7.2.2 Schwache Felder

H„ufigster anzutreffender Effekt (name historisch)

Vorherrschende Wechselwirkung: Spin-Bahn-Kopplung:

H₀ = - $\displaystyle {\hbar^2 \over 2m}$ $\displaystyle \nabla^{2}_{}$ - $\displaystyle {Z e^2 \over 4\pi \varepsilon_0 r}$ + $\displaystyle \xi$ L^.S	Ungest”rter Hamilton	(198)
$\displaystyle \mathcal {Y}$ _ls^jm_j = $\displaystyle \sum_{m_l, m_s}^{}$ $\displaystyle \underbrace{\langle l s m_l m_s \vert j m_j\rangle }_{\mbox{Clebsch-Gordon-Koeffizient \index{Clebsch-Gordon-Koeffizient}}}^{}\,$ Y_{lm_l}( $\displaystyle \theta$ , $\displaystyle \phi$ ) $\displaystyle \chi_{s, m_s}^{}$	EF von L², S², J², J_z	(199)
H₁' = $\displaystyle {\mu_B \over \hbar}$ (L_z + 2S_z)B_z = $\displaystyle {\mu_B \over \hbar}$ (J_z + S_z)B_z	St”rung, Magnetfeld in z	(200)
$\displaystyle \Rightarrow$ $\displaystyle \Delta$ E = $\displaystyle \mu_{B}^{}$ B_zm_j + $\displaystyle {\mu_B \over \hbar}$ B_z $\displaystyle \sum_{\mbox{Spin}}^{}$ $\displaystyle \int$ d $\displaystyle \Omega$ $\displaystyle \overline{\mathcal{Y}_{l,{1\over 2}}^{jm_j}}$ S_z $\displaystyle \mathcal {Y}$ _l,^jm_j		(201)

Berechnung des Integrals:

1.

Explizites Einsetzen der Clebsch-Gordon-Koeffizienten:

$\displaystyle \mathcal {Y}$ _l,^{(l + )m_j} = $\displaystyle \sqrt{{l+m_j + {1\over 2} \over 2l+1}}$ Y_{l, m_j -} $\displaystyle \chi_{{1\over 2},{1\over 2}}^{}$ + $\displaystyle \sqrt{{l-m_j + {1\over 2} \over 2l+1}}$ Y_{l, m_j +} $\displaystyle \chi_{{1\over 2},-{1\over 2}}^{}$			(202)
$\displaystyle \mathcal {Y}$ _{l, -}^{(l + )m_j} = $\displaystyle \sqrt{{l-m_j + {1\over 2} \over 2l+1}}$ Y_{l, m_j -} $\displaystyle \chi_{{1\over 2},{1\over 2}}^{}$ + $\displaystyle \sqrt{{l+m_j + {1\over 2} \over 2l+1}}$ Y_{l, m_j +} $\displaystyle \chi_{{1\over 2},-{1\over 2}}^{}$			(203)
$\displaystyle \Rightarrow$ $\displaystyle \sum_{\mbox{Spin}}^{}$ $\displaystyle \int$ d $\displaystyle \Omega$ $\displaystyle \overline{\mathcal{Y}_{l,{1\over 2}}^{l+{1\over 2} m_j}}$ S_z $\displaystyle \mathcal {Y}$ _{l + ,}^jm_j = $\displaystyle {m_j \over 2l+1}$ $\displaystyle \hbar$			(204)
$\displaystyle \Rightarrow$ $\displaystyle \sum_{\mbox{Spin}}^{}$ $\displaystyle \int$ d $\displaystyle \Omega$ $\displaystyle \overline{\mathcal{Y}_{l,{1\over 2}}^{l-{1\over 2} m_j}}$ S_z $\displaystyle \mathcal {Y}$ _l,^{l - m_j} = - $\displaystyle {m_j \over 2l+1}$ $\displaystyle \hbar$			(205)

2.

Operatoralgebra (mit Wiegner-Eckard-Theorem) fhrt zu

$\displaystyle \langle$ lsjm_j| S_z| lsjm_j $\displaystyle \rangle$ = m_j $\displaystyle \hbar$ $\displaystyle \left[\vphantom{ {j(j+1) + s (s+1) - l(l+1) \over 2 j(j+1)} }\right.$ $\displaystyle {j(j+1) + s (s+1) - l(l+1) \over 2 j(j+1)}$ $\displaystyle \left.\vphantom{ {j(j+1) + s (s+1) - l(l+1) \over 2 j(j+1)} }\right]$

(206)

was mit dem ersten Ergebnis identisch ist, da s = ${1\over 2}$ und j = l $\pm$ ${1\over 2}$

Energieverschiebung

$\displaystyle \Delta$ E = $\displaystyle \underbrace{1+ \left[ {j(j+1) + s (s+1) - l(l+1) \over 2 j(j+1)} \right]}_{\mbox{= g: Land‚scher g-Faktor \index{Land‚scher g-Faktor}}}^{}\,$ $\displaystyle \mu_{B}^{}$ B_zm_j

(207)

Gesamtenergieverschiebung:

E_{n, j, m_j} = $\displaystyle \underbrace{E_n}_{\mbox{nicht-rel.}}^{}\,$ + $\displaystyle \underbrace{\Delta E_{n,j}}_{\mbox{Feinstruktur}}^{}\,$ + $\displaystyle \underbrace{\Delta E_{m_j}}_{Zeemann}^{}\,$

(208)

Der anomale Zeemann-Effekt zeigt mehr Linien als der normale

Mit zunehmendem Feld geht der anomale Zeemann-Effekt zun„chst in den normalen und dann den Paschen-Back-Effekt ber