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Addition von Drehimpulsen
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<noinclude>{{Scripthinweis|Quantenmechanik|4|4}}</noinclude> Der Gesamtdrehimpuls kann folgendermaßen dargestellt werden: :<math>\hat{\bar{J}}=\hat{\bar{L}}+\hat{\bar{S}}</math> Die Vertauschungsrelationen: :<math>\left[ {{{\hat{L}}}_{j}},{{{\hat{S}}}_{k}} \right]=0</math> Beide Operatoren wirken in verschiedenen Räumen. Wäre der Operator nicht Null, so wären die zugehörigen Eigenzustände nicht separabel. :<math>\begin{align} & \Rightarrow \left[ {{{\hat{J}}}_{j}},{{{\hat{J}}}_{k}} \right]=\left[ {{{\hat{L}}}_{j}},{{{\hat{L}}}_{k}} \right]+\left[ {{{\hat{S}}}_{j}},{{{\hat{S}}}_{k}} \right] \\ & \left[ {{{\hat{L}}}_{j}},{{{\hat{L}}}_{k}} \right]=i\hbar {{\varepsilon }_{jkl}}{{{\hat{L}}}_{l}} \\ & \left[ {{{\hat{S}}}_{j}},{{{\hat{S}}}_{k}} \right]=i\hbar {{\varepsilon }_{jkl}}{{{\hat{S}}}_{l}} \\ & \Rightarrow \left[ {{{\hat{J}}}_{j}},{{{\hat{J}}}_{k}} \right]=i\hbar {{\varepsilon }_{jkl}}{{{\hat{J}}}_{l}} \\ \end{align}</math> Drehimpuls Vertauschungsrelationen! :<math>\left[ {{{\hat{J}}}^{2}},{{{\hat{L}}}_{3}} \right]=\left[ {{{\hat{L}}}^{2}}+{{{\hat{\bar{S}}}}^{2}}+2\hat{\bar{L}}\cdot \hat{\bar{S}},{{{\hat{L}}}_{3}} \right]=2{{\hat{\bar{S}}}_{j}}\left[ {{{\hat{L}}}_{j}},{{{\hat{L}}}_{3}} \right]=2i\hbar \left( {{{\hat{S}}}_{2}}{{{\hat{L}}}_{1}}-{{{\hat{S}}}_{1}}{{{\hat{L}}}_{2}} \right)\ne 0</math> Ebenso: :<math>\left[ {{{\hat{J}}}^{2}},{{{\hat{S}}}_{3}} \right]\ne 0</math> Also: Die <math>2(2l+1)</math> Produktzustände <math>\left| lm{{m}_{S}} \right\rangle =\left| lm \right\rangle \left| {{m}_{s}} \right\rangle </math> sind Eigenzustände zu <math>{{\hat{L}}^{2}},{{\hat{L}}_{3}},{{\hat{\bar{S}}}^{2}},{{\hat{S}}_{3}}</math> aber nicht zu <math>{{\hat{J}}^{2}}</math>, da <math>\left[ {{{\hat{J}}}^{2}},{{{\hat{L}}}_{3}} \right]\ne 0</math> bzw. <math>\left[ {{{\hat{J}}}^{2}},{{{\hat{S}}}_{3}} \right]\ne 0</math> '''Ziel: Suche gemeinsame Eigenzustände zu '''<math>{{\hat{J}}^{2}}</math> , <math>{{\hat{J}}_{3}}</math> , <math>{{\hat{L}}^{2}},{{\hat{\bar{S}}}^{2}}</math> . Dies muss möglich sein, da :<math>\begin{align} & \left[ {{{\hat{J}}}^{2}},{{{\hat{L}}}^{2}} \right]=\left[ {{{\hat{L}}}^{2}}+{{{\hat{\bar{S}}}}^{2}}+2\hat{\bar{L}}\cdot \hat{\bar{S}},{{{\hat{L}}}^{2}} \right]=0 \\ & \left[ {{{\hat{J}}}^{2}},{{{\hat{\bar{S}}}}^{2}} \right]=\left[ {{{\hat{L}}}^{2}}+{{{\hat{\bar{S}}}}^{2}}+2\hat{\bar{L}}\cdot \hat{\bar{S}},{{{\hat{\bar{S}}}}^{2}} \right]=0 \\ & \left[ {{{\hat{J}}}_{3}},{{{\hat{L}}}^{2}} \right]=\left[ {{{\hat{L}}}_{3}}+{{{\hat{\bar{S}}}}_{3}},{{{\hat{L}}}^{2}} \right]=0 \\ & \left[ {{{\hat{J}}}_{3}},{{{\hat{\bar{S}}}}^{2}} \right]=\left[ {{{\hat{L}}}_{3}}+{{{\hat{\bar{S}}}}_{3}},{{{\hat{\bar{S}}}}^{2}} \right]=0 \\ \end{align}</math> Die Eigenwertgleichungen lauten: :<math>\begin{align} & {{{\hat{J}}}^{2}}\left| j{{m}_{j}}ls \right\rangle ={{\hbar }^{2}}(j(j+1))\left| j{{m}_{j}}ls \right\rangle \\ & {{{\hat{J}}}_{3}}\left| j{{m}_{j}}ls \right\rangle =\hbar {{m}_{j}}\left| j{{m}_{j}}ls \right\rangle \\ & {{{\hat{L}}}^{2}}\left| j{{m}_{j}}ls \right\rangle ={{\hbar }^{2}}(l(l+1)\left| j{{m}_{j}}ls \right\rangle \\ & {{{\hat{\bar{S}}}}^{2}}\left| j{{m}_{j}}ls \right\rangle ={{\hbar }^{2}}(s(s+1)\left| j{{m}_{j}}ls \right\rangle \\ \end{align}</math> Durch Einschub eines Vollständigen Satzes orthonormierter Eigenfunktionen, durch Einschub eines Projektors auf diesen vollständigen atz, also durch Einschub einer "1" kann der neue Eigenzustand <math>\left| j{{m}_{j}}ls \right\rangle </math> bezüglich des alten Zustandes <math>\left| lms{{m}_{s}} \right\rangle </math> entwickelt werden: :<math>\left| j{{m}_{j}}ls \right\rangle =\sum\limits_{\begin{smallmatrix} m \\ {{m}_{S}}={{m}_{j}}-m \end{smallmatrix}}{{}}\left| lms{{m}_{s}} \right\rangle \left\langle lms{{m}_{s}} | j{{m}_{j}}ls \right\rangle </math> Zu beachten ist: Es wird ausschließlich über die Komponenten der alten Basis summiert, die sich von der neuen Basis unterscheiden (das heißt: Nur dieser Teil der Basis wird transformiert)! Dabei heißen die Entwicklungskoeffizienten der neuen Basis bezüglich der alten Basisvektoren, also die Koordinaten der neuen Basis in der alten Basis {{FB|Clebsch-Gordan-Koeffizienten}}! :<math>\left\langle lms{{m}_{s}} | j{{m}_{j}}ls \right\rangle </math> Dabei gilt: {| class="wikitable" border="1" |-! <math>s=\frac{1}{2}</math>!!<math>{{m}_{s}}=\frac{1}{2}</math>!!<math>{{m}_{s}}=-\frac{1}{2}</math> |- |<math>j=l+\frac{1}{2}</math>||<math>{{\left( \frac{l+{{m}_{j}}+\frac{1}{2}}{2l+1} \right)}^{\frac{1}{2}}}</math>||<math>{{\left( \frac{l-{{m}_{j}}+\frac{1}{2}}{2l+1} \right)}^{\frac{1}{2}}}</math> |- |<math>j=l-\frac{1}{2}</math>||<math>-{{\left( \frac{l-{{m}_{j}}+\frac{1}{2}}{2l+1} \right)}^{\frac{1}{2}}}</math>||<math>{{\left( \frac{l+{{m}_{j}}+\frac{1}{2}}{2l+1} \right)}^{\frac{1}{2}}}</math> |} Wobei: :<math>\begin{align} & j=l\pm \frac{1}{2} \\ & {{m}_{j}}=m+{{m}_{S}} \\ & m=-l,...,+l \\ & {{m}_{S}}=-\frac{1}{2},+\frac{1}{2} \\ \end{align}</math>
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