Skript / lecture notes - Universität Paderborn

Prof. Dr. Wolf Gero Schmidt
Universität Paderborn, Lehrstuhl für Theoretische Physik
7 Greenfunktion der Einteilchen–Schrödingergleichung
Green’s Function of the Single-Particle Schrödinger Equation
In the theory of interacting systems the Green’s function, or the propagator, plays a
crucial role. In its basic definition it is a much more complex function than the ”simple”
Green’s function, familiar from the theory of partial differential equations, but many of its
properties do bear a very close relationship to the simple function. It is worthwhile, therefore,
to review the theory of Green’s functions within the framework of the Schrödinger
equation and perturbation theory.
7.1 Definition und Darstellung
Definition and Representation
Es gelte Suppose we have a partial differential equation of the general form
{Ĥ(r) − E}ψ(r) = 0 (7.1)
mit H . . . hermitescher Operator. Dann ist die zugehörige Greenfunktion definiert durch
where Ĥ(r) is a general Hermitian operator. The Green’s function is defined as the
solution of the equation
{Ĥ(r) − E}G(r, r′ ; E) = −δ(r − r ′ ) (7.2)
wobei G(r, r ′ ; E) und ψ(r) den selben Randbedingungen genügen. which also satisfies the
same boundary conditions imposed on the original problem. In other words it satisfies the
same equation and boundary conditions as the wave function ψ(r) but with an additional
”source” at an arbitrary position r ′ . It is this extra degree of freedom which makes the
Green’s function so very useful.
Darstellung mittels EV und EW von Ĥ
Green’s function in terms of the eigenvalues E n and eigenfunctions ψ n (r) of the defining
operator
{Ĥ(r) − E n}ψ n (r) = 0,
Ĥ hermitesch hermitian
⇒ {ψ n } bilden VONS form a complete set
⇒ {ψ n } können als Basis zur Darstellung von G benutzt werden may be used as a basis
to represent G
Ansatz: ansatz
G(r, r ′ ; E) = ∑ n,n ′ G n,n ′ · ψ n (r) · ψ ∗ n ′(r′ ) (7.3)
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