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5. Operatoralgebra


[A, B] = AB - BA (5.1)
{A, B} = AB + BA (5.2)
$\displaystyle \left[\vphantom{A,B}\right.$A, B$\displaystyle \left.\vphantom{A,B}\right]^{\dagger}_{}$ = $\displaystyle \left[\vphantom{B^\dagger, A^\dagger}\right.$B$\scriptstyle \dagger$, A$\scriptstyle \dagger$$\displaystyle \left.\vphantom{B^\dagger, A^\dagger}\right]$ (5.3)
$\displaystyle \left[\vphantom{A,B}\right.$A, B$\displaystyle \left.\vphantom{A,B}\right]$ = - $\displaystyle \left[\vphantom{B,A}\right.$B, A$\displaystyle \left.\vphantom{B,A}\right]$ (5.4)
$\displaystyle \left[\vphantom{AB,C}\right.$AB, C$\displaystyle \left.\vphantom{AB,C}\right]$ = A[B, C] + [A, C]B (5.5)
$\displaystyle \left[\vphantom{A,BC}\right.$A, BC$\displaystyle \left.\vphantom{A,BC}\right]$ = B[A, C] + [A, B]C (5.6)
$\displaystyle \left[\vphantom{\left[A,B\right],C}\right.$$\displaystyle \left[\vphantom{A,B}\right.$A, B$\displaystyle \left.\vphantom{A,B}\right]$, C$\displaystyle \left.\vphantom{\left[A,B\right],C}\right]$ = {A,{B, C}} - {B,{A, C}} (5.7)
$\displaystyle \left[\vphantom{AB,CD}\right.$AB, CD$\displaystyle \left.\vphantom{AB,CD}\right]$ = AC[B, D] + A[B, C]D + C[A, D]B + [AC]DB (5.8)
$\displaystyle \left[\vphantom{AB,CD}\right.$AB, CD$\displaystyle \left.\vphantom{AB,CD}\right]$ = A{A, C}D - AC{B, D} - C{A, D}B + {C, A}DB (5.9)
eAt = $\displaystyle \sum_{n=0}^{\infty}$$\displaystyle {(At)^n \over n!}$ (5.10)
eAeB = eA + Be$\scriptstyle \einhalb$[A, B] (5.11)


next up previous contents
Next: 6. Quantenmechanik Up: Formelsammlung zur Theoretischen Physik Previous: 4.7 Fouriereihen   Inhalt
Alexander Wagner
2000-04-14