Skript / lecture notes - Universität Paderborn

Prof. Dr. Wolf Gero Schmidt
Universität Paderborn, Lehrstuhl für Theoretische Physik
einsetzen in Definitionsgleichung use in definition
{Ĥ(r) − E}G(r, r′ ; E) (7.4)
= ∑ n,n ′ G n,n ′{Ĥ(r) − E}ψ n(r)ψ ∗ n ′(r′ ) (7.5)
= ∑ G n,n ′{E n − E}ψ n (r)ψn ∗ ′(r′ )
n,n
} ′ {{ }
!
=−δ(r−r ′ )= − ∑ n ψn(r)ψ∗ n (r′ )
−→
Vollständigkeit von completeness of {ψ n}
(7.6)
⇒G n,n ′ = δ nn ′
E − E n
(7.7)
⇒
G(r, r ′ ; E) = ∑ n
ψ n (r)ψ ∗ n(r ′ )
E − E n
(7.8)
Nutzen der Greenfunktion? Use of Green’s function?
Einfache Vorschrift zur Berechnung der Lösung des inhomogenen Problems Having obtained
the Green’s function, what are its uses? The most common one in the theory of
differential equations is that it tells us about the solution to the inhomogeneous problem.
If we have an equation of the form
{Ĥ(r) − E}ψ(r) = f(r) (7.9)
falls Greenfunktion des homogenen Problems bekannt where f(r) is a known function.
Provided we know the Green’s function of the homogeneous problem
{Ĥ(r) − E}G(r, r′ ; E) = −δ(r − r ′ ) (7.10)
dann gilt then ψ(r) is given by
∫
ψ(r) = −
f(r ′ )G(r, r ′ ; E) dr ′ (7.11)
Beweis Proof: Anwendung von {Ĥ − E} von links
This is easily verified by operating on both sides with (Ĥ(r) − E):
∫
{Ĥ(r) − E}ψ(r) = − {Ĥ(r) − E}G(r, r′ ; E) f(r ′ ) dr ′ (7.12)
} {{ }
−δ(r−r ′ )
∫
= f(r ′ )δ(r − r ′ ) dr ′ (7.13)
✷
= f(r) (7.14)
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