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L.2 Drehimpulse in Kugelkoordinaten
Hier werden die Drehimpulsoperatoren definiert:
operLx := - I hbar (Sin[\[Phi]] D[#1, \[Theta]] + Cot[\[Theta]] Cos[\[Phi]] D[#1, \[Phi]]) &;
operLy := - I hbar (Cos[\[Phi]] D[#1, \[Theta]] - Cot[\[Theta]] Sin[\[Phi]] D[#1, \[Phi]]) &;
operLz := - I hbar D[#1, \[Phi]] &;
x2 ist dann
operLx[operLx[\[Psi][r, \[Theta], \[Phi]]]]
Simplify[%]
-I hbar (-I hbar Cos[\[Phi]] Cot[\[Theta]] (-Cot[\[Theta]] Sin[\[Phi]]
(\[Psi]^(0,0,1))[r,\[Theta],\[Phi]]+Cos[\[Phi]] Cot[\[Theta]]
(\[Psi]^(0,0,2))[r,\[Theta],\[Phi]]+Cos[\[Phi]]
(\[Psi]^(0,1,0))[r,\[Theta],\[Phi]]+Sin[\[Phi]] (\[Psi]^(0,1,1))[r,\[Theta],\[Phi]])-I
hbar Sin[\[Phi]] (-Cos[\[Phi]] Csc[\[Theta]]^2
(\[Psi]^(0,0,1))[r,\[Theta],\[Phi]]+Cos[\[Phi]] Cot[\[Theta]]
(\[Psi]^(0,1,1))[r,\[Theta],\[Phi]]+Sin[\[Phi]] (\[Psi]^(0,2,0))[r,\[Theta],\[Phi]]))
hbar^2 (-(1/4) (3+Cos[2 \[Theta]]) Csc[\[Theta]]^2 Sin[2 \[Phi]]
(\[Psi]^(0,0,1))[r,\[Theta],\[Phi]]+Cos[\[Phi]]^2 Cot[\[Theta]]^2
(\[Psi]^(0,0,2))[r,\[Theta],\[Phi]]+Cos[\[Phi]]^2 Cot[\[Theta]]
(\[Psi]^(0,1,0))[r,\[Theta],\[Phi]]+Cot[\[Theta]] Sin[2 \[Phi]]
(\[Psi]^(0,1,1))[r,\[Theta],\[Phi]]+Sin[\[Phi]]^2 (\[Psi]^(0,2,0))[r,\[Theta],\[Phi]])
operLy[operLy[\[Psi][r,\[Theta],\[Phi]]]]
Simplify[%]
-I hbar (I hbar Cot[\[Theta]] Sin[\[Phi]] (-Cos[\[Phi]] Cot[\[Theta]]
(\[Psi]^(0,0,1))[r,\[Theta],\[Phi]]-Cot[\[Theta]] Sin[\[Phi]]
(\[Psi]^(0,0,2))[r,\[Theta],\[Phi]]-Sin[\[Phi]]
(\[Psi]^(0,1,0))[r,\[Theta],\[Phi]]+Cos[\[Phi]] (\[Psi]^(0,1,1))[r,\[Theta],\[Phi]])-I hbar
Cos[\[Phi]] (Csc[\[Theta]]^2 Sin[\[Phi]] (\[Psi]^(0,0,1))[r,\[Theta],\[Phi]]-Cot[\[Theta]]
Sin[\[Phi]] (\[Psi]^(0,1,1))[r,\[Theta],\[Phi]]+Cos[\[Phi]]
(\[Psi]^(0,2,0))[r,\[Theta],\[Phi]]))
-hbar^2 (1/4 (3+Cos[2 \[Theta]]) Csc[\[Theta]]^2 Sin[2 \[Phi]]
(\[Psi]^(0,0,1))[r,\[Theta],\[Phi]]+Cot[\[Theta]]^2 Sin[\[Phi]]^2
(\[Psi]^(0,0,2))[r,\[Theta],\[Phi]]+Cot[\[Theta]] Sin[\[Phi]]^2
(\[Psi]^(0,1,0))[r,\[Theta],\[Phi]]-2 Cos[\[Phi]] Cot[\[Theta]] Sin[\[Phi]]
(\[Psi]^(0,1,1))[r,\[Theta],\[Phi]]+Cos[\[Phi]]^2 (\[Psi]^(0,2,0))[r,\[Theta],\[Phi]])
und
operLz[operLz[\[Psi][r,\[Theta],\[Phi]]]]
-hbar^2 (\[Psi]^(0,0,2))[r,\[Theta],\[Phi]]
Damit wird 
2
operLx[operLx[\[Psi][r,\[Theta],\[Phi]]]]+operLy[operLy[\[Psi][r,\[Theta],\[Phi]]]]
+operLz[operLz[\[Psi][r,\[Theta],\[Phi]]]]
-hbar^2 (\[Psi]^(0,0,2))[r,\[Theta],\[Phi]]-I hbar (I hbar Cot[\[Theta]] Sin[\[Phi]]
(-Cos[\[Phi]] Cot[\[Theta]] (\[Psi]^(0,0,1))[r,\[Theta],\[Phi]]-Cot[\[Theta]] Sin[\[Phi]]
(\[Psi]^(0,0,2))[r,\[Theta],\[Phi]]-Sin[\[Phi]]
(\[Psi]^(0,1,0))[r,\[Theta],\[Phi]]+Cos[\[Phi]] (\[Psi]^(0,1,1))[r,\[Theta],\[Phi]])-I
hbar Cos[\[Phi]] (Csc[\[Theta]]^2 Sin[\[Phi]]
(\[Psi]^(0,0,1))[r,\[Theta],\[Phi]]-Cot[\[Theta]] Sin[\[Phi]]
(\[Psi]^(0,1,1))[r,\[Theta],\[Phi]]+Cos[\[Phi]] (\[Psi]^(0,2,0))[r,\[Theta],\[Phi]]))-I
hbar (-I hbar Cos[\[Phi]] Cot[\[Theta]] (-Cot[\[Theta]] Sin[\[Phi]]
(\[Psi]^(0,0,1))[r,\[Theta],\[Phi]]+Cos[\[Phi]] Cot[\[Theta]]
(\[Psi]^(0,0,2))[r,\[Theta],\[Phi]]+Cos[\[Phi]]
(\[Psi]^(0,1,0))[r,\[Theta],\[Phi]]+Sin[\[Phi]] (\[Psi]^(0,1,1))[r,\[Theta],\[Phi]])-I
hbar Sin[\[Phi]] (-Cos[\[Phi]] Csc[\[Theta]]^2
(\[Psi]^(0,0,1))[r,\[Theta],\[Phi]]+Cos[\[Phi]] Cot[\[Theta]]
(\[Psi]^(0,1,1))[r,\[Theta],\[Phi]]+Sin[\[Phi]] (\[Psi]^(0,2,0))[r,\[Theta],\[Phi]]))
Simplify[%]
-hbar^2 (Csc[\[Theta]]^2 (\[Psi]^(0,0,2))[r,\[Theta],\[Phi]]+Cot[\[Theta]]
(\[Psi]^(0,1,0))[r,\[Theta],\[Phi]]+(\[Psi]^(0,2,0))[r,\[Theta],\[Phi]])
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©2005-2019 Ulm University, Othmar
Marti, released under the Creative Commons License cc-by-sa
4.0 Lizenzinformationen