Vorlesung_33_20171218

$\hat{L}^2 Y=\hbar^2 l ( l+ 1 )Y$

\to | L | = \sqrt{ℓ (ℓ + 1)} ℏ

$m\leq \ell$ sein muss.

L_{z, max} = ℓ ℏ < | L | = \sqrt{ℓ (ℓ + 1)} ℏ

$\vec{L}$ genau in z-Richtung zeigen?

$\ell = 3$ $z-$ Achse einschließen kann

\begin{matrix} ℓ = 3 \Rightarrow | L | = \sqrt{12} ℏ \\ L_{z, max} = 3 ℏ \\ \cos (ϑ) = \frac{L_{z}}{| L |} = \frac{3 ℏ}{\sqrt{12} ℏ} \Rightarrow ϑ = 30^{\circ} \end{matrix}

$\Theta(\vartheta)$ :

\sin (ϑ) \frac{\partial}{\partial ϑ} (\sin (ϑ) \frac{\partial Θ (ϑ)}{\partial ϑ}) + [ℓ (ℓ + 1) \sin^{2} (ϑ) - m^{2}] Θ (ϑ) = 0

Lösung: Zugeordnete Legendre-Funktionen (analog zu Multipolentwicklung)

\begin{matrix} Θ (ϑ) = A P_{ℓ}^{m} (\cos (ϑ)) \\ P_{ℓ}^{m} (x) \equiv {(1 - x^{2})}^{\frac{| m |}{2}} {(\frac{d}{d x})}^{| m |} P_{ℓ} (x) \end{matrix}

$\ell$ -te Legendre-Polynom definiert durch Rodriguez-Formel

\begin{matrix} P_{ℓ} (x) \equiv \frac{1}{2^{ℓ} ℓ!} {(\frac{d}{d x})}^{ℓ} (x^{2} - 1)^{ℓ} \\ Beispiele: P_{0} (x) = 1 \\ P_{1} (x) = \frac{1}{2} \frac{d}{d x} (x^{2} - 1) = x \\ P_{2} (x) = \frac{1}{4 \cdot 2} {(\frac{d}{d x})}^{2} (x^{2} - 1)^{2} = \frac{1}{2} (3 x^{2} - 1) \end{matrix}

$P_\ell(x)$ $\ell$

$\ell$ $P_\ell(x)=P_\ell(-x)$

$P_\ell(x)=-P_\ell(-x)$

$P_\ell^m(x)$ $m$ $\Rightarrow$ $\sqrt{1-x^2}$ $\Rightarrow$ kein Polynom

$x=cos(\theta)$ $\Rightarrow$ $\sqrt{1-\cos^2\theta}=\sin{\theta}$

Bsp.:

\begin{matrix} P_{0} = 1 \\ P_{1} = x \\ P_{2} = \frac{1}{2} (3 x^{2} - 1) \\ P_{0}^{0} (\cos (ϑ)) = 1 \\ P_{1}^{1} (\cos (ϑ)) = \sin (ϑ) \\ P_{1}^{0} (\cos (ϑ)) = \cos (ϑ) \\ P_{2}^{2} (\cos (ϑ)) = 3 \sin^{3} (ϑ) \\ P_{2}^{1} (\cos (ϑ)) = 3 \sin (ϑ) \cos (ϑ) \\ P_{2}^{0} (\cos (ϑ)) = \frac{1}{2} (3 \cos^{2} (ϑ) - 1) \end{matrix}

$\ell$ muss nicht negativ und ganzzahlig sein wegen Ableitung in Rodriguez-Formel

P_{ℓ}^{m} (x) = 0 für | m | > ℓ, m = - ℓ, - ℓ + 1, . . . ℓ - 1, ℓ

$m$ $\ell$ $|m|\gt\ell$ .

Normierung:

\begin{matrix} d^{3} r = r^{2} \sin (ϑ) d r d ϑ d φ \\ \int {| Ψ (r, ϑ, φ) |}^{2} r^{2} \sin (ϑ) d r d ϑ d φ = \int {| R (r) |}^{2} r^{2} d r \int {| Y (ϑ, φ) |}^{2} \sin (ϑ) d ϑ d φ = 1 \\ \int_{0}^{\infty} {| R (r) |}^{2} r^{2} d r = 1 \\ \int_{0}^{2 π} d φ \int_{0}^{π} d ϑ {| Y (ϑ, φ) |}^{2} \sin (ϑ) = 1 \end{matrix}

$\rightarrow$ Normierungskonstante

$\Rightarrow$ Endergebnis: Normierte Kugelflächenfunktion

Y_{ℓ}^{m} (ϑ, φ) = (- 1)^{m} \sqrt{\frac{(2 ℓ + 1)}{4 π} \frac{(ℓ - | m |)!}{(ℓ + | m |)!}} \cdot e^{i m φ} P_{ℓ}^{m} (\cos (ϑ))

Orthogonal

\int_{0}^{2 π} d φ \int_{0}^{π} d ϑ \sin (θ) {Y_{ℓ}^{m}}^{*} (ϑ, φ) Y_{ℓ^{'}}^{m^{'}} (ϑ, φ) = δ_{ℓ, ℓ^{'}} δ_{m, m^{'}}

es gilt außerdem:

$Y_l^m=(-1)^m(Y_l^m)^*$

Beispiele:

\begin{matrix} Y_{0}^{0} = {(\frac{1}{4 π})}^{\frac{1}{2}} \\ Y_{1}^{0} = {(\frac{3}{4 π})}^{\frac{1}{2}} \cos (ϑ) \\ Y_{1}^{\pm 1} = \mp {(\frac{3}{8 π})}^{\frac{1}{2}} \sin (ϑ) e^{\pm i φ} \\ Y_{2}^{0} = {(\frac{5}{16 π})}^{\frac{1}{2}} (3 \cos^{2} (ϑ) - 1) \\ Y_{2}^{\pm 1} = \mp {(\frac{15}{8 π})}^{\frac{1}{2}} \sin (ϑ) \cos (ϑ) e^{\pm i φ} \\ Y_{2}^{\pm 2} = {(\frac{15}{32 π})}^{\frac{1}{2}} \sin^{2} ϑ e^{\pm 2 i φ} \end{matrix}