7.6: Spherical Harmonics


The simultaneous eigenstates, $$Y_{l,m}(\theta,\phi)$$, of $$L^2$$ and $$L_z$$ are known as the spherical harmonics . Let us investigate their functional form.

We know that $L_+\,Y_{l,l}(\theta,\phi) = 0,$ because there is no state for which $$m$$ has a larger value than $$+l$$. Writing $Y_{l,l}(\theta,\phi) = {\mit\Theta}_{l,l}(\theta)\,{\rm e}^{\,{\rm i}\,l\,\phi}$ [see Equations ([e8.34]) and ([e8.38])], and making use of Equation ([e8.28]), we obtain

$\hbar\,{\rm e}^{\,{\rm i}\,\phi}\left(\frac{\partial}{\partial\theta} + {\rm i}\,\cot\theta\,\frac{\partial}{\partial\phi}\right){\mit\Theta}_{l,l}(\theta)\,{\rm e}^{\,i\,l\,\phi}=0.$

This equation yields $\frac{d{\mit\Theta}_{l,l}}{d\theta} - l\,\cot\theta\,{\mit\Theta}_{l,l} = 0.$ which can easily be solved to give ${\mit\Theta}_{l,l}\sim (\sin\theta)^{\,l}.$ Hence, we conclude that

$\label{e8.59} Y_{l,l}(\theta,\phi) \sim (\sin\theta)^{\,l}\,{\rm e}^{\,{\rm i}\,l\,\phi}.$

Likewise, it is easy to demonstrate that

$\label{e8.60} Y_{l,-l}(\theta,\phi) \sim (\sin\theta)^{\,l}\,{\rm e}^{-{\rm i}\,l\,\phi}.$

Once we know $$Y_{l,l}$$, we can obtain $$Y_{l,l-1}$$ by operating on $$Y_{l,l}$$ with the lowering operator $$L_-$$. Thus,

$Y_{l,l-1} \sim L_-\,Y_{l,l} \sim {\rm e}^{-{\rm i}\,\phi}\left(-\frac{\partial}{\partial\theta} + {\rm i}\,\cot\theta\,\frac{\partial}{\partial\phi}\right) (\sin\theta)^{\,l}\,{\rm e}^{\,{\rm i}\,l\,\phi},$

where use has been made of Equation ([e8.28]). The previous equation yields $Y_{l,l-1}\sim {\rm e}^{\,{\rm i}\,(l-1)\,\phi}\left(\frac{d}{d\theta} +l\,\cot\theta\right)(\sin\theta)^{\,l}.$

Now, $\label{e8.64} \left(\frac{d}{d\theta}+l\,\cot\theta\right)f(\theta)\equiv \frac{1}{(\sin\theta)^{\,l}}\frac{d}{d\theta}\left[ (\sin\theta)^{\,l}\,f(\theta)\right],$ where $$f(\theta)$$ is a general function. Hence, we can write

$\label{e8.64a} Y_{l,l-1}(\theta,\phi)\sim \frac{ {\rm e}^{\,{\rm i}\,(l-1)\,\phi}}{(\sin\theta)^{\,l-1}}\left(\frac{1}{\sin\theta}\frac{d}{d\theta}\right) (\sin\theta)^{2\,l}.$

ikewise, we can show that

$\label{e8.65} Y_{l,-l+1}(\theta,\phi)\sim L_+\,Y_{l,-l}\sim \frac{ {\rm e}^{-{\rm i}\,(l-1)\,\phi}}{(\sin\theta)^{\,l-1}}\left(\frac{1}{\sin\theta}\frac{d}{d\theta}\right) (\sin\theta)^{2\,l}.$

We can now obtain $$Y_{l,l-2}$$ by operating on $$Y_{l,l-1}$$ with the lowering operator. We get

$Y_{l,l-2}\sim L_-\,Y_{l,l-1}\sim {\rm e}^{-{\rm i}\,\phi}\left(-\frac{\partial}{\partial\theta} + {\rm i}\,\cot\theta\,\frac{\partial}{\partial\phi}\right) \frac{ {\rm e}^{\,{\rm i}\,(l-1)\,\phi}}{(\sin\theta)^{\,l-1}}\left(\frac{1}{\sin\theta}\frac{d}{d\theta}\right) (\sin\theta)^{2\,l},$

which reduces to $Y_{l,l-2}\sim {\rm e}^{-{\rm i}\,(l-2)\,\phi}\left[\frac{d}{d\theta} +(l-1)\,\cot\theta\right] \frac{1}{(\sin\theta)^{\,l-1}}\left(\frac{1}{\sin\theta}\frac{d}{d\theta}\right) (\sin\theta)^{2\,l}.$ Finally, making use of Equation ([e8.64]), we obtain

$\label{e8.68} Y_{l,l-2}(\theta,\phi) \sim \frac{ {\rm e}^{\,{\rm i}\,(l-2)\,\phi}}{(\sin\theta)^{\,l-2}}\left(\frac{1}{\sin\theta}\frac{d}{d\theta}\right)^2 (\sin\theta)^{2\,l}.$ Likewise, we can show that

$\label{e8.69} Y_{l,-l+2}(\theta,\phi) \sim L_+\,Y_{l,-l+1}\sim \frac{ {\rm e}^{-{\rm i}\,(l-2)\,\phi}}{(\sin\theta)^{\,l-2}}\left(\frac{1}{\sin\theta}\frac{d}{d\theta}\right)^2 (\sin\theta)^{2\,l}.$

A comparison of Equations ([e8.59]), ([e8.64a]), and ([e8.68]) reveals the general functional form of the spherical harmonics:

$Y_{l,m}(\theta,\phi)\sim \frac{ {\rm e}^{\,{\rm i}\,m\,\phi}}{(\sin\theta)^{\,m}}\left(\frac{1}{\sin\theta}\frac{d}{d\theta}\right)^{l-m} (\sin\theta)^{2\,l}.$

Here, $$m$$ is assumed to be non-negative. Making the substitution $$u=\cos\theta$$, we can also write

$Y_{l,m}(u,\phi)\sim {\rm e}^{\,{\rm i}\,m\,\phi}\,(1-u^{\,2})^{-m/2}\left(\frac{d}{d u}\right)^{l-m} (1-u^{\,2})^{\,l}.$

Finally, it is clear from Equations ([e8.60]), ([e8.65]), and ([e8.69]) that

$Y_{l,-m} \sim Y^{\,\ast}_{l,m}.$

Figure 18: The $$$$\left|Y_{l, m}(\theta, \phi)\right|^{2}$$$$ plotted as a functions of $$\theta$$. The solid, short-dashed, and long-dashed curves correspond to $$$$l, m=0,0, \text { and } 1,0, \text { and } 1,\pm 1$$$$, respectively.

We now need to normalize our spherical harmonic functions so as to ensure that $\oint |Y_{l,m}(\theta,\phi)|^{\,2}\,d{\mit\Omega} = 1.$ After a great deal of tedious analysis, the normalized spherical harmonic functions are found to take the form $Y_{l,m}(\theta,\phi) =(-1)^m\, \left[\frac{2\,l+1}{4\pi}\,\frac{(l-m)!}{(l+m)!}\right]^{1/2} P_{l,m}(\cos\theta)\,{\rm e}^{\,{\rm i}\,m\,\phi}$ for $$m\geq 0$$, where the $$P_{l,m}$$ are known as associated Legendre polynomials , and are written $P_{l,m}(u) = (-1)^{l+m}\,\frac{(l+m)!}{(l-m)!}\,\frac{(1-u^{\,2})^{-m/2}}{2^l\,l!}\left(\frac{d}{du}\right)^{l-m} (1-u^{\,2})^{\,l}$ for $$m\geq 0$$. Alternatively, $P_{l,m}(u) = (-1)^{l}\,\frac{(1-u^{\,2})^{m/2}}{2^l\,l!}\left(\frac{d}{du}\right)^{l+m} (1-u^{\,2})^{\,l},$ for $$m\geq 0$$. The spherical harmonics characterized by $$m<0$$ can be calculated from those characterized by $$m>0$$ via the identity $Y_{l,-m} = (-1)^m\,Y^{\,\ast}_{l,m}.$ The spherical harmonics are orthonormal: that is, $\label{spho} \oint Y_{l',m'}^{\,\ast}\,Y_{l,m}\,d{\mit\Omega} = \delta_{ll'}\,\delta_{mm'},$ and also form a complete set. In other words, any well-behaved function of $$\theta$$ and $$\phi$$ can be represented as a superposition of spherical harmonics. Finally, and most importantly, the spherical harmonics are the simultaneous eigenstates of $$L_z$$ and $$L^2$$ corresponding to the eigenvalues $$m\,\hbar$$ and $$l\,(l+1)\,\hbar^{\,2}$$, respectively.

Figure 19: The $$$$\left|Y_{l, m}(\theta, \phi)\right|^{2}$$$$ plotted as a functions of  $$\theta$$. The solid, short-dashed, and long-dashed curves correspond to  $$$$l, m=2,0, \text { and } 2,\pm 1, \text { and } 2,\pm 2$$$$ respectively.

All of the $$l=0$$, $$l=1$$, and $$l=2$$ spherical harmonics are listed below: \begin{aligned} Y_{0,0} &=\frac{1}{\sqrt{4\pi}},\\[0.5ex] Y_{1,0} &= \sqrt{\frac{3}{4\pi}}\,\cos\theta,\\[0.5ex] Y_{1,\pm1} &= \mp \sqrt{\frac{3}{8\pi}}\,\sin\theta\,{\rm e}^{\pm{\rm i}\,\phi},\\[0.5ex] Y_{2,0} &= \sqrt{\frac{5}{16\pi}}\,(3\,\cos^2\theta - 1),\\[0.5ex] Y_{2,\pm 1}&=\mp\sqrt{\frac{15}{8\pi}}\,\sin\theta\,\cos\theta\,{\rm e}^{\pm{\rm i}\,\phi},\\[0.5ex] Y_{2,\pm 2}&= \sqrt{\frac{15}{32\pi}}\,\sin^2\theta\,{\rm e}^{\pm 2\,{\rm i}\,\phi}.\end{aligned} The $$\theta$$ variation of these functions is illustrated in Figures [ylm1] and [ylm2].
