By Heitler W.
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Extra resources for The Quantum Theory of Radiation 2nd.Ed.
Sample text
52b) ∂t F¨ ur einen linearen Operator A ist der Mittelwert (= Erwartungswert) im Zustand ψ definiert durch −i A = d3 x ψ ∗ (x, t)Aψ(x, t) . 53) Dieser ¨ andert sich mit der Zeit wie folgt: d A = dt ∂A d3 x ψ˙ ∗ Aψ + ψ ∗ ψ + ψ ∗ Aψ˙ ∂t . 52a,b) ergibt sich daraus d i A = [H, A] + dt ∂A ∂t . 54) Bemerkungen: (i) Hermitizit¨at von H: Unter der Voraussetzung, daß die Wellenfunktionen ψ und ϕ im Unendlichen verschwinden, erh¨alt man durch zweimalige partielle Integration d3 x − 2 2m ∗ ∇2 ψ ϕ= d3 x ψ ∗ − 2 2m ∇2 ϕ .
2 2mN ∇2N + V (x1 , . . , xN ) ψ . 51) Hier bedeutet ∇i , i = 1, . . N , den Nabla-Operator bez¨ uglich xi . 6 Das Ehrenfestsche Theorem Die klassische Newtonsche Mechanik muß als Grenzfall in der Quantenmechanik enthalten sein. Wir wollen in diesem Abschnitt untersuchen, in welchem Sinn dies der Fall ist. 52a) ∂ ∗ ψ = Hψ ∗ . 52b) ∂t F¨ ur einen linearen Operator A ist der Mittelwert (= Erwartungswert) im Zustand ψ definiert durch −i A = d3 x ψ ∗ (x, t)Aψ(x, t) . 53) Dieser ¨ andert sich mit der Zeit wie folgt: d A = dt ∂A d3 x ψ˙ ∗ Aψ + ψ ∗ ψ + ψ ∗ Aψ˙ ∂t .
35) definiert. 36c) (Das Skalarprodukt ist linear im 2. Faktor und antilinear im 1. ) Weiters gilt: (ϕ, ϕ) ≥ 0 , und somit (ϕ, ϕ) = 0 ⇔ ϕ ≡ 0 . 37) Operatoren im Skalarprodukt: (ϕ, Aψ) = d3 x ϕ∗ (x)Aψ(x) . 38) Definition. A† heißt zu A adjungierter Operator“, wenn gilt ” (A† ϕ, ψ) = (ϕ, Aψ) , d. h. 39b) f¨ ur beliebige ϕ und ψ. 26 2. Wellenfunktion und Schr¨ odinger-Gleichung Definition. 40) wir schreiben dann2 auch A† = A. 39a) folgt (AB)† = B † A† . 43) und erw¨ ahnen auch die Baker-Hausdorff-Identit¨at eA Be−A = B + [A, B] + 1 2!
The Quantum Theory of Radiation 2nd.Ed. by Heitler W.
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