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== Kwadrat operatora momentu pędu we współrzędnych kulistych ==
Bardzo ważną wielkością jest kwadrat operatora całkowitego momentu pędu, przedstawia się on podobnie jak kwadrat operatora pędu, jako suma kwadratów współrzędnych operatora momentu pędu. Odpowiednie współrzędne operatora momentu pędu są to operatory zdefiniowane przez wzory {{LinkWzór|5.35}} (współrzędna iksowa operatora momentu pędu), {{LinkWzór|5.36}} (współrzędna igrekowa operatora momentu pędu), {{LinkWzór|5.37}} (współrzędna zetowa operatora momentu pędu), zatem nasz omawiany obiekt zdefiniujmy rozpisując go w sposób:
{{IndexWzór|<MATH>\hat{l}^2=\hat{l}^2_x+\hat{l}^2_y+\hat{l}^2_z=-\hbar^2\Bigg\{\left(y{{\partial}\over{\partial z}}-z{{\partial}\over{\partial y}}\right)^2+\left( z{{\partial}\over{\partial x}}-x{{\partial}\over{\partial z}}\right)^2+\left(x{{\partial}\over{\partial y}}-y{{\partial}\over{\partial x}}\right)^2\Bigg\}=</MATH><br><MATH>=-\hbar^2\left\{y^2{{\partial^2}\over{\partial z^2}}+z^2{{\partial^2}\over{\partial y^2}}-y{{\partial}\over{\partial z}}z{{\partial}\over{\partial y}}- z{{\partial}\over{\partial y}}y{{\partial}\over{\partial z}}+z^2{{\partial^2}\over{\partial x^2}}+x^2{{\partial^2}\over{\partial z^2}}-z{{\partial}\over{\partial x}}x{{\partial}\over{\partial z}}-x{{\partial}\over{\partial z}}z{{\partial}\over{\partial x}} \right\}+</MATH><BR>
<MATH>-\hbar^2\left\{x^2{{\partial^2}\over{\partial y^2}}+y^2{{\partial^2}\over{\partial x^2}}-x{{\partial}\over{\partial y}}y{{\partial}\over{\partial x}}-y{{\partial}\over{\partial x}}x{{\partial}\over{\partial y}}\right\}=-\hbar\left\{y^2{{\partial^2}\over{\partial z^2}}+z^2{{\partial^2}\over{\partial y^2}}-yz{{\partial^2}\over{\partial y\partial z}}-y{{\partial}\over{\partial y}}-zy{{\partial^2}\over{\partial y\partial z}}-z{{\partial}\over{\partial z}}\right\}+</MATH><BR><MATH>-\hbar\left\{ z^2{{\partial^2}\over{\partial x^2}}+x^2{{\partial^2}\over{\partial z^2}}-zx{{\partial^2}\over{\partial z\partial x}}-z{{\partial}\over{\partial z}}-xz{{\partial^2}\over{\partial x\partial z}}-x{{\partial}\over{\partial x}} \right\}-\hbar\left\{x^2{{\partial^2}\over{\partial y^2}}+y^2{{\partial^2}\over{\partial x^2}}-xy{{\partial^2}\over{\partial x\partial y}}-x{{\partial }\over{\partial x}}-yx{{\partial^2}\over{\partial x\partial y}}-y{{\partial}\over{\partial y}} \right\}=</MATH><BR><MATH>=-\hbar\left\{\left(y^2+z^2\right){{\partial^2}\over{\partial x^2}}+\left(x^2+y^2\right){{\partial^2}\over{\partial z^2}}+\left(x^2+z^2\right){{\partial^2}\over{\partial y^2}}\right\}-\hbar\left\{x^2{{\partial^2}\over{\partial x^2}}+z^2{{\partial^2}\over{\partial x^2}}+y^2{{\partial}\over{\partial y^2}}-x^2{{\partial^2}\over{\partial x^2}}-z^2{{\partial^2}\over{\partial x^2}}-y^2{{\partial}\over{\partial y^2}} \right\}+</MATH><BR><MATH>-\hbar\left\{-yz{{\partial^2}\over{\partial y\partial z}}-y{{\partial}\over{\partial y}}-zy{{\partial^2}\over{\partial y\partial z}}-z{{\partial}\over{\partial z}} \right\}-\hbar\left\{-zx{{\partial^2}\over{\partial z\partial x}}-z{{\partial}\over{\partial z}}-xz{{\partial^2}\over{\partial x\partial z}}-x{{\partial}\over{\partial x}} \right\}+</MATH><BR><MATH>-\hbar\left\{ -xy{{\partial^2}\over{\partial x\partial y}}-x{{\partial }\over{\partial x}}-yx{{\partial^2}\over{\partial x\partial y}}-y{{\partial}\over{\partial y}} \right\}=-\hbar\left\{\left(x^2+y^2+z^2\right)\Delta -x^2{{\partial^2}\over{\partial x^2}}-z^2{{\partial^2}\over{\partial x^2}}-y^2{{\partial}\over{\partial y^2}}\right\}+</MATH><BR><MATH>-\hbar\left\{-yz{{\partial^2}\over{\partial y\partial z}}-y{{\partial}\over{\partial y}}-zy{{\partial^2}\over{\partial y\partial z}}-z{{\partial}\over{\partial z}} \right\}-\hbar\left\{-zx{{\partial^2}\over{\partial z\partial x}}-z{{\partial}\over{\partial z}}-xz{{\partial^2}\over{\partial x\partial z}}-x{{\partial}\over{\partial x}} \right\}+</MATH><BR><MATH>-\hbar\left\{ -xy{{\partial^2}\over{\partial x\partial y}}-x{{\partial }\over{\partial x}}-yx{{\partial^2}\over{\partial x\partial y}}-y{{\partial}\over{\partial y}} \right\}=-\hbar\left\{(x^2+y^2+z^2)\Delta-(x{{\partial }\over{\partial x}}+y{{\partial }\over{\partial y}}+z{{\partial }\over{\partial z}})-(x{{\partial}\over{\partial x}}x{{\partial}\over{\partial x}}+y{{\partial}\over{\partial y}}y{{\partial}\over{\partial y}}+z{{\partial}\over{\partial z}}y{{\partial}\over{\partial z}})\right\}+</MATH><BR><MATH>-\hbar\left({-2xy{{\partial^2}\over{\partial x\partial y}}-2xz{{\partial^2}\over{\partial x\partial z}}-2yz{{\partial^2}\over{\partial y\partial z}}}\right)=-\hbar\left\{(x^2+y^2+z^2)\Delta-\left(x{{\partial}\over{\partial x}}+y{{\partial}\over{\partial y}}+z{{\partial}\over{\partial z}}\right)-\left(x{{\partial}\over{\partial x}}+y{{\partial}\over{\partial y}}+z{{\partial}\over{\partial z}}\right)^2\right\}</MATH>}}
: <MATH>-\hbar^2\left\{x^2{{\partial^2}\over{\partial y^2}}+y^2{{\partial^2}\over{\partial x^2}}-x{{\partial}\over{\partial y}}y{{\partial}\over{\partial x}}-y{{\partial}\over{\partial x}}x{{\partial}\over{\partial y}}\right\}=</MATH>
: <MATH>=-\hbar\left\{y^2{{\partial^2}\over{\partial z^2}}+z^2{{\partial^2}\over{\partial y^2}}-yz{{\partial^2}\over{\partial y\partial z}}-y{{\partial}\over{\partial y}}-zy{{\partial^2}\over{\partial y\partial z}}-z{{\partial}\over{\partial z}}\right\}+</MATH>
: <MATH>-\hbar\left\{ z^2{{\partial^2}\over{\partial x^2}}+x^2{{\partial^2}\over{\partial z^2}}-zx{{\partial^2}\over{\partial z\partial x}}-z{{\partial}\over{\partial z}}-xz{{\partial^2}\over{\partial x\partial z}}-x{{\partial}\over{\partial x}} \right\}+</MATH>
: <MATH>-\hbar\left\{x^2{{\partial^2}\over{\partial y^2}}+y^2{{\partial^2}\over{\partial x^2}}-xy{{\partial^2}\over{\partial x\partial y}}-x{{\partial }\over{\partial x}}-yx{{\partial^2}\over{\partial x\partial y}}-y{{\partial}\over{\partial y}} \right\}=</MATH>
: <MATH>=-\hbar\left\{\left(y^2+z^2\right){{\partial^2}\over{\partial x^2}}+\left(x^2+y^2\right){{\partial^2}\over{\partial z^2}}+\left(x^2+z^2\right){{\partial^2}\over{\partial y^2}}\right\}+</MATH>
<MATH>-\hbar\left\{x^2{{\partial^2}\over{\partial x^2}}+z^2{{\partial^2}\over{\partial x^2}}+y^2{{\partial}\over{\partial y^2}}-x^2{{\partial^2}\over{\partial x^2}}-z^2{{\partial^2}\over{\partial x^2}}-y^2{{\partial}\over{\partial y^2}} \right\}+</MATH>
: <MATH>-\hbar\left\{-yz{{\partial^2}\over{\partial y\partial z}}-y{{\partial}\over{\partial y}}-zy{{\partial^2}\over{\partial y\partial z}}-z{{\partial}\over{\partial z}} \right\}+</MATH>
: <MATH>-\hbar\left\{-zx{{\partial^2}\over{\partial z\partial x}}-z{{\partial}\over{\partial z}}-xz{{\partial^2}\over{\partial x\partial z}}-x{{\partial}\over{\partial x}} \right\}+</MATH>
: <MATH>-\hbar\left\{ -xy{{\partial^2}\over{\partial x\partial y}}-x{{\partial }\over{\partial x}}-yx{{\partial^2}\over{\partial x\partial y}}-y{{\partial}\over{\partial y}} \right\}=</MATH>
: <MATH>=-\hbar\left\{\left(x^2+y^2+z^2\right)\Delta -x^2{{\partial^2}\over{\partial x^2}}-z^2{{\partial^2}\over{\partial x^2}}-y^2{{\partial}\over{\partial y^2}}\right\}+</MATH>
: <MATH>-\hbar\left\{-yz{{\partial^2}\over{\partial y\partial z}}-y{{\partial}\over{\partial y}}-zy{{\partial^2}\over{\partial y\partial z}}-z{{\partial}\over{\partial z}} \right\}</MATH>
: <MATH>-\hbar\left\{-zx{{\partial^2}\over{\partial z\partial x}}-z{{\partial}\over{\partial z}}-xz{{\partial^2}\over{\partial x\partial z}}-x{{\partial}\over{\partial x}} \right\}+</MATH>
: <MATH>-\hbar\left\{ -xy{{\partial^2}\over{\partial x\partial y}}-x{{\partial }\over{\partial x}}-yx{{\partial^2}\over{\partial x\partial y}}-y{{\partial}\over{\partial y}} \right\}=</MATH>
: <MATH>=-\hbar\left\{(x^2+y^2+z^2)\Delta-(x{{\partial }\over{\partial x}}+y{{\partial }\over{\partial y}}+z{{\partial }\over{\partial z}})-(x{{\partial}\over{\partial x}}x{{\partial}\over{\partial x}}+y{{\partial}\over{\partial y}}y{{\partial}\over{\partial y}}+z{{\partial}\over{\partial z}}y{{\partial}\over{\partial z}})\right\}+</MATH>
: <MATH>-\hbar\left({-2xy{{\partial^2}\over{\partial x\partial y}}-2xz{{\partial^2}\over{\partial x\partial z}}-2yz{{\partial^2}\over{\partial y\partial z}}}\right)=</MATH>
: <MATH>=-\hbar\left\{(x^2+y^2+z^2)\Delta-\left(x{{\partial}\over{\partial x}}+y{{\partial}\over{\partial y}}+z{{\partial}\over{\partial z}}\right)-\left(x{{\partial}\over{\partial x}}+y{{\partial}\over{\partial y}}+z{{\partial}\over{\partial z}}\right)^2\right\}</MATH>}}
Z obliczeń powyższych dostajemy, że kwadrat operatora całkowitego momentu pędu jest zapisany przy pomocy operatora położenia <Math>\vec{r}\;</math> i operatora różniczkowania cząstkowego &nabla; i operatora &Delta;, i ten nasz operator jest równy do równoważnego powyżej przedstawienia:
{{IndexWzór|<MATH>\hat{l}^2=-\hbar^2\left\{r^2\Delta-(\vec{r}\nabla)^2-\vec{r}\nabla\right\}</MATH>|5.38}}
Linia 188 ⟶ 173:
{{IndexWzór|<MATH>\hat{l}^2=-\hbar^2\left\{r^2\Delta -(r{{\partial}\over{\partial r}})^2-r{{\partial}\over{\partial r}} \right\}=-\hbar^2\left\{
r^2\Delta-r^2{{\partial^2}\over{\partial r^2}}-r{{\partial}\over{\partial r}}-r{{\partial}\over{\partial r}}
\right\}=</MATH><br><MATH>=-\hbar\left\{ r^2\Delta-r^2{{\partial^2}\over{\partial r^2}}-2r{{\partial}\over{\partial r}}\right\}=-\hbar\left\{r^2\Delta-r{{\partial^2}\over{\partial r^2}}(r) \right\}\;</MATH>|5.40}}
Z definicji laplasjanu mamy wzór operatorowy zdefiniowanej poprzez operator &Lambda;, który z kolei jest definiowany poprzez wielkości kątowe, które to będą nam potrzebne przy definicji rozważanego tutaj operatora:
{{IndexWzór|<MATH>\Delta={{1}\over{r}}{{\partial^2}\over{\partial r^2}}(r)+{{1}\over{r^2}}\Lambda</MATH>|5.41}}
Linia 201 ⟶ 186:
 
Mając operator momentu pędu iksowy znając jego definicję we współrzędnych kartezjańskich wedle wzoru operatorowego {{LinkWzór|5.35}} i wyznaczmy czemu jest równy ten operator po podzieleniu go przez liczbę urojoną <MATH>-i\hbar\;</MATH>:
{{IndexWzór|<MATH>{{\hat{l}_x}\over{-i\hbar}}=y{{\partial}\over{\partial z}}-z{{\partial}\over{\partial y}}=\underbrace{r\sin\theta\sin\phi}_{y}\underbrace{\left[\cos\phi{{\partial}\over{\partial r}}-{{\sin\phi}\over {r}}{{\partial}\over{\partial\phi}}\right]}_{{{\partial}\over{\partial z}}}-\underbrace{r\cos\phi}_{z}\underbrace{\left[\sin\theta\sin\phi{{\partial}\over{\partial r}}+{{\cos\theta}\over{r\sin\phi}}{{\partial}\over{\partial\theta}}+{{\sin\theta\cos\phi}\over{r}}{{\partial}\over{\partial\phi}}\right]}_{{{\partial}\over{\partial y}}}=</MATH><br>
<MATH>-\underbrace{=r\sin\theta\sin\phi\cos\phi}_{z{\partial}\underbraceover{\left[partial r}}-\sin\theta\sin^2\phi{{\partial}\over{\partial r\phi}}+{{-r\cos\theta}phi\over{rsin\theta\sin\phi}}{{\partial }\over{\partial\theta r}}+-{{\sincos\thetaphi\cos\phitheta}\over{r\sin\phi}}{{\partial}\over{\partial\phitheta}}-\right]}_{cos^2\phi\sin\theta{{\partial}\over{\partial y}\phi}}=</MATH><br>
<MATH>=r\sin\theta\sin\phi\cos\phi{{\partial}\over{\partial r}}-\sin\theta\sin^2\phi{{\partial}\over{\partial\phi}}-r\cos\phi\sin\theta\sin\phi{{\partial }\over{\partial r}}-{{\cos\phi\cos\theta}\over{\sin\phi}}{{\partial}\over{\partial\theta}}-\cos^2\phi\sin\theta{{\partial}\over{\partial\phi}}=</MATH>br>
<MATH>=-\sin\theta(\sin^2\phi+\cos^2\phi){{\partial}\over{\partial\phi}}-\operatorname{ctg}\phi\cos\theta{{\partial}\over{\partial\theta}}=
-\sin\theta{{\partial}\over{\partial\phi}}-\operatorname{ctg}\phi\cos\theta{{\partial}\over{\partial\theta}}</MATH>}}
Linia 211 ⟶ 195:
 
Mając operator igrekowy momentu pędu zdefiniowanej w postaci operatorowej we współrzędnych kartezjańskich wedle {{LinkWzór|5.36}}, przedstawmy go we współrzędnych kulistych zamieniając wszystkie te współrzędne kartezjańskie oraz operatory cząstkowe zdefiniowane we współrzędnych kartezjańskich na współrzędne kuliste, napiszemy go po podzieleniu przez liczbę urojoną: <MATH>-i\hbar\;</MATH>:
{{IndexWzór|<MATH>{{\hat{l}_y}\over{-i\hbar}}=z{{\partial}\over{\partial x}}-x{{\partial}\over{\partial z}}=\underbrace{r\cos\phi}_{z}\underbrace{\left[\cos\theta\sin\phi{{\partial}\over{\partial r}}-{{\sin\theta}\over{r\sin\phi}}{{\partial}\over{\partial\theta}}+{{\cos\theta\cos\phi}\over{r}}{{\partial}\over{\partial\phi}}\right]}_{{{\partial}\over{\partial x}}}+-\underbrace{r\cos\theta\sin\phi}_{x}\underbrace{\left[\cos\phi{{\partial }\over{\partial r}}-{{\sin\phi}\over{r}}{{\partial}\over{\partial\phi}}\right]}_{{{\partial}\over{\partial z}}}=</MATH><br>
<MATH>-\underbrace{r\cos\theta\sin\phi}_{x}\underbrace{\left[\cos\phi{{\partial }\over{\partial r}}-{{\sin\phi}\over{r}}{{\partial}\over{\partial\phi}}\right]}_{{{\partial}\over{\partial z}}}=</MATH><br>
<MATH>=r\cos\theta\cos\phi\sin\phi{{\partial}\over{\partial r}}-\operatorname{ctg}\phi\sin\theta{{\partial}\over{\partial\theta}}+
\cos^2\phi\cos\theta{{\partial}\over{\partial\phi}}-
Linia 222 ⟶ 205:
 
Mając operator momentu pędu zetowy przedstawiony we współrzędnych kartezjańskich w postaci operatorowej wedle {{LinkWzór|5.37}}, w nim zamieńmy wszystkie jego współrzędne kartezjańskie i operatory pochodnych cząstkowych zdefiniowanych we współrzędnych karteziańskich na współrzędne kuliste, dalej wyznaczmy ten operator po podzieleniu go przez liczbę urojoną <MATH>-i\hbar\;</MATH>:
{{IndexWzór|<MATH>{{\hat{l}_z}\over{-i\hbar}}=x{{\partial}\over{\partial y}}-y{{\partial}\over{\partial x}}=\;</MATH><BR><MATH>=\underbrace{r\cos\theta\sin\phi}_{x}\underbrace{\left[\sin\theta\sin\phi{{\partial}\over{\partial r}}+
{{\cos\theta}\over{r\sin\phi}}{{\partial}\over{\partial\theta}}+{{\sin\theta\cos\phi}\over r}{{\partial}\over{\partial \phi}}\right]}_{{{\partial}\over{\partial y}}}-\underbrace{r\sin\theta\sin\phi}_{y}\underbrace{\left[\cos\theta\sin\phi{{\partial}\over{\partial r}}-{{\sin\theta}\over{r\sin\phi}}{{\partial}\over{\partial\theta}}+{{\cos\theta\cos\phi}\over{r}}{{\partial}\over{\partial\phi}}\right]}_{{{\partial}\over{\partial x}}}=</MATH><br><MATH>=r\cos\theta\sin\theta\sin^2\phi{{\partial}\over{\partial r}}+\cos^2\theta{{\partial}\over{\partial\theta}}+\sin\theta\cos\theta\sin\phi\cos\phi{{\partial}\over{\partial\phi}}-r\sin\theta\cos\theta\sin\phi^2{{\partial}\over{\partial r}}+\sin\theta^2{{\partial}\over{\partial\theta}}-\sin\theta\cos\theta\sin\phi\cos\phi{{\partial}\over{\partial\phi}}=</MATH><br> <MATH>=\cos^2\theta{{\partial}\over{\partial\theta}}+
<MATH>-\underbrace{r\sin\theta\sin\phi}_{y}\underbrace{\left[\cos\theta\sin\phi{{\partial}\over{\partial r}}-{{\sin\theta}\over{r\sin\phi}}{{\partial}\over{\partial\theta}}+{{\cos\theta\cos\phi}\over{r}}{{\partial}\over{\partial\phi}}\right]}_{{{\partial}\over{\partial x}}}=</MATH><br><MATH>=r\cos\theta\sin\theta\sin^2\phi{{\partial}\over{\partial r}}+\cos^2\theta{{\partial}\over{\partial\theta}}+\sin\theta\cos\theta\sin\phi\cos\phi{{\partial}\over{\partial\phi}}+</MATH><br>
<MATH>-r\sin\theta\cos\theta\sin\phi^2{{\partial}\over{\partial r}}+\sin\theta^2{{\partial}\over{\partial\theta}}-\sin\theta\cos\theta\sin\phi\cos\phi{{\partial}\over{\partial\phi}}=</MATH><br> <MATH>=\cos^2\theta{{\partial}\over{\partial\theta}}+
\sin^2\theta{{\partial}\over{\partial\theta}}
= (\cos^2\theta+\sin^2\theta){{\partial}\over{\partial\theta}} ={{\partial}\over{\partial\theta}}</MATH>}}
Linia 266 ⟶ 247:
\operatorname{ctg}\phi\cos\theta{{\partial}\over{\partial\theta}}\right]-
ii\hbar\left(-\cos\theta{{\partial}\over{\partial\phi}}+
\operatorname{ctg}\phi\sin\theta{{\partial}\over{\partial\theta}}\right)\right]=-\left[\cos\theta-i\sin\theta\right]{{\partial}\over{\partial\phi}}+\operatorname{ctg}\phi\left[\sin\theta+i\cos\theta\right]{{\partial}\over{\partial\theta}}</MATH><br>|5.55}}
<MATH>=-\left[\cos\theta-i\sin\theta\right]{{\partial}\over{\partial\phi}}+\operatorname{ctg}\phi\left[\sin\theta+i\cos\theta\right]{{\partial}\over{\partial\theta}}</MATH>|5.55}}
Operator {{LinkWzór|5.48}} we współrzędnych kulistych na postawie {{LinkWzór|5.55}} jest zdefiniowany w sposób:
{{IndexWzór|<MATH>\hat{l}_{-}=-e^{-i\theta}{{\partial}\over{\partial\phi}}+\operatorname{ctg}\phi e^{i({{\pi}\over{2}}-\theta)}{{\partial}\over{\partial\theta}}</MATH>|5.56|Obramuj}}
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