Hermitesche Operatoren Beispiel Essay

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Essay Successful writing and unsuccessful students

hermitescher Operator, selbstadjungierter Operator, ein Operator in einem HilbertraumH, der mit seinem adjungierten Operator identisch ist. Ein Operator in H ist eine lineare Abbildung A: D

H eines Unterraums D von H in H, sie heißt symmetrisch, wenn gilt:



für alle x,yD. Die dicht definierten symmetrischen Operatoren sind genau diejenigen, deren Graph im Graphen ihres adjungierten Operators

enthalten ist. Gilt sogar A =

, so heißt Ahermitesch oder selbstadjungiert. Auf endlichdimensionalen Hilberträumen, D = H, wird ein hermitescher Operator durch eine hermitesche Matrix dargestellt. Das Spektrum (d.h. seine Eigenwerte) eines hermiteschen Operators ist reell. In der quantentheoretischen Beschreibung physikalischer Systeme werden den Observablen hermitesche Operatoren zugeordnet, z.B. die Ortsoperatorenxa, Impulsoperatorenpa und der Hamilton-OperatorH in der Quantenmechanik.

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