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.

Now, we know that

$$L_+\,Y_{l,l}(\theta,\phi) = 0,$$ (570)

since there is no state for which $m$ has a larger value than $+l$. Writing

$$Y_{l,l}(\theta,\phi) = \Theta_{l,l}(\theta)\,{\rm e}^{\,{\rm i}\,l\,\phi}$$ (571)

[see Eqs. (549) and (553)], and making use of Eq. (534), we obtain

$$\hbar\,{\rm e}^{\,{\rm i}\,\phi}\left(\frac{\partial}{\parti... ...ial\phi}\right)\Theta_{l,l}(\theta)\,{\rm e}^{\,i\,l\,\phi}=0.$$ (572)

This equation yields

$$\frac{d\Theta_{l,l}}{d\theta} - l\,\cot\theta\,\Theta_{l,l} = 0.$$ (573)

which can easily be solved to give

$$\Theta_{l,l}\sim (\sin\theta)^l.$$ (574)

Hence, we conclude that

$$Y_{l,l}(\theta,\phi) \sim (\sin\theta)^l\,{\rm e}^{\,{\rm i}\,l\,\phi}.$$ (575)

Likewise, it is easy to demonstrate that

$$Y_{l,-l}(\theta,\phi) \sim (\sin\theta)^l\,{\rm e}^{-{\rm i}\,l\,\phi}.$$ (576)

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,

$$Y_{l,l-1} \sim L_-\,Y_{l,l} \sim {\rm e}^{-{\rm i}\,\phi}\le... ...tial\phi}\right) (\sin\theta)^l\,{\rm e}^{\,{\rm i}\,l\,\phi},$$ (577)

where use has been made of Eq. (534). The above equation yields

$$Y_{l,l-1}\sim {\rm e}^{\,{\rm i}\,(l-1)\,\phi}\left(\frac{d}{d\theta} +l\,\cot\theta\right)(\sin\theta)^l.$$ (578)

Now,

$$\left(\frac{d}{d\theta}+l\,\cot\theta\right)f(\theta)\equiv ... ...a)^l}\frac{d}{d\theta}\left[ (\sin\theta)^l\,f(\theta)\right],$$ (579)

where $f(\theta)$ is a general function. Hence, we can write

$$Y_{l,l-1}(\theta,\phi)\sim \frac{{\rm e}^{\,{\rm i}\,(l-1)\,... ...ac{1}{\sin\theta}\frac{d}{d\theta}\right) (\sin\theta)^{2\,l}.$$ (580)

Likewise, we can show that

$$Y_{l,-l+1}(\theta,\phi)\sim L_+\,Y_{l,-l}\sim \frac{{\rm e}^... ...ac{1}{\sin\theta}\frac{d}{d\theta}\right) (\sin\theta)^{2\,l}.$$ (581)

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

$$Y_{l,l-2}\sim L_-\,Y_{l,l-1}\sim {\rm e}^{-{\rm i}\,\phi}\le... ...ac{1}{\sin\theta}\frac{d}{d\theta}\right) (\sin\theta)^{2\,l},$$ (582)

which reduces to

$$Y_{l,l-2}\sim {\rm e}^{-{\rm i}\,(l-2)\,\phi}\left[\frac{d}{... ...ac{1}{\sin\theta}\frac{d}{d\theta}\right) (\sin\theta)^{2\,l}.$$ (583)

Finally, making use of Eq. (579), we obtain

$$Y_{l,l-2}(\theta,\phi) \sim \frac{{\rm e}^{\,{\rm i}\,(l-2)\... ...{1}{\sin\theta}\frac{d}{d\theta}\right)^2 (\sin\theta)^{2\,l}.$$ (584)

Likewise, we can show that

$$Y_{l,-l+2}(\theta,\phi) \sim L_+\,Y_{l,-l+1}\sim \frac{{\rm ... ...{1}{\sin\theta}\frac{d}{d\theta}\right)^2 (\sin\theta)^{2\,l}.$$ (585)

A comparison of Eqs. (575), (580), and (584) reveals the general functional form of the spherical harmonics:

$$Y_{l,m}(\theta,\phi)\sim \frac{{\rm e}^{\,{\rm i}\,m\,\phi}}... ...\sin\theta}\frac{d}{d\theta}\right)^{l-m} (\sin\theta)^{2\,l}.$$ (586)

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

$$Y_{l,m}(u,\phi)\sim {\rm e}^{\,{\rm i}\,m\,\phi}\,(1-u^2)^{-m/2}\left(\frac{d}{d u}\right)^{l-m} (1-u^2)^l.$$ (587)

Finally, it is clear from Eqs. (576), (581), and (585) that

$$Y_{l,-m} \sim Y^{\,\ast}_{l,m}.$$ (588)

We now need to normalize our spherical harmonic functions so as to ensure that

$$\oint \vert Y_{l,m}(\theta,\phi)\vert^2\,d\Omega = 1.$$ (589)

After a great deal of tedious analysis, the normalized spherical harmonic functions are found to take the form

$$Y_{l,m}(\theta,\phi) =(-1)^m\, \left[\frac{2\,l+1}{4\,\pi}\,... ...right]^{1/2} P_{l,m}(\cos\theta)\,{\rm e}^{\,{\rm i}\,m\,\phi}$$ (590)

for $m\geq 0$, where the $P_{l,m}$ are known as associated Legendre polynomials, and are written

$$P_{l,m}(u) = (-1)^{l+m}\,\frac{(l+m)!}{(l-m)!}\,\frac{(1-u^2)^{-m/2}}{2^l\,l!}\left(\frac{d}{du}\right)^{l-m} (1-u^2)^l$$ (591)

for $m\geq 0$. Alternatively,

$$P_{l,m}(u) = (-1)^{l}\,\frac{(1-u^2)^{m/2}}{2^l\,l!}\left(\frac{d}{du}\right)^{l+m} (1-u^2)^l,$$ (592)

for $m\geq 0$. The spherical harmonics characterized by $m<0$ can be calculated from those characterized by $m>0$ via the identity

$$Y_{l,-m} = (-1)^m\,Y^{\,\ast}_{l,m}.$$ (593)

The spherical harmonics are orthonormal: i.e.,

$$\oint Y_{l',m'}^{\,\ast}\,Y_{l,m}\,d\Omega = \delta_{ll'}\,\delta_{mm'},$$ (594)

and also form a complete set. In other words, any 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 15: The $\vert Y_{l,m}(\theta ,\phi )\vert^{\,2}$ plotted as a functions of $\theta$. The solid, short-dashed, and long-dashed curves correspond to $l,m=0,0$, and $1,0$, and $1,\pm 1$, respectively.

All of the $l=0$, $l=1$, and $l=2$ spherical harmonics are listed below:

$$Y_{0,0} = \frac{1}{\sqrt{4\pi}},$$ (595)

$$Y_{1,0} = \sqrt{\frac{3}{4\pi}}\,\cos\theta,$$ (596)

$$Y_{1,\pm1} = \mp \sqrt{\frac{3}{8\pi}}\,\sin\theta\,{\rm e}^{\pm{\rm i}\,\phi},$$ (597)

$$Y_{2,0} = \sqrt{\frac{5}{16\pi}}\,(3\,\cos^2\theta - 1),$$ (598)

$$Y_{2,\pm 1} = \mp\sqrt{\frac{15}{8\pi}}\,\sin\theta\,\cos\theta\,{\rm e}^{\pm{\rm i}\,\phi},$$ (599)

$$Y_{2,\pm 2} = \sqrt{\frac{15}{32\pi}}\,\sin^2\theta\,{\rm e}^{\pm 2\,{\rm i}\,\phi}.$$ (600)

The $\theta$ variation of these functions is illustrated in Figs. 15 and 16.

Figure 16: The $\vert Y_{l,m}(\theta ,\phi )\vert^{\,2}$ plotted as a functions of $\theta$. The solid, short-dashed, and long-dashed curves correspond to $l,m=2,0$, and $2,\pm 1$, and $2,\pm 2$, respectively.