The Angular Part

The angular part of (4.62) in fact is again an eigenvalue problem, because the equation $\hat{\Omega}Y = c Y$ is an eigenvalue equation for the eigenvectors (eigenfunctions, remember that a function is a vector in a Hilbert space) and possible eigenvalues $c$ of the operator $\hat{\Omega}$. Let us write down again this equation:

$$\left[\frac{1}{\sin \theta}\frac{\partial}{\partial \theta} \left... ...\partial^2 Y(\theta,\varphi)}{\partial \varphi^2}\right] = c Y(\theta,\varphi).$$ (286)

We do not explicitely construct the eigenfunctions $Y$ of the operator $\hat{\Omega}$ here but only give the results. In fact, this operator is closely related to the angular momentum operator which we will discuss in the next session. Similar to what we have found for the harmonic oscillator, it turns out that solutions of (4.63) are possible only for $c=-l(l+1)$, where $l=0,1,2,3,...$ is an integer. All the solutions can be labeled by two quantum numbers $l$ and $m$, where $m$ is an integer that can take the values $-l, -l+1,...,l-1,l$. The solutions are called spherical harmonics and have the explicit form

$$Y_{lm}(\theta,\varphi) = (-1)^{(m+\vert m\vert)/2}i^l \left[\frac{2l+1}{4\pi}\frac{(l-\ver... ...}{(l+\vert m\vert)!}\right]^{1/2} P_l^{\vert m\vert}(\cos \theta) e^{im\varphi}$$

$$P_l^{\vert m\vert}(x) := \frac{1}{2^ll!}(1-x^2)^{\vert m\vert/2}\frac{d^{l+\vert m\vert}}{d x^{l+\vert m\vert}}(x^2 -1)^l$$

$$l = 0,1,2,3,...;\quad m= -l,-l+1,-l+2,...,l-1,l.$$ (287)

The $P_l^{\vert m\vert}$ are called associated Legendre polynomials. The spherical harmonics are an orthonormal function system on the surface of the unit sphere $\vert{\bf x}\vert=1$. We write the orthonormality relation both in our abstract bra -ket and in explicit form:

$$\vert lm\rangle \longleftrightarrow Y_{lm}(\theta,\varphi)$$ (288)

$$\langle l'm'\vert lm \rangle = \delta_{ll'}\delta_{mm'} \longleftrightarrow \int_0^{2\pi} \int_0^{\pi} Y_{l'm'}^*(\theta,\varphi)Y_{lm}(\theta,\varphi)\sin \theta d\theta d \varphi = \delta_{ll'}\delta{mm'}.$$

The spherical harmonics with $l=0,1,2,3,4,...$ are denoted as $s$-, $p$-, $d$-, $f$-, $g$-,... functions which you might know already from chemistry (`orbitals'). The explicit forms for some of the first sphericals are

$$Y_{00}=\frac{1}{\sqrt{4\pi}},\quad Y_{10}= i \sqrt{\frac{3}{4\pi}... ...quad Y_{1\pm 1}= \mp i \sqrt{\frac{3}{8\pi}}\sin \theta \cdot e^{\pm i\varphi}.$$ (289)