Skript / lecture notes - Universität Paderborn

Prof. Dr. Wolf Gero Schmidt
Universität Paderborn, Lehrstuhl für Theoretische Physik
Bem: Note: Insbesondere gilt In particular it holds
∫
Ĥ(t) = dr ˆψ + (r, t)Ĥ(r) ψ(r, ˆ t) (8.75)
Bewegungsgleichung der Feldoperatoren Equation of motion
̂= Heisenberg’sche Bewegungsgleichung Like any other operator, the field operator must
satisfy the Heisenberg equation of motion
∂ ˆψ(r, t)
]
=
[Ĥ, ˆψ(r, t)
i ∂t
− ˆψ(r,t) ˆψ(r ′ ,t)
(8.76)
∫
{ }} {
= dr ′ [ ˆψ + (r ′ , t)Ĥ(r′ ) ˆψ(r ′ , t) ˆψ(r, t) − ˆψ(r, t) ˆψ + (r ′ , t)Ĥ(r′ )ψ(r ′ , t)] (8.77)
∫
= − dr ′ [ ˆψ + (r ′ , t)Ĥ(r′ ) ˆψ(r, t) ˆψ(r ′ , t) + ˆψ(r, t)ψ + (r ′ , t)Ĥ(r′ )ψ(r ′ , t)] (8.78)
∫
= − dr ′ [ ˆψ + (r ′ , t) ˆψ(r, t) + ˆψ(r, t) ˆψ + (r ′ , t)
} {{ }
)ψ(r, t) (8.79)
δ(r−r ′ )
= − Ĥ(r) ˆψ(r, t) (8.80)
⇒ i ∂ ∂t ˆψ(r, t) = Ĥ(r) ˆψ(r, t) (8.81)
D.h. Feldoperator genügt der Schrödingergleichung, erinnert damit an Schrödingerwellenfunktion,
aber ist ein Operator, der im Bild des Besetzungszahlenformalismus wirkt.
Großer Vorteil im Vergleich zur “gewöhnlichen” Wellenfunktionen ist, daß die Statistik
des Systems über die Kommutatorbeziehungen bereits im Feldoperator korrekt enthalten
ist. Hier Fermionenstatistik, aber für Bosonen läßt sich der Formalismus ähnlich
darstellen. Thus, the field operator satisfies Schrödinger’s equation but acts upon the
occupation number representation. This is obviously an important development. Let us
consider the implications further. Since the vacuum state is an arbitrary framework and
is not used explicitly, it means that all of the spatial and time dependence of the system
now resides in the field operator whose equation of motion is the Schrödinger equation.
In fact, we have come full circle, in a sense, because the field operator in many respects
appears to be the Schrödinger wave function interpreted as an operator. The fact that it
is an operator which acts in the occupation number representation is, however, of vital
importance. The Schrödinger wave functions do not, for instance, include the statistics
of the system inherent in the cummutator relations. Furthermore, from the form of the
field operator, its action upon a system is to add or to subtract a particle to it. In case
of interacting systems this is important because by studying how the system reacts to
that extra particle a great deal may be learned about it.
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