l}}^{\mathcal{l}}}{\partial {\varphi }^2}=\left({\mathcal{l}}^2{\cos }^2\theta -\mathcal{l}{\sin }^2\theta \right)Y_{\mathcal{l}}^{\mathcal{l}}-{\mathcal{l}}^2{Y}_{\mathcal{l}}^{\mathcal{l}}\\ =-\mathcal{l}\left(\mathcal{l}+1\right){\sin }^2\theta Y_{\mathcal{l}}^{\mathcal{l}}\end{array}$        (XIII)

The equation (XIII) satisfies with the equation given in (4.18).

Consider $B=\frac{1}{4}\sqrt{\frac{105}{2\pi }}$ and the spherical harmonics $Y_3^2\left(\theta ,\varphi \right)$ can be written as

    $Y_3^2=B{e}^{2i\varphi }{\sin }^2\theta \cos \theta$        (XIV)

Take the partial derivative of equation (XIV) with respect to $\theta$ and it becomes,

    $\frac{\partial Y_3^2}{\partial \theta }=B{e}^{2i\varphi }\left(2\sin \theta {\cos }^2\theta -{\sin }^2\theta \right)$        (XV)

Use equation (XV) in (XIII) and it can be written as

    $\begin{array}{c}\sin \theta \frac{\partial }{\partial \theta }\left(\sin \theta \frac{\partial Y_3^2}{\partial \theta }\right)=B{e}^{2i\varphi }\sin \theta \frac{\partial }{\partial \theta }\left(2{\sin }^2\theta {\cos }^2\theta -{\sin }^4\theta \right)\\ =B{e}^{2i\varphi }\sin \theta \left(\begin{array}{l}4\sin \theta {\cos }^2\theta \\ -4{\sin }^3\theta -4{\sin }^3\theta \cos \theta \end{array}\right)\\ =4B{e}^{2i\varphi }{\sin }^2\theta \cos \theta \left({\cos }^2\theta -2{\sin }^2\theta \right)\\ =4\left({\cos }^2-2{\sin }^2\theta \right)Y_3^2\end{array}$        (XVI)

Take the second partial derivative of $Y_3^2$ with respect to $\varphi$.

    $\frac{{\partial }^2{Y}_3^2}{\partial {\varphi }^2}=4Y_3^2$        (XVII)

Use equation (XVII) and (XVI) in (XIII) and compare,

    $\begin{array}{c}4\left({\cos }^2\theta -2{\sin }^2\theta -1\right)Y_3^2=4\left(-3{\sin }^2\theta \right)Y_3^2\\ =-\mathcal{l}\left(\mathcal{l}+1\right){\sin }^2\theta Y_3^2\end{array}$        (XVIII)

In equation (XVIII), where $\mathcal{l}=3$ fits the RHS of equation 4.18 and hence it is proved.

Conclusion:

Therefore, The spherical harmonics $Y_{\mathcal{l}}^{\mathcal{l}}\left(\theta ,\varphi \right)$ is $\underset{\_}{\frac{1}{\mathcal{l}!}\sqrt{\frac{\left(2\mathcal{l}+1\right)!}{4\pi }}{\left(-\frac{1}{2}{e}^{i\varphi }\sin \theta \right)}^{\mathcal{l}}}$ and $Y_3^2\left(\theta ,\varphi \right)$ is $\underset{\_}{\frac{1}{4}\sqrt{\frac{105}{2\pi }}{e}^{2i\varphi }{\sin }^2\theta \cos \theta }$ and it satisfies with the angular equation for appropriate value of $m$ and $\mathcal{l}$.