Orbital Angular Momentum

The central potential has rotational symmetry and therefore a conserved quantity, the angular momentum (Nöther's theorem ). Here, we introduce polar coordinates and realise that the Laplacian can be written as

$$\Delta = \frac{\partial^2}{\partial r^2}+\frac{2}{r}\frac{\partial}{\partial r} -\frac{{\bf L}^2}{\hbar^2 r^2},$$ (1.9)

where the angular momentum is

$$\hat{L}_x = -i\hbar \left( -\sin \varphi \frac{\partial}{\partial \theta} -\cos \varphi \cot \theta \frac{\partial}{\partial \varphi} \right)$$

$$\hat{L}_y = -i\hbar \left( \cos \varphi \frac{\partial}{\partial \theta} -\sin \varphi \cot \theta \frac{\partial}{\partial \varphi} \right)$$

$$\hat{L}_z = -i\hbar\frac{\partial}{\partial \varphi}.$$ (1.10)

and its square is given by

$${\bf\hat{L}}^2= -\hbar^2 \left[\frac{1}{\sin \theta}\frac{\partia... ...a}\right) +\frac{1}{\sin^2 \theta}\frac{\partial^2}{\partial \varphi^2}\right].$$ (1.11)

The eigenvalue equations for ${\bf\hat{L}}^2$ and $\hat{L}_z$ are

$${\bf\hat{L}}^2 Y_{lm}(\theta,\varphi) = \hbar^2 l(l+1)Y_{lm}(\theta,\varphi),\quad l=0,1,2,3,...$$ (1.12)

$$\hat{L}_z Y_{lm}(\theta,\varphi) = \hbar m Y_{lm}(\theta,\varphi),$$ (1.13)

where the spherical harmonics have quantum numbers ${\color{col1}l}$ and ${\color{col1}m}$ and the explicit form

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

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

$$l = 0,1,2,3,...;\quad m= -l,-l+1,-l+2,...,l-1,l.$$ (1.14)

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:

$$\vert lm\rangle \longleftrightarrow Y_{lm}(\theta,\varphi)$$ (1.15)

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

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

$$Y_{00}=\frac{1}{\sqrt{4\pi}},\quad Y_{10}= i \sqrt{\frac{3}{4\pi}... ...quad Y_{1\pm 1}= \mp i \sqrt{\frac{3}{8\pi}}\sin \theta \cdot e^{\pm i\varphi}.$$ (1.16)

Figure: Absolute squares of various spherical harmonics. From http://mathworld.wolfram.com/SphericalHarmonic.html

Further information on spherical harmonics in various books and under

http://mathworld.wolfram.com/SphericalHarmonic.htmlhttp://mathworld.wolfram.com/SphericalHarmonic.html.

The Spherical harmonics are used in many areas of science, ranging from nuclear physics up to computer vision tasks. If you like online physics teaching, have a look at

http://scienceworld.wolfram.com/physics/HydrogenAtom.htmlhttp://scienceworld.wolfram.com/physics/HydrogenAtom.html .