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Cell["\<\
Gegeben seien zwei Drehimpulsoperatoren, J1 und J2, die miteinander \
kommutieren. Die Eigenwertgleichungen lauten
J1^2|j1,m1> = j1(j1+1)|j1,m1>,   J1z|j1,m1> = m1|j1,m1>
 und   
 J2^2|j2,m2> = j2(j2+1)|j2,m2>,   J2z|j2,m2> = m2|j2,m2>.

Dann gibt es eine gemeinsame Eigenbasis der Operatoren 
J1^2, J1z, J2^2,J2z, die mit |j1,m1;j2,m2> 
bezeichnet werden soll. 

Auf der anderen Seite gibt es den Drehimpulsoperator J=J1+J2 mit den \
Eigenvektoren |j,m>.
Daher muss es m\[ODoubleDot]glich sein, f\[UDoubleDot]r gegebenes j1 und j2, \
die Zust\[ADoubleDot]nde |j,m> als Linearkombination von 
|j1,m1; j2,m2> zu w\[ADoubleDot]hlen welche dann Eigenzust\[ADoubleDot]nde \
von J^2 und Jz sind. Dieses Problem wird als
\"Addition von Drehimpulsen\" bezeichnet.\
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Cell["\<\
Es gilt:  m=m1+m2, da Jz = J1z+J2z.

Die folgende Routine stellt in der   m-m2-Ebene (m=m1+m2) die 
m\[ODoubleDot]glichen Zust\[ADoubleDot]nde dar, die bei der Addition von J1 \
und J2 
eingenommen werden k\[ODoubleDot]nnen (oBdA: j1>=j2).\
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Es gibt einen Zustand mit  m=j1+j2,
zwei Zust\[ADoubleDot]nde mit  m=j1+j2-1, etc.\
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Die Anzahl der Zust\[ADoubleDot]nde nimmt zu bis  m=|j1-j2| erreicht ist;
danach bleibt die Anzahl konstant bis m=-|j1-j2|.\
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Danach nimmt die Anzahl der Zust\[ADoubleDot]nde ab, bei m=-j1-j2  gibt es \
wieder nur einen Zustand.\
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Cell["\<\
Der maximale Drehimpuls, den man mit der Addition von J1 und J2 erreichen \
kann, ist j=j1+j2.\
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Betrachten wir nun die zwei Zust\[ADoubleDot]nde f\[UDoubleDot]r m=j1+j2-1:
Einer davon geh\[ODoubleDot]rt zu j=j1+j2, der andere zur maximalen \
Projektion (d.h. max. m)
f\[UDoubleDot]r den Drehimpuls j=j1+j2-1.\
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Insgesamt kann der Drehimpuls die Werte j=j1+j2 bis j=|j1-j2| mit \
Schrittweite \"1\" annehmen.\
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Startpunkt m = j1 + j2.
Das entspricht dem Zustand | j1,j1; j2,j2>   mit   j=j1+j2.
Damit ist der Zustand gleich  |j1+j2, j1+j2>.\
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Wir operieren nun wiederholt mit dem Leiteroperator  J- = J1- + J2- und \
konstruieren somit
Schritt f\[UDoubleDot]r Schritt alle Zust\[ADoubleDot]nde mit j=j1+j2.
Normierung beachten:   J-|j,m+1>  =  Sqrt[(j+m)(j-m+1)] |j,m> 
Im Folgenden: 
ket[m1,m2] == |j1,m1; j2,m2> 
KET[j,m] == |j,m> \
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Danach erniedrigen wir j um \"1\".
W\[ADoubleDot]hle den Startpunkt |j,j> so, dass er orthogonal zu allen schon \
konstruierten Zust\[ADoubleDot]nden 
mit gleichem m ist. Das wird folgendermassen erreicht:
1. W\[ADoubleDot]hle |j,j-j2; j2,j2> = ket[j-j2,j2].
2. Subtrahiere die Komponenten in die Richtungen aller bisher konstruierten \
Zust\[ADoubleDot]nde.
3. Normiere den Zustand.
Wende im n\[ADoubleDot]chsten Schritt wieder J-  an, um die \
Zust\[ADoubleDot]nde mit niedrigerem m zu konstruieren.\
\>", "Subsubsection",
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AddJ: Funktion mit Drehimpuls j1 und j2 als Argument.
            Ausgabe:  KET[j,m] == |j,m> als Linearkombination von  ket[m1,m2] \
== |j1,m1; j2,m2>
 Jm: lokale Funktion. Operator J-
 sp:  Skalarprodukt von zwei  ket[m1,m2] == |j1,m1; j2,m2> 
\
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