Editing Drehimpuls- Eigenzustände
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In Komponenten:<math>{{\hat{L}}_{j}}={{\varepsilon }_{jkl}}{{\hat{r}}_{k}}{{\hat{p}}_{l}}</math> | In Komponenten:<math>{{\hat{L}}_{j}}={{\varepsilon }_{jkl}}{{\hat{r}}_{k}}{{\hat{p}}_{l}}</math> | ||
<math>\hat{\bar{L}}=\hat{\bar{r}}\times \hat{\bar{p}}</math> | |||
ist hermitesch:<math>{{\hat{L}}_{j}}^{+}={{\varepsilon }_{jkl}}{{\left( {{{\hat{r}}}_{k}}{{{\hat{p}}}_{l}} \right)}^{+}}={{\varepsilon }_{jkl}}{{\hat{p}}_{l}}^{+}{{\hat{r}}_{k}}^{+}={{\varepsilon }_{jkl}}{{\hat{p}}_{l}}{{\hat{r}}_{k}}={{\varepsilon }_{jkl}}{{\hat{r}}_{k}}{{\hat{p}}_{l}}</math> | ist hermitesch:<math>{{\hat{L}}_{j}}^{+}={{\varepsilon }_{jkl}}{{\left( {{{\hat{r}}}_{k}}{{{\hat{p}}}_{l}} \right)}^{+}}={{\varepsilon }_{jkl}}{{\hat{p}}_{l}}^{+}{{\hat{r}}_{k}}^{+}={{\varepsilon }_{jkl}}{{\hat{p}}_{l}}{{\hat{r}}_{k}}={{\varepsilon }_{jkl}}{{\hat{r}}_{k}}{{\hat{p}}_{l}}</math> | ||
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<u>'''Vertauschungs- Relationen:'''</u> | <u>'''Vertauschungs- Relationen:'''</u> | ||
<math>\begin{align} | |||
& \left[ {{{\hat{L}}}_{1}},{{{\hat{L}}}_{2}} \right]=\left[ \left( {{{\hat{r}}}_{2}}{{{\hat{p}}}_{3}}-{{{\hat{r}}}_{3}}{{{\hat{p}}}_{2}} \right),\left( {{{\hat{r}}}_{3}}{{{\hat{p}}}_{1}}-{{{\hat{r}}}_{1}}{{{\hat{p}}}_{3}} \right) \right]={{{\hat{r}}}_{2}}{{{\hat{p}}}_{3}}{{{\hat{r}}}_{3}}{{{\hat{p}}}_{1}}-{{{\hat{r}}}_{2}}{{{\hat{p}}}_{3}}{{{\hat{r}}}_{1}}{{{\hat{p}}}_{3}}-{{{\hat{r}}}_{3}}{{{\hat{p}}}_{2}}{{{\hat{r}}}_{3}}{{{\hat{p}}}_{1}}+{{{\hat{r}}}_{3}}{{{\hat{p}}}_{2}}{{{\hat{r}}}_{1}}{{{\hat{p}}}_{3}}-{{{\hat{r}}}_{3}}{{{\hat{p}}}_{1}}{{{\hat{r}}}_{2}}{{{\hat{p}}}_{3}}+{{{\hat{r}}}_{3}}{{{\hat{p}}}_{1}}{{{\hat{r}}}_{3}}{{{\hat{p}}}_{2}}+{{{\hat{r}}}_{1}}{{{\hat{p}}}_{3}}{{{\hat{r}}}_{2}}{{{\hat{p}}}_{3}}-{{{\hat{r}}}_{1}}{{{\hat{p}}}_{3}}{{{\hat{r}}}_{3}}{{{\hat{p}}}_{2}} \\ | & \left[ {{{\hat{L}}}_{1}},{{{\hat{L}}}_{2}} \right]=\left[ \left( {{{\hat{r}}}_{2}}{{{\hat{p}}}_{3}}-{{{\hat{r}}}_{3}}{{{\hat{p}}}_{2}} \right),\left( {{{\hat{r}}}_{3}}{{{\hat{p}}}_{1}}-{{{\hat{r}}}_{1}}{{{\hat{p}}}_{3}} \right) \right]={{{\hat{r}}}_{2}}{{{\hat{p}}}_{3}}{{{\hat{r}}}_{3}}{{{\hat{p}}}_{1}}-{{{\hat{r}}}_{2}}{{{\hat{p}}}_{3}}{{{\hat{r}}}_{1}}{{{\hat{p}}}_{3}}-{{{\hat{r}}}_{3}}{{{\hat{p}}}_{2}}{{{\hat{r}}}_{3}}{{{\hat{p}}}_{1}}+{{{\hat{r}}}_{3}}{{{\hat{p}}}_{2}}{{{\hat{r}}}_{1}}{{{\hat{p}}}_{3}}-{{{\hat{r}}}_{3}}{{{\hat{p}}}_{1}}{{{\hat{r}}}_{2}}{{{\hat{p}}}_{3}}+{{{\hat{r}}}_{3}}{{{\hat{p}}}_{1}}{{{\hat{r}}}_{3}}{{{\hat{p}}}_{2}}+{{{\hat{r}}}_{1}}{{{\hat{p}}}_{3}}{{{\hat{r}}}_{2}}{{{\hat{p}}}_{3}}-{{{\hat{r}}}_{1}}{{{\hat{p}}}_{3}}{{{\hat{r}}}_{3}}{{{\hat{p}}}_{2}} \\ | ||
& ={{{\hat{r}}}_{2}}{{{\hat{p}}}_{3}}{{{\hat{r}}}_{3}}{{{\hat{p}}}_{1}}-{{{\hat{r}}}_{1}}{{{\hat{p}}}_{3}}{{{\hat{r}}}_{3}}{{{\hat{p}}}_{2}}+{{{\hat{r}}}_{3}}{{{\hat{p}}}_{2}}{{{\hat{r}}}_{1}}{{{\hat{p}}}_{3}}-{{{\hat{r}}}_{3}}{{{\hat{p}}}_{1}}{{{\hat{r}}}_{2}}{{{\hat{p}}}_{3}}={{{\hat{r}}}_{2}}{{{\hat{p}}}_{3}}{{{\hat{r}}}_{3}}{{{\hat{p}}}_{1}}-{{{\hat{r}}}_{2}}{{{\hat{r}}}_{3}}{{{\hat{p}}}_{3}}{{{\hat{p}}}_{1}}+{{{\hat{r}}}_{1}}{{{\hat{r}}}_{3}}{{{\hat{p}}}_{3}}{{{\hat{p}}}_{2}}-{{{\hat{r}}}_{1}}{{{\hat{p}}}_{3}}{{{\hat{r}}}_{3}}{{{\hat{p}}}_{2}} \\ | & ={{{\hat{r}}}_{2}}{{{\hat{p}}}_{3}}{{{\hat{r}}}_{3}}{{{\hat{p}}}_{1}}-{{{\hat{r}}}_{1}}{{{\hat{p}}}_{3}}{{{\hat{r}}}_{3}}{{{\hat{p}}}_{2}}+{{{\hat{r}}}_{3}}{{{\hat{p}}}_{2}}{{{\hat{r}}}_{1}}{{{\hat{p}}}_{3}}-{{{\hat{r}}}_{3}}{{{\hat{p}}}_{1}}{{{\hat{r}}}_{2}}{{{\hat{p}}}_{3}}={{{\hat{r}}}_{2}}{{{\hat{p}}}_{3}}{{{\hat{r}}}_{3}}{{{\hat{p}}}_{1}}-{{{\hat{r}}}_{2}}{{{\hat{r}}}_{3}}{{{\hat{p}}}_{3}}{{{\hat{p}}}_{1}}+{{{\hat{r}}}_{1}}{{{\hat{r}}}_{3}}{{{\hat{p}}}_{3}}{{{\hat{p}}}_{2}}-{{{\hat{r}}}_{1}}{{{\hat{p}}}_{3}}{{{\hat{r}}}_{3}}{{{\hat{p}}}_{2}} \\ | ||
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'''Vertauschungsrelationen<math>'''\left[ {{{\hat{L}}}_{+}},{{{\hat{L}}}_{3}} \right]=\left[ {{{\hat{L}}}_{1}},{{{\hat{L}}}_{3}} \right]+i\left[ {{{\hat{L}}}_{2}},{{{\hat{L}}}_{3}} \right]=-i\hbar {{\hat{L}}_{2}}-\hbar {{\hat{L}}_{1}}=-\hbar \left( {{{\hat{L}}}_{1}}+i{{{\hat{L}}}_{2}} \right)</math> | '''Vertauschungsrelationen<math>'''\left[ {{{\hat{L}}}_{+}},{{{\hat{L}}}_{3}} \right]=\left[ {{{\hat{L}}}_{1}},{{{\hat{L}}}_{3}} \right]+i\left[ {{{\hat{L}}}_{2}},{{{\hat{L}}}_{3}} \right]=-i\hbar {{\hat{L}}_{2}}-\hbar {{\hat{L}}_{1}}=-\hbar \left( {{{\hat{L}}}_{1}}+i{{{\hat{L}}}_{2}} \right)</math> | ||
<math>\begin{align} | |||
& \left[ {{{\hat{L}}}_{+}},{{{\hat{L}}}_{3}} \right]=-\hbar {{{\hat{L}}}_{+}} \\ | & \left[ {{{\hat{L}}}_{+}},{{{\hat{L}}}_{3}} \right]=-\hbar {{{\hat{L}}}_{+}} \\ | ||
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<math>{{\hat{L}}^{2}}={{\hat{L}}_{1}}^{2}+{{\hat{L}}_{2}}^{2}+{{\hat{L}}_{3}}^{2}={{\hat{L}}_{3}}^{2}+{{\hat{L}}_{+}}{{\hat{L}}_{-}}-\hbar {{\hat{L}}_{3}}</math> | |||
Warum ? | Warum ? | ||
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gehorchen den Eigenwertgleichungen<math>{{\hat{L}}^{2}}\left| a,b \right\rangle =a\left| a,b \right\rangle </math> | gehorchen den Eigenwertgleichungen<math>{{\hat{L}}^{2}}\left| a,b \right\rangle =a\left| a,b \right\rangle </math> | ||
<math>{{\hat{L}}_{3}}\left| a,b \right\rangle =b\left| a,b \right\rangle </math> | |||
Prinzipiell: Für alle Observablen müssen wir Quantenzahlen einführen. Zum formalen Vorgehen schreibt man diese Quantenzahlen einfach in einen Zustandsvektor. Diese Quantenzahlen sind Eigenwerte der Observablen, also mögliche Messwerte. Unser formaler Zustand aus Quantenzahlen ist per Definition ein Eigenvektor zu diesen Quantenzahlen. | Prinzipiell: Für alle Observablen müssen wir Quantenzahlen einführen. Zum formalen Vorgehen schreibt man diese Quantenzahlen einfach in einen Zustandsvektor. Diese Quantenzahlen sind Eigenwerte der Observablen, also mögliche Messwerte. Unser formaler Zustand aus Quantenzahlen ist per Definition ein Eigenvektor zu diesen Quantenzahlen. | ||
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'''0''' | '''0''' | ||
'''0''' | '''0''' | ||
<math>\frac{1}{2}</math> | |||
<math>\hbar \sqrt{\frac{3}{4}}</math> | <math>\hbar \sqrt{\frac{3}{4}}</math> | ||
<math>-\frac{1}{2},+\frac{1}{2}</math> | <math>-\frac{1}{2},+\frac{1}{2}</math> | ||
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<math>-1,0,1</math> | <math>-1,0,1</math> | ||
<math>\frac{3}{2}</math> | |||
<math>\hbar \sqrt{\frac{15}{4}}</math> | <math>\hbar \sqrt{\frac{15}{4}}</math> | ||
<math>-\frac{3}{2},-\frac{1}{2},\frac{1}{2},\frac{3}{2}</math> | <math>-\frac{3}{2},-\frac{1}{2},\frac{1}{2},\frac{3}{2}</math> | ||
<math>\begin{align} | |||
& {{{\hat{L}}}^{2}}\left| l,m \right\rangle ={{\hbar }^{2}}l(l+1)\left| l,m \right\rangle \\ | & {{{\hat{L}}}^{2}}\left| l,m \right\rangle ={{\hbar }^{2}}l(l+1)\left| l,m \right\rangle \\ |