Mechanika kwantowa/Postulat pierwszy mechaniki kwantowej: Różnice pomiędzy wersjami

Usunięta treść Dodana treść
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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\leftBigg\{ \left(x^2+y^2+z^2\right)\Delta-\left(x{{\partial }\over{\partial x}}+y{{\partial }\over{\partial y}}+z{{\partial }\over{\partial z}}\right)+\;</MATH><BR><MATH>-\left(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)\rightBigg\}+</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)=\;</MATH><BR><MATH>=-\hbar\left\{\left(x^2+y^2+z^2\right)\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}}