A T-mátrix és a szórási amplitúdó kapcsolata
Vezessünk be egy hullámfüggvényt a közegbeli szórásra,
ψ
k
(
q
)
MathType@MTEF@5@5@+=feaagCart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbdfgBPjMCPbqefmuyTjMCPfgarmqr1ngBPrgitLxBI9gBamrr1ngBPrwtHrhAYaqeguuDJXwAKbstHrhAGq1DVbWefv3ySLgznfgDOfdarCqr1ngBPrginfgDObYtUvgarqqr1ngBPrgifHhDYfgasaacH8qrps0lbbf9q8WrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0RYxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaacaGacmGadaWaaiqacaabauaagaaakeaacqaHipqEdaWgaaWcbaGaaC4AaaqabaGcdaqadaqaaiaahghaaiaawIcacaGLPaaaaaa@4E76@
-val analóg mennyiséget a közegre, ez legyen
Γ
(
k
,
k
'
,
K
,
z
)
=
∫
d
3
k
(
2
π
)
3
v
(
q
)
χ
(
k
−
q
,
k
'
,
K
,
z
)
MathType@MTEF@5@5@+=feaagCart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbdfgBPjMCPbqefmuyTjMCPfgarmqr1ngBPrgitLxBI9gBamrr1ngBPrwtHrhAYaqeguuDJXwAKbstHrhAGq1DVbWefv3ySLgznfgDOfdarCqr1ngBPrginfgDObYtUvgarqqr1ngBPrgifHhDYfgasaacH8qrps0lbbf9q8WrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0RYxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaacaGacmGadaWaaiqacaabauaagaaakeaacqqHtoWrdaqadaqaaiaahUgacaGGSaGaaC4AaiaacEcacaGGSaGaaC4saiaacYcacaWG6baacaGLOaGaayzkaaGaeyypa0Zaa8qaaeaadaWcaaqaaiaadsgadaahaaWcbeqaaiaaiodaaaGccaWGRbaabaWaaeWaaeaacaaIYaGaeqiWdahacaGLOaGaayzkaaWaaWbaaSqabeaacaaIZaaaaaaakiaadAhadaqadaqaaiaahghaaiaawIcacaGLPaaacqaHhpWydaqadaqaaiaahUgacqGHsislcaWHXbGaaiilaiaahUgacaGGNaGaaiilaiaahUeacaGGSaGaamOEaaGaayjkaiaawMcaaaWcbeqab0Gaey4kIipaaaa@6C51@
χ
(
k
,
k
'
,
K
,
z
)
=
(
2
π
)
3
δ
(
k
−
k
'
)
+
F
(
+
)
(
K
,
k
)
z
−
2
(
e
k
−
μ
)
∫
d
3
q
(
2
π
)
3
v
(
q
)
χ
(
k
−
q
,
k
'
,
K
,
z
)
MathType@MTEF@5@5@+=feaagCart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbdfgBPjMCPbqefmuyTjMCPfgarmqr1ngBPrgitLxBI9gBamrr1ngBPrwtHrhAYaqeguuDJXwAKbstHrhAGq1DVbWefv3ySLgznfgDOfdarCqr1ngBPrginfgDObYtUvgarqqr1ngBPrgifHhDYfgasaacH8qrps0lbbf9q8WrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0RYxir=Jbba9q8aq0=yq=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@8991@
T = 0
MathType@MTEF@5@5@+=feaagCart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbdfgBPjMCPbqefmuyTjMCPfgarmqr1ngBPrgitLxBI9gBamrr1ngBPrwtHrhAYaqeguuDJXwAKbstHrhAGq1DVbWefv3ySLgznfgDOfdarCqr1ngBPrginfgDObYtUvgarqqr1ngBPrgifHhDYfgasaacH8qrps0lbbf9q8WrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0RYxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaacaGacmGadaWaaiqacaabauaagaaakeaacaWGubGaeyypa0JaaGimaaaa@4B93@
esetén a diagram
T
=
0
MathType@MTEF@5@5@+=feaagCart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbdfgBPjMCPbqefmuyTjMCPfgarmqr1ngBPrgitLxBI9gBamrr1ngBPrwtHrhAYaqeguuDJXwAKbstHrhAGq1DVbWefv3ySLgznfgDOfdarCqr1ngBPrginfgDObYtUvgarqqr1ngBPrgifHhDYfgasaacH8qrps0lbbf9q8WrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0RYxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaacaGacmGadaWaaiqacaabauaagaaakeaacaWGubGaeyypa0JaaGimaaaa@4B94@
esete, ekkor
F
(
+
)
(
K
,
k
)
=
1
MathType@MTEF@5@5@+=feaagCart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbdfgBPjMCPbqefmuyTjMCPfgarmqr1ngBPrgitLxBI9gBamrr1ngBPrwtHrhAYaqeguuDJXwAKbstHrhAGq1DVbWefv3ySLgznfgDOfdarCqr1ngBPrginfgDObYtUvgarqqr1ngBPrgifHhDYfgasaacH8qrps0lbbf9q8WrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0RYxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaacaGacmGadaWaaiqacaabauaagaaakeaacaWGgbWaaSbaaSqaamaabmaabaGaey4kaScacaGLOaGaayzkaaaabeaakmaabmaabaGaaC4saiaacYcacaWHRbaacaGLOaGaayzkaaGaeyypa0JaaGymaaaa@5229@
.Ekkor
χ
MathType@MTEF@5@5@+=feaagCart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbdfgBPjMCPbqefmuyTjMCPfgarmqr1ngBPrgitLxBI9gBamrr1ngBPrwtHrhAYaqeguuDJXwAKbstHrhAGq1DVbWefv3ySLgznfgDOfdarCqr1ngBPrginfgDObYtUvgarqqr1ngBPrgifHhDYfgasaacH8qrps0lbbf9q8WrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0RYxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaacaGacmGadaWaaiqacaabauaagaaakeaacqaHhpWyaaa@4AB2@
az alábbi egyenletnek tesz eleget:
χ
0
(
k
,
k
'
,
K
,
z
)
=
(
2
π
)
3
δ
(
k
−
k
'
)
+
1
z
−
2
(
e
k
−
μ
)
∫
d
3
q
(
2
π
)
3
v
(
q
)
χ
0
(
k
−
q
,
k
'
,
K
,
z
)
MathType@MTEF@5@5@+=feaagCart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbdfgBPjMCPbqefmuyTjMCPfgarmqr1ngBPrgitLxBI9gBamrr1ngBPrwtHrhAYaqeguuDJXwAKbstHrhAGq1DVbWefv3ySLgznfgDOfdarCqr1ngBPrginfgDObYtUvgarqqr1ngBPrgifHhDYfgasaacH8qrps0lbbf9q8WrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0RYxir=Jbba9q8aq0=yq=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@84BF@
(4)
Γ
0
(
k
,
k
'
,
K
,
z
)
=
∫
d
3
q
(
2
π
)
3
v
(
q
)
χ
(
k
−
q
,
k
'
,
K
,
z
)
MathType@MTEF@5@5@+=feaagCart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbdfgBPjMCPbqefmuyTjMCPfgarmqr1ngBPrgitLxBI9gBamrr1ngBPrwtHrhAYaqeguuDJXwAKbstHrhAGq1DVbWefv3ySLgznfgDOfdarCqr1ngBPrginfgDObYtUvgarqqr1ngBPrgifHhDYfgasaacH8qrps0lbbf9q8WrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0RYxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaacaGacmGadaWaaiqacaabauaagaaakeaacqqHtoWrdaWgaaWcbaGaaGimaaqabaGcdaqadaqaaiaahUgacaGGSaGaaC4AaiaacEcacaGGSaGaaC4saiaacYcacaWG6baacaGLOaGaayzkaaGaeyypa0Zaa8qaaeaadaWcaaqaaiaadsgadaahaaWcbeqaaiaaiodaaaGccaWGXbaabaWaaeWaaeaacaaIYaGaeqiWdahacaGLOaGaayzkaaWaaWbaaSqabeaacaaIZaaaaaaakiaadAhadaqadaqaaiaahghaaiaawIcacaGLPaaacqaHhpWydaqadaqaaiaahUgacqGHsislcaWHXbGaaiilaiaahUgacaGGNaGaaiilaiaahUeacaGGSaGaamOEaaGaayjkaiaawMcaaaWcbeqab0Gaey4kIipaaaa@6D47@
Az (5)-ös egyenletből:
z
−
2
(
e
k
−
μ
)
χ
0
(
k
,
k
'
,
K
,
z
)
−
∫
d
3
q
(
2
π
)
3
v
(
q
)
χ
0
(
k
−
q
,
k
'
,
K
,
z
)
=
(
2
π
)
3
[
z
−
2
(
e
k
−
μ
)
]
δ
(
k
−
k
1
)
MathType@MTEF@5@5@+=feaagCart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbdfgBPjMCPbqefmuyTjMCPfgarmqr1ngBPrgitLxBI9gBamrr1ngBPrwtHrhAYaqeguuDJXwAKbstHrhAGq1DVbWefv3ySLgznfgDOfdarCqr1ngBPrginfgDObYtUvgarqqr1ngBPrgifHhDYfgasaacH8qrps0lbbf9q8WrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0RYxir=Jbba9q8aq0=yq=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@8F29@
(5a)
(
2
e
k
1
−
2
e
k
+
i
η
)
ψ
k
1
(
k
)
−
∫
d
3
q
(
2
π
)
3
v
(
q
)
ψ
k
1
(
k
−
q
)
=
(
2
π
)
3
[
2
e
k
1
−
2
e
k
+
i
η
]
δ
(
k
−
k
'
)
MathType@MTEF@5@5@+=feaagCart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbdfgBPjMCPbqefmuyTjMCPfgarmqr1ngBPrgitLxBI9gBamrr1ngBPrwtHrhAYaqeguuDJXwAKbstHrhAGq1DVbWefv3ySLgznfgDOfdarCqr1ngBPrginfgDObYtUvgarqqr1ngBPrgifHhDYfgasaacH8qrps0lbbf9q8WrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0RYxir=Jbba9q8aq0=yq=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@8C18@
ez abból jött, hogy még előző órán
ψ
k
(
q
)
=
(
2
π
)
3
δ
(
k
−
q
)
−
1
q
2
−
k
2
−
i
η
∫
d
3
p
(
2
π
)
3
V
(
p
)
ψ
k
(
q
−
p
)
MathType@MTEF@5@5@+=feaagCart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbdfgBPjMCPbqefmuyTjMCPfgarmqr1ngBPrgitLxBI9gBamrr1ngBPrwtHrhAYaqeguuDJXwAKbstHrhAGq1DVbWefv3ySLgznfgDOfdarCqr1ngBPrginfgDObYtUvgarqqr1ngBPrgifHhDYfgasaacH8qrps0lbbf9q8WrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0RYxir=Jbba9q8aq0=yq=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@7904@
Ha most
k
≠
k
1
MathType@MTEF@5@5@+=feaagCart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbdfgBPjMCPbqefmuyTjMCPfgarmqr1ngBPrgitLxBI9gBamrr1ngBPrwtHrhAYaqeguuDJXwAKbstHrhAGq1DVbWefv3ySLgznfgDOfdarCqr1ngBPrginfgDObYtUvgarqqr1ngBPrgifHhDYfgasaacH8qrps0lbbf9q8WrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0RYxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaacaGacmGadaWaaiqacaabauaagaaakeaacaWHRbGaeyiyIKRaaC4AamaaBaaaleaacaaIXaaabeaaaaa@4D91@
, akkor
ψ
k
1
∗
(
k
)
MathType@MTEF@5@5@+=feaagCart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbdfgBPjMCPbqefmuyTjMCPfgarmqr1ngBPrgitLxBI9gBamrr1ngBPrwtHrhAYaqeguuDJXwAKbstHrhAGq1DVbWefv3ySLgznfgDOfdarCqr1ngBPrginfgDObYtUvgarqqr1ngBPrgifHhDYfgasaacH8qrps0lbbf9q8WrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0RYxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaacaGacmGadaWaaiqacaabauaagaaakeaacqaHipqEdaqhaaWcbaGaaC4AamaaBaaameaacaaIXaaabeaaaSqaaiabgEHiQaaakmaabmaabaGaaC4AaaGaayjkaiaawMcaaaaa@5053@
-vel szorozva (5a)-t, majd
∫
d
3
k
(
2
π
)
3
MathType@MTEF@5@5@+=feaagCart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbdfgBPjMCPbqefmuyTjMCPfgarmqr1ngBPrgitLxBI9gBamrr1ngBPrwtHrhAYaqeguuDJXwAKbstHrhAGq1DVbWefv3ySLgznfgDOfdarCqr1ngBPrginfgDObYtUvgarqqr1ngBPrgifHhDYfgasaacH8qrps0lbbf9q8WrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0RYxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaacaGacmGadaWaaiqacaabauaagaaakeaadaWdbaqaamaalaaabaGaamizamaaCaaaleqabaGaaG4maaaakiaadUgaaeaadaqadaqaaiaaikdacqaHapaCaiaawIcacaGLPaaadaahaaWcbeqaaiaaiodaaaaaaaqabeqaniabgUIiYdaaaa@52B4@
-val kiintegrálva kapjuk az (5b) egyenletet:
∫
d
3
k
(
2 π
)
3
[
z − 2 (
e
k
− μ
)
]
χ
0
(
k , k ' , K , z
)
ψ
k
1
∗
(
k
)
−
∫
d
3
k
(
2 π
)
3
∫
d
3
q
(
2 π
)
3
v (
q
)
χ
0
(
k − q , k ' , K , z
)
ψ
k
1
∗
(
k
)
=
= [
z − 2 (
e
k '
− μ
)
]
ψ
k
1
∗
(
k '
)
MathType@MTEF@5@5@+=feaagCart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbdfgBPjMCPbqefmuyTjMCPfgarmqr1ngBPrgitLxBI9gBamrr1ngBPrwtHrhAYaqeguuDJXwAKbstHrhAGq1DVbWefv3ySLgznfgDOfdarCqr1ngBPrginfgDObYtUvgarqqr1ngBPrgifHhDYfgasaacH8qrps0lbbf9q8WrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0RYxir=Jbba9q8aq0=yq=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@B142@
(5b)
A bal oldal második tagjában térjünk át
k
"
=
k
−
q
MathType@MTEF@5@5@+=feaagCart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbdfgBPjMCPbqefmuyTjMCPfgarmqr1ngBPrgitLxBI9gBamrr1ngBPrwtHrhAYaqeguuDJXwAKbstHrhAGq1DVbWefv3ySLgznfgDOfdarCqr1ngBPrginfgDObYtUvgarqqr1ngBPrgifHhDYfgasaacH8qrps0lbbf9q8WrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0RYxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaacaGacmGadaWaaiqacaabauaagaaakeaacaWHRbGaaiOiaiabg2da9iaahUgacqGHsislcaWHXbaaaa@4E76@
szerinti integrálásra, ekkor
∫
d
3
k
(
2
π
)
3
∫
d
3
q
(
2
π
)
3
v
(
q
)
χ
0
(
k
−
q
,
k
'
,
K
,
z
)
ψ
k
1
∗
(
k
)
=
−
∫
d
3
k
"
(
2
π
)
3
χ
0
(
k
"
,
k
'
,
K
,
z
)
⋅
∫
d
3
q
(
2
π
)
3
v
(
q
)
ψ
k
1
∗
(
k
"
+
q
)
︸
[
2
e
k
1
−
2
e
k
"
−
i
η
]
ψ
k
1
∗
(
k
"
)
MathType@MTEF@5@5@+=feaagCart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbdfgBPjMCPbqefmuyTjMCPfgarmqr1ngBPrgitLxBI9gBamrr1ngBPrwtHrhAYaqeguuDJXwAKbstHrhAGq1DVbWefv3ySLgznfgDOfdarCqr1ngBPrginfgDObYtUvgarqqr1ngBPrgifHhDYfgasaacH8qrps0lbbf9q8WrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0RYxir=Jbba9q8aq0=yq=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@BCA7@
ahol a kapcsos kifejezéshez elvégeztünk egy
q
→
−
q
MathType@MTEF@5@5@+=feaagCart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbdfgBPjMCPbqefmuyTjMCPfgarmqr1ngBPrgitLxBI9gBamrr1ngBPrwtHrhAYaqeguuDJXwAKbstHrhAGq1DVbWefv3ySLgznfgDOfdarCqr1ngBPrginfgDObYtUvgarqqr1ngBPrgifHhDYfgasaacH8qrps0lbbf9q8WrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0RYxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaacaGacmGadaWaaiqacaabauaagaaakeaacaWHXbGaeyOKH4QaeyOeI0IaaCyCaaaa@4DC9@
trafót. Mivel
v
MathType@MTEF@5@5@+=feaagCart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbdfgBPjMCPbqefmuyTjMCPfgarmqr1ngBPrgitLxBI9gBamrr1ngBPrwtHrhAYaqeguuDJXwAKbstHrhAGq1DVbWefv3ySLgznfgDOfdarCqr1ngBPrginfgDObYtUvgarqqr1ngBPrgifHhDYfgasaacH8qrps0lbbf9q8WrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0RYxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaacaGacmGadaWaaiqacaabauaagaaakeaacaWG2baaaa@49F6@
szimmetrikus, így az marad maga.
Ekkor az (5b) egyenlet:
∫
d
3
k
(
2
π
)
3
[
z
−
2
(
e
k
1
−
μ
)
+
i
η
]
χ
0
(
k
,
k
'
,
K
,
z
)
ψ
k
1
∗
(
k
)
=
[
z
−
2
(
e
k
'
−
μ
)
]
ψ
k
1
∗
(
k
'
)
MathType@MTEF@5@5@+=feaagCart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbdfgBPjMCPbqefmuyTjMCPfgarmqr1ngBPrgitLxBI9gBamrr1ngBPrwtHrhAYaqeguuDJXwAKbstHrhAGq1DVbWefv3ySLgznfgDOfdarCqr1ngBPrginfgDObYtUvgarqqr1ngBPrgifHhDYfgasaacH8qrps0lbbf9q8WrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0RYxir=Jbba9q8aq0=yq=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@88A8@
∫
d
3
k
(
2
π
)
3
χ
0
(
k
,
k
'
,
K
,
z
)
ψ
k
1
∗
(
k
)
=
(
z
−
2
e
k
'
+
2
μ
)
ψ
k
1
∗
(
k
'
)
z
−
2
e
k
1
+
2
μ
MathType@MTEF@5@5@+=feaagCart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbdfgBPjMCPbqefmuyTjMCPfgarmqr1ngBPrgitLxBI9gBamrr1ngBPrwtHrhAYaqeguuDJXwAKbstHrhAGq1DVbWefv3ySLgznfgDOfdarCqr1ngBPrginfgDObYtUvgarqqr1ngBPrgifHhDYfgasaacH8qrps0lbbf9q8WrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0RYxir=Jbba9q8aq0=yq=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@8131@
ahol
η
→
0
MathType@MTEF@5@5@+=feaagCart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbdfgBPjMCPbqefmuyTjMCPfgarmqr1ngBPrgitLxBI9gBamrr1ngBPrwtHrhAYaqeguuDJXwAKbstHrhAGq1DVbWefv3ySLgznfgDOfdarCqr1ngBPrginfgDObYtUvgarqqr1ngBPrgifHhDYfgasaacH8qrps0lbbf9q8WrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0RYxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaacaGacmGadaWaaiqacaabauaagaaakeaacqaH3oaAcqGHsgIRcaaIWaaaaa@4D4E@
határátmenetet elvégeztük, azaz
η
=
0
MathType@MTEF@5@5@+=feaagCart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbdfgBPjMCPbqefmuyTjMCPfgarmqr1ngBPrgitLxBI9gBamrr1ngBPrwtHrhAYaqeguuDJXwAKbstHrhAGq1DVbWefv3ySLgznfgDOfdarCqr1ngBPrginfgDObYtUvgarqqr1ngBPrgifHhDYfgasaacH8qrps0lbbf9q8WrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0RYxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaacaGacmGadaWaaiqacaabauaagaaakeaacqaH3oaAcqGH9aqpcaaIWaaaaa@4C67@
-t behelyettesítettünk (
z
MathType@MTEF@5@5@+=feaagCart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbdfgBPjMCPbqefmuyTjMCPfgarmqr1ngBPrgitLxBI9gBamrr1ngBPrwtHrhAYaqeguuDJXwAKbstHrhAGq1DVbWefv3ySLgznfgDOfdarCqr1ngBPrginfgDObYtUvgarqqr1ngBPrgifHhDYfgasaacH8qrps0lbbf9q8WrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0RYxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaacaGacmGadaWaaiqacaabauaagaaakeaacaWG6baaaa@49FA@
úgyis tartalmaz még egy képzetes részt a Matsubara-frekvencia miatt). Használjuk fel a
∫
d
3
q
(
2
π
)
3
ψ
q
(
k
1
)
ψ
q
∗
(
k
2
)
=
(
2
π
)
3
⋅
δ
(
k
1
−
k
2
)
MathType@MTEF@5@5@+=feaagCart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbdfgBPjMCPbqefmuyTjMCPfgarmqr1ngBPrgitLxBI9gBamrr1ngBPrwtHrhAYaqeguuDJXwAKbstHrhAGq1DVbWefv3ySLgznfgDOfdarCqr1ngBPrginfgDObYtUvgarqqr1ngBPrgifHhDYfgasaacH8qrps0lbbf9q8WrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0RYxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaacaGacmGadaWaaiqacaabauaagaaakeaadaWdbaqaamaalaaabaGaamizamaaCaaaleqabaGaaG4maaaakiaadghaaeaadaqadaqaaiaaikdacqaHapaCaiaawIcacaGLPaaadaahaaWcbeqaaiaaiodaaaaaaOGaeqiYdK3aaSbaaSqaaiaahghaaeqaaOWaaeWaaeaacaWHRbWaaSbaaSqaaiaaigdaaeqaaaGccaGLOaGaayzkaaGaeqiYdK3aa0baaSqaaiaahghaaeaacqGHxiIkaaGcdaqadaqaaiaahUgadaWgaaWcbaGaaGOmaaqabaaakiaawIcacaGLPaaaaSqabeqaniabgUIiYdGccqGH9aqpdaqadaqaaiaaikdacqaHapaCaiaawIcacaGLPaaadaahaaWcbeqaaiaaiodaaaGccqGHflY1cqaH0oazdaqadaqaaiaahUgadaWgaaWcbaGaaGymaaqabaGccqGHsislcaWHRbWaaSbaaSqaaiaaikdaaeqaaaGccaGLOaGaayzkaaaaaa@70CE@
összefüggést, azaz integráljuk mindkét oldalt
∫
ψ
k
1
(
k
"
)
d
3
k
1
(
2
π
)
3
MathType@MTEF@5@5@+=feaagCart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbdfgBPjMCPbqefmuyTjMCPfgarmqr1ngBPrgitLxBI9gBamrr1ngBPrwtHrhAYaqeguuDJXwAKbstHrhAGq1DVbWefv3ySLgznfgDOfdarCqr1ngBPrginfgDObYtUvgarqqr1ngBPrgifHhDYfgasaacH8qrps0lbbf9q8WrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0RYxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaacaGacmGadaWaaiqacaabauaagaaakeaadaWdbaqaaiabeI8a5naaBaaaleaacaWHRbWaaSbaaWqaaiaaigdaaeqaaaWcbeaakmaabmaabaGaaC4AaiaackcaaiaawIcacaGLPaaadaWcaaqaaiaadsgadaahaaWcbeqaaiaaiodaaaGccaWGRbWaaSbaaSqaaiaaigdaaeqaaaGcbaWaaeWaaeaacaaIYaGaeqiWdahacaGLOaGaayzkaaWaaWbaaSqabeaacaaIZaaaaaaaaeqabeqdcqGHRiI8aaaa@5AB3@
szerint:
χ
0
(
k
"
,
k
'
,
K
,
z
)
=
(
z
−
2
e
k
'
+
2
μ
)
∫
d
3
k
1
(
2
π
)
3
ψ
k
1
∗
(
k
'
)
ψ
k
1
(
k
)
z
−
2
e
k
1
+
2
μ
MathType@MTEF@5@5@+=feaagCart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbdfgBPjMCPbqefmuyTjMCPfgarmqr1ngBPrgitLxBI9gBamrr1ngBPrwtHrhAYaqeguuDJXwAKbstHrhAGq1DVbWefv3ySLgznfgDOfdarCqr1ngBPrginfgDObYtUvgarqqr1ngBPrgifHhDYfgasaacH8qrps0lbbf9q8WrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0RYxir=Jbba9q8aq0=yq=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@81CE@
Ne felejtsük, hogy
ψ
k
1
∗
(
k
'
)
=
(
2
π
)
3
δ
(
k
'
−
k
1
)
+
f
∗
˜
(
k
1
,
k
'
)
2
e
k
1
−
2
e
k
'
−
i
η
MathType@MTEF@5@5@+=feaagCart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbdfgBPjMCPbqefmuyTjMCPfgarmqr1ngBPrgitLxBI9gBamrr1ngBPrwtHrhAYaqeguuDJXwAKbstHrhAGq1DVbWefv3ySLgznfgDOfdarCqr1ngBPrginfgDObYtUvgarqqr1ngBPrgifHhDYfgasaacH8qrps0lbbf9q8WrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0RYxir=Jbba9q8aq0=yq=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@73CD@
ezt beírva és elvégezve az integrálást, majd parciális törtekre való bontást:
χ
0
(
k
"
,
k
'
,
K
,
z
)
=
ψ
k
'
(
k
"
)
+
∫
d
3
k
1
(
2
π
)
3
[
1
2
e
k
1
−
2
e
k
'
−
i
η
+
1
z
−
2
e
k
+
2
μ
]
ψ
k
1
(
k
"
)
⋅
f
˜
∗
(
k
1
,
k
)
MathType@MTEF@5@5@+=feaagCart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbdfgBPjMCPbqefmuyTjMCPfgarmqr1ngBPrgitLxBI9gBamrr1ngBPrwtHrhAYaqeguuDJXwAKbstHrhAGq1DVbWefv3ySLgznfgDOfdarCqr1ngBPrginfgDObYtUvgarqqr1ngBPrgifHhDYfgasaacH8qrps0lbbf9q8WrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0RYxir=Jbba9q8aq0=yq=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@9127@
Mindkét oldalt
∫
d
3
k
(
2
π
)
3
v
(
k
−
k
"
)
MathType@MTEF@5@5@+=feaagCart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbdfgBPjMCPbqefmuyTjMCPfgarmqr1ngBPrgitLxBI9gBamrr1ngBPrwtHrhAYaqeguuDJXwAKbstHrhAGq1DVbWefv3ySLgznfgDOfdarCqr1ngBPrginfgDObYtUvgarqqr1ngBPrgifHhDYfgasaacH8qrps0lbbf9q8WrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0RYxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaacaGacmGadaWaaiqacaabauaagaaakeaadaWdbaqaamaalaaabaGaamizamaaCaaaleqabaGaaG4maaaakiaadUgaaeaadaqadaqaaiaaikdacqaHapaCaiaawIcacaGLPaaadaahaaWcbeqaaiaaiodaaaaaaOGaamODamaabmaabaGaaC4AaiabgkHiTiaahUgacaGGIaaacaGLOaGaayzkaaaaleqabeqdcqGHRiI8aaaa@58C8@
szerint integrálva:
Γ
0
(
k
,
k
'
,
K
,
z
)
=
f
˜
(
k
,
k
'
)
+
∫
d
3
q
(
2
π
)
3
[
1
2
e
q
−
2
e
k
'
−
i
η
+
1
z
−
2
e
q
+
2
μ
]
f
˜
(
k
,
q
)
⋅
f
˜
∗
(
k
'
,
q
)
MathType@MTEF@5@5@+=feaagCart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbdfgBPjMCPbqefmuyTjMCPfgarmqr1ngBPrgitLxBI9gBamrr1ngBPrwtHrhAYaqeguuDJXwAKbstHrhAGq1DVbWefv3ySLgznfgDOfdarCqr1ngBPrginfgDObYtUvgarqqr1ngBPrgifHhDYfgasaacH8qrps0lbbf9q8WrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0RYxir=Jbba9q8aq0=yq=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@8CE5@
T
>
0
MathType@MTEF@5@5@+=feaagCart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbdfgBPjMCPbqefmuyTjMCPfgarmqr1ngBPrgitLxBI9gBamrr1ngBPrwtHrhAYaqeguuDJXwAKbstHrhAGq1DVbWefv3ySLgznfgDOfdarCqr1ngBPrginfgDObYtUvgarqqr1ngBPrgifHhDYfgasaacH8qrps0lbbf9q8WrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0RYxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaacaGacmGadaWaaiqacaabauaagaaakeaacaWGubGaeyOpa4JaaGimaaaa@4B96@
esetén a (4) -es egyenlet mindkét oldaláról levonunk
Γ
(
k
,
k
'
,
K
,
z
)
z
−
2
e
k
+
2
μ
MathType@MTEF@5@5@+=feaagCart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbdfgBPjMCPbqefmuyTjMCPfgarmqr1ngBPrgitLxBI9gBamrr1ngBPrwtHrhAYaqeguuDJXwAKbstHrhAGq1DVbWefv3ySLgznfgDOfdarCqr1ngBPrginfgDObYtUvgarqqr1ngBPrgifHhDYfgasaacH8qrps0lbbf9q8WrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0RYxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaacaGacmGadaWaaiqacaabauaagaaakeaadaWcaaqaaiabfo5ahnaabmaabaGaaC4AaiaacYcacaWHRbGaai4jaiaacYcacaWHlbGaaiilaiaadQhaaiaawIcacaGLPaaaaeaacaWG6bGaeyOeI0IaaGOmaiaadwgadaWgaaWcbaGaaC4AaaqabaGccqGHRaWkcaaIYaGaeqiVd0gaaaaa@5A82@
-t, majd beírjuk
Γ
MathType@MTEF@5@5@+=feaagCart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbdfgBPjMCPbqefmuyTjMCPfgarmqr1ngBPrgitLxBI9gBamrr1ngBPrwtHrhAYaqeguuDJXwAKbstHrhAGq1DVbWefv3ySLgznfgDOfdarCqr1ngBPrginfgDObYtUvgarqqr1ngBPrgifHhDYfgasaacH8qrps0lbbf9q8WrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0RYxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaacaGacmGadaWaaiqacaabauaagaaakeaacqqHtoWraaa@4A63@
definícióját:
χ (
k , k ' , K , z
) −
1
z − 2
e
k
+ 2 μ
∫
d
3
q
(
2 π
)
3
v (
q
) χ (
k − q , k ' , K , z
)
=
=
(
2 π
)
3
δ (
k − k '
) +
F
(
+
)
(
K , k
) − 1
z − 2 (
e
k
− μ
)
Γ (
k , k ' , K , z
)
MathType@MTEF@5@5@+=feaagCart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbdfgBPjMCPbqefmuyTjMCPfgarmqr1ngBPrgitLxBI9gBamrr1ngBPrwtHrhAYaqeguuDJXwAKbstHrhAGq1DVbWefv3ySLgznfgDOfdarCqr1ngBPrginfgDObYtUvgarqqr1ngBPrgifHhDYfgasaacH8qrps0lbbf9q8WrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0RYxir=Jbba9q8aq0=yq=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@9F7B@
ekkor
χ
MathType@MTEF@5@5@+=feaagCart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbdfgBPjMCPbqefmuyTjMCPfgarmqr1ngBPrgitLxBI9gBamrr1ngBPrwtHrhAYaqeguuDJXwAKbstHrhAGq1DVbWefv3ySLgznfgDOfdarCqr1ngBPrginfgDObYtUvgarqqr1ngBPrgifHhDYfgasaacH8qrps0lbbf9q8WrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0RYxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaacaGacmGadaWaaiqacaabauaagaaakeaacqaHhpWyaaa@4AB2@
-t felírva, mint
χ
(
k
,
k
'
,
K
,
z
)
=
∫
d
3
q
2
π
3
(
2
π
)
3
δ
(
k
−
q
)
⋅
χ
(
q
,
k
'
,
K
,
z
)
MathType@MTEF@5@5@+=feaagCart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbdfgBPjMCPbqefmuyTjMCPfgarmqr1ngBPrgitLxBI9gBamrr1ngBPrwtHrhAYaqeguuDJXwAKbstHrhAGq1DVbWefv3ySLgznfgDOfdarCqr1ngBPrginfgDObYtUvgarqqr1ngBPrgifHhDYfgasaacH8qrps0lbbf9q8WrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0RYxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaacaGacmGadaWaaiqacaabauaagaaakeaacqaHhpWydaqadaqaaiaahUgacaGGSaGaaC4AaiaacEcacaGGSaGaaC4saiaacYcacaWG6baacaGLOaGaayzkaaGaeyypa0Zaa8qaaeaadaWcaaqaaiaadsgadaahaaWcbeqaaiaaiodaaaGccaWGXbaabaGaaGOmaiabec8aWnaaCaaaleqabaGaaG4maaaaaaGcdaqadaqaaiaaikdacqaHapaCaiaawIcacaGLPaaadaahaaWcbeqaaiaaiodaaaGccqaH0oazdaqadaqaaiaahUgacqGHsislcaWHXbaacaGLOaGaayzkaaGaeyyXICTaeq4Xdm2aaeWaaeaacaWHXbGaaiilaiaahUgacaGGNaGaaiilaiaahUeacaGGSaGaamOEaaGaayjkaiaawMcaaaWcbeqab0Gaey4kIipaaaa@7307@
∫
d
3
q
(
2 π
)
3
[
(
2 π
)
3
δ (
k − k '
) −
v (
k − q
)
z − 2
e
k
+ 2 μ
] χ (
q , k ' , K , z
)
=
=
∫
d
3
k
(
2 π
)
3
[
(
2 π
)
3
δ (
k − q
) −
v (
k − q
)
z − 2
e
k
+ 2 μ
]
χ
0
(
k , k " , K , z
)
=
=
(
2 π
)
3
δ (
q − k "
)
MathType@MTEF@5@5@+=feaagCart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbdfgBPjMCPbqefmuyTjMCPfgarmqr1ngBPrgitLxBI9gBamrr1ngBPrwtHrhAYaqeguuDJXwAKbstHrhAGq1DVbWefv3ySLgznfgDOfdarCqr1ngBPrginfgDObYtUvgarqqr1ngBPrgifHhDYfgasaacH8qrps0lbbf9q8WrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0RYxir=Jbba9q8aq0=yq=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@B831@
Integrálva mindkét oldalt
∫
d
3
k
(
2
π
)
3
χ
0
(
k
"
,
k
,
K
,
z
)
MathType@MTEF@5@5@+=feaagCart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbdfgBPjMCPbqefmuyTjMCPfgarmqr1ngBPrgitLxBI9gBamrr1ngBPrwtHrhAYaqeguuDJXwAKbstHrhAGq1DVbWefv3ySLgznfgDOfdarCqr1ngBPrginfgDObYtUvgarqqr1ngBPrgifHhDYfgasaacH8qrps0lbbf9q8WrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0RYxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaacaGacmGadaWaaiqacaabauaagaaakeaadaWdbaqaamaalaaabaGaamizamaaCaaaleqabaGaaG4maaaakiaadUgaaeaadaqadaqaaiaaikdacqaHapaCaiaawIcacaGLPaaadaahaaWcbeqaaiaaiodaaaaaaOGaeq4Xdm2aaSbaaSqaaiaaicdaaeqaaOWaaeWaaeaacaWHRbGaaiOiaiaacYcacaWHRbGaaiilaiaahUeacaGGSaGaamOEaaGaayjkaiaawMcaaaWcbeqab0Gaey4kIipaaaa@5D6A@
szerint:
χ
(
k
"
,
k
'
,
K
,
z
)
=
χ
0
(
k
"
,
k
'
,
K
,
z
)
+
∫
d
3
k
(
2
π
)
3
χ
0
(
k
"
,
k
,
K
,
z
)
F
(
+
)
(
K
,
k
)
−
1
z
−
2
(
e
k
−
μ
)
Γ
(
k
,
k
'
,
K
,
z
)
MathType@MTEF@5@5@+=feaagCart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbdfgBPjMCPbqefmuyTjMCPfgarmqr1ngBPrgitLxBI9gBamrr1ngBPrwtHrhAYaqeguuDJXwAKbstHrhAGq1DVbWefv3ySLgznfgDOfdarCqr1ngBPrginfgDObYtUvgarqqr1ngBPrgifHhDYfgasaacH8qrps0lbbf9q8WrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0RYxir=Jbba9q8aq0=yq=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@906E@
Most integrálva
∫
d
3
k
(
2
π
)
3
v
(
k
−
k
"
)
MathType@MTEF@5@5@+=feaagCart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbdfgBPjMCPbqefmuyTjMCPfgarmqr1ngBPrgitLxBI9gBamrr1ngBPrwtHrhAYaqeguuDJXwAKbstHrhAGq1DVbWefv3ySLgznfgDOfdarCqr1ngBPrginfgDObYtUvgarqqr1ngBPrgifHhDYfgasaacH8qrps0lbbf9q8WrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0RYxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaacaGacmGadaWaaiqacaabauaagaaakeaadaWdbaqaamaalaaabaGaamizamaaCaaaleqabaGaaG4maaaakiaadUgaaeaadaqadaqaaiaaikdacqaHapaCaiaawIcacaGLPaaadaahaaWcbeqaaiaaiodaaaaaaOGaamODamaabmaabaGaaC4AaiabgkHiTiaahUgacaGGIaaacaGLOaGaayzkaaaaleqabeqdcqGHRiI8aaaa@58C8@
szerint:
Γ
(
k
,
k
'
,
K
,
z
)
=
Γ
0
(
k
,
k
'
,
K
,
z
)
+
∫
d
3
q
(
2
π
)
3
Γ
0
(
k
,
q
,
K
,
z
)
F
(
+
)
(
K
,
k
)
−
1
z
−
2
(
e
q
−
μ
)
Γ
(
q
,
k
'
,
K
,
z
)
MathType@MTEF@5@5@+=feaagCart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbdfgBPjMCPbqefmuyTjMCPfgarmqr1ngBPrgitLxBI9gBamrr1ngBPrwtHrhAYaqeguuDJXwAKbstHrhAGq1DVbWefv3ySLgznfgDOfdarCqr1ngBPrginfgDObYtUvgarqqr1ngBPrgifHhDYfgasaacH8qrps0lbbf9q8WrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0RYxir=Jbba9q8aq0=yq=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@8DA7@
Alacsony energiás szórásnál, azaz ha
|
k
a
|
≪
1
MathType@MTEF@5@5@+=feaagCart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbdfgBPjMCPbqefmuyTjMCPfgarmqr1ngBPrgitLxBI9gBamrr1ngBPrwtHrhAYaqeguuDJXwAKbstHrhAGq1DVbWefv3ySLgznfgDOfdarCqr1ngBPrginfgDObYtUvgarqqr1ngBPrgifHhDYfgasaacH8qrps0lbbf9q8WrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0RYxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaacaGacmGadaWaaiqacaabauaagaaakeaadaabdaqaaiaahUgacaWGHbaacaGLhWUaayjcSdGaeSOAI0JaaGymaaaa@500C@
:
f
˜
(
k
,
k
'
)
≈
4
π
ℏ
2
a
m
[
1
+
O
(
|
k
a
|
)
]
⇒
Γ
0
(
k
,
k
'
,
K
,
z
)
≈
Γ
0
(
0,0,0,0
)
=
4
π
ℏ
2
a
m
⇒
Γ
(
k
,
k
'
,
K
,
z
)
≈
Γ
(
0,0,0,0
)
=
4
π
ℏ
2
a
m
MathType@MTEF@5@5@+=feaagCart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbdfgBPjMCPbqefmuyTjMCPfgarmqr1ngBPrgitLxBI9gBamrr1ngBPrwtHrhAYaqeguuDJXwAKbstHrhAGq1DVbWefv3ySLgznfgDOfdarCqr1ngBPrginfgDObYtUvgarqqr1ngBPrgifHhDYfgasaacH8qrps0lbbf9q8WrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0RYxir=Jbba9q8aq0=yq=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@9E32@
v
(
k
)
←
Γ
(
0,0,0,0
)
=
4
π
ℏ
2
a
/
m
MathType@MTEF@5@5@+=feaagCart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbdfgBPjMCPbqefmuyTjMCPfgarmqr1ngBPrgitLxBI9gBamrr1ngBPrwtHrhAYaqeguuDJXwAKbstHrhAGq1DVbWefv3ySLgznfgDOfdarCqr1ngBPrginfgDObYtUvgarqqr1ngBPrgifHhDYfgasaacH8qrps0lbbf9q8WrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0RYxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaacaGacmGadaWaaiqacaabauaagaaakeaacaWG2bWaaeWaaeaacaWHRbaacaGLOaGaayzkaaGaeyiKHWQaeu4KdC0aaeWaaeaacaaIWaGaaiilaiaaicdacaGGSaGaaGimaiaacYcacaaIWaaacaGLOaGaayzkaaGaeyypa0JaaGinaiabec8aWjabl+qiOnaaCaaaleqabaGaaGOmaaaakiaadggacaGGVaGaamyBaaaa@5E6D@
,
v
(
r
)
=
4
π
ℏ
2
a
m
δ
(
r
)
MathType@MTEF@5@5@+=feaagCart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbdfgBPjMCPbqefmuyTjMCPfgarmqr1ngBPrgitLxBI9gBamrr1ngBPrwtHrhAYaqeguuDJXwAKbstHrhAGq1DVbWefv3ySLgznfgDOfdarCqr1ngBPrginfgDObYtUvgarqqr1ngBPrgifHhDYfgasaacH8qrps0lbbf9q8WrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0RYxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaacaGacmGadaWaaiqacaabauaagaaakeaacaWG2bWaaeWaaeaacaWHYbaacaGLOaGaayzkaaGaeyypa0ZaaSaaaeaacaaI0aGaeqiWdaNaeS4dHG2aaWbaaSqabeaacaaIYaaaaOGaamyyaaqaaiaad2gaaaGaeqiTdq2aaeWaaeaacaWHYbaacaGLOaGaayzkaaaaaa@5828@
Az utolsó tagját a diagramnak lecseréljük a következőre:
Azt szoktuk mondani, hogy
a
>
0
MathType@MTEF@5@5@+=feaagCart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbdfgBPjMCPbqefmuyTjMCPfgarmqr1ngBPrgitLxBI9gBamrr1ngBPrwtHrhAYaqeguuDJXwAKbstHrhAGq1DVbWefv3ySLgznfgDOfdarCqr1ngBPrginfgDObYtUvgarqqr1ngBPrgifHhDYfgasaacH8qrps0lbbf9q8WrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0RYxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaacaGacmGadaWaaiqacaabauaagaaakeaacaWGHbGaeyOpa4JaaGimaaaa@4BA3@
esetén egy kölcsönhatás taszító,
a
<
0
MathType@MTEF@5@5@+=feaagCart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbdfgBPjMCPbqefmuyTjMCPfgarmqr1ngBPrgitLxBI9gBamrr1ngBPrwtHrhAYaqeguuDJXwAKbstHrhAGq1DVbWefv3ySLgznfgDOfdarCqr1ngBPrginfgDObYtUvgarqqr1ngBPrgifHhDYfgasaacH8qrps0lbbf9q8WrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0RYxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaacaGacmGadaWaaiqacaabauaagaaakeaacaWGHbGaeyipaWJaaGimaaaa@4B9F@
esetén viszont vonzó. Ez a hétköznapi képünknek nem teljesen fog megfelelni. Ennek árnyalásához tekintsük a következőket.
Alakrezonancia
Ez már egy igen egyszerű potenciál,
erre a S. egyenletet is meg tudjuk oldani.
A Schrödinger egyenlet ekkor:
−
ℏ
2
2
m
d
2
χ
(
r
)
d
r
2
+
V
(
r
)
χ
(
r
)
=
E
⋅
χ
(
r
)
MathType@MTEF@5@5@+=feaagCart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbdfgBPjMCPbqefmuyTjMCPfgarmqr1ngBPrgitLxBI9gBamrr1ngBPrwtHrhAYaqeguuDJXwAKbstHrhAGq1DVbWefv3ySLgznfgDOfdarCqr1ngBPrginfgDObYtUvgarqqr1ngBPrgifHhDYfgasaacH8qrps0lbbf9q8WrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0RYxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaacaGacmGadaWaaiqacaabauaagaaakeaadaWcaaqaaiabgkHiTiabl+qiOnaaCaaaleqabaGaaGOmaaaaaOqaaiaaikdacaWGTbaaamaalaaabaGaamizamaaCaaaleqabaGaaGOmaaaakiabeE8aJnaabmaabaGaamOCaaGaayjkaiaawMcaaaqaaiaadsgacaWGYbWaaWbaaSqabeaacaaIYaaaaaaakiabgUcaRiaadAfadaqadaqaaiaadkhaaiaawIcacaGLPaaacqaHhpWydaqadaqaaiaadkhaaiaawIcacaGLPaaacqGH9aqpcaWGfbGaeyyXICTaeq4Xdm2aaeWaaeaacaWGYbaacaGLOaGaayzkaaaaaa@677D@
ahol
χ
(
r
)
=
r
⋅
R
(
r
)
MathType@MTEF@5@5@+=feaagCart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbdfgBPjMCPbqefmuyTjMCPfgarmqr1ngBPrgitLxBI9gBamrr1ngBPrwtHrhAYaqeguuDJXwAKbstHrhAGq1DVbWefv3ySLgznfgDOfdarCqr1ngBPrginfgDObYtUvgarqqr1ngBPrgifHhDYfgasaacH8qrps0lbbf9q8WrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0RYxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaacaGacmGadaWaaiqacaabauaagaaakeaacqaHhpWydaqadaqaaiaadkhaaiaawIcacaGLPaaacqGH9aqpcaWGYbGaeyyXICTaamOuamaabmaabaGaamOCaaGaayjkaiaawMcaaaaa@54D0@
alakú. Szeretnénk majd az alacsony energiás szórásokat tekinteni. De előtte:
vegyük az
E
<
0
MathType@MTEF@5@5@+=feaagCart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbdfgBPjMCPbqefmuyTjMCPfgarmqr1ngBPrgitLxBI9gBamrr1ngBPrwtHrhAYaqeguuDJXwAKbstHrhAGq1DVbWefv3ySLgznfgDOfdarCqr1ngBPrginfgDObYtUvgarqqr1ngBPrgifHhDYfgasaacH8qrps0lbbf9q8WrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0RYxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaacaGacmGadaWaaiqacaabauaagaaakeaacaWGfbGaeyipaWJaaGimaaaa@4B83@
esetet. Ekkor a megoldás
r
<
r
0
MathType@MTEF@5@5@+=feaagCart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbdfgBPjMCPbqefmuyTjMCPfgarmqr1ngBPrgitLxBI9gBamrr1ngBPrwtHrhAYaqeguuDJXwAKbstHrhAGq1DVbWefv3ySLgznfgDOfdarCqr1ngBPrginfgDObYtUvgarqqr1ngBPrgifHhDYfgasaacH8qrps0lbbf9q8WrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0RYxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaacaGacmGadaWaaiqacaabauaagaaakeaacaWGYbGaeyipaWJaamOCamaaBaaaleaacaaIWaaabeaaaaa@4CD3@
tartományban
χ
(
r
)
=
sin
(
q
⋅
r
)
MathType@MTEF@5@5@+=feaagCart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbdfgBPjMCPbqefmuyTjMCPfgarmqr1ngBPrgitLxBI9gBamrr1ngBPrwtHrhAYaqeguuDJXwAKbstHrhAGq1DVbWefv3ySLgznfgDOfdarCqr1ngBPrginfgDObYtUvgarqqr1ngBPrgifHhDYfgasaacH8qrps0lbbf9q8WrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0RYxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaacaGacmGadaWaaiqacaabauaagaaakeaacqaHhpWydaqadaqaaiaadkhaaiaawIcacaGLPaaacqGH9aqpciGGZbGaaiyAaiaac6gadaqadaqaaiaadghacqGHflY1caWGYbaacaGLOaGaayzkaaaaaa@56D0@
, illetve az
r
>
r
0
MathType@MTEF@5@5@+=feaagCart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbdfgBPjMCPbqefmuyTjMCPfgarmqr1ngBPrgitLxBI9gBamrr1ngBPrwtHrhAYaqeguuDJXwAKbstHrhAGq1DVbWefv3ySLgznfgDOfdarCqr1ngBPrginfgDObYtUvgarqqr1ngBPrgifHhDYfgasaacH8qrps0lbbf9q8WrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0RYxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaacaGacmGadaWaaiqacaabauaagaaakeaacaWGYbGaeyOpa4JaamOCamaaBaaaleaacaaIWaaabeaaaaa@4CD7@
tartományon:
χ
(
r
)
=
e
−
κ
r
MathType@MTEF@5@5@+=feaagCart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbdfgBPjMCPbqefmuyTjMCPfgarmqr1ngBPrgitLxBI9gBamrr1ngBPrwtHrhAYaqeguuDJXwAKbstHrhAGq1DVbWefv3ySLgznfgDOfdarCqr1ngBPrginfgDObYtUvgarqqr1ngBPrgifHhDYfgasaacH8qrps0lbbf9q8WrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0RYxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaacaGacmGadaWaaiqacaabauaagaaakeaacqaHhpWydaqadaqaaiaadkhaaiaawIcacaGLPaaacqGH9aqpcaWGLbWaaWbaaSqabeaacqGHsislcqaH6oWAcaWGYbaaaaaa@52E5@
(a hullámfüggvényt később, ha akarjuk – de nem fogjuk – normálhatjuk).
χ
MathType@MTEF@5@5@+=feaagCart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbdfgBPjMCPbqefmuyTjMCPfgarmqr1ngBPrgitLxBI9gBamrr1ngBPrwtHrhAYaqeguuDJXwAKbstHrhAGq1DVbWefv3ySLgznfgDOfdarCqr1ngBPrginfgDObYtUvgarqqr1ngBPrgifHhDYfgasaacH8qrps0lbbf9q8WrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0RYxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaacaGacmGadaWaaiqacaabauaagaaakeaacqaHhpWyaaa@4AB2@
és a deriváltja a határon menjen át simán, azaz
χ
'
(
r
)
χ
(
r
)
|
r
<
r
0
=
q
ctg
(
q
r
)
=
−
κ
MathType@MTEF@5@5@+=feaagCart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbdfgBPjMCPbqefmuyTjMCPfgarmqr1ngBPrgitLxBI9gBamrr1ngBPrwtHrhAYaqeguuDJXwAKbstHrhAGq1DVbWefv3ySLgznfgDOfdarCqr1ngBPrginfgDObYtUvgarqqr1ngBPrgifHhDYfgasaacH8qrps0lbbf9q8WrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0RYxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaacaGacmGadaWaaiqacaabauaagaaakeaadaabcaqaamaalaaabaGaeq4XdmMaai4jamaabmaabaGaamOCaaGaayjkaiaawMcaaaqaaiabeE8aJnaabmaabaGaamOCaaGaayjkaiaawMcaaaaaaiaawIa7amaaBaaaleaacaWGYbGaeyipaWJaamOCamaaBaaameaacaaIWaaabeaaaSqabaGccqGH9aqpcaWGXbGaci4yaiaacshacaGGNbWaaeWaaeaacaWGXbGaamOCaaGaayjkaiaawMcaaiabg2da9iabgkHiTiabeQ7aRbaa@63B7@
.
Ha ennek van megoldása, akkor létezik kötött állapot.
ℏ
q
=
2
m
(
V
0
−
E
)
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MathType@MTEF@5@5@+=feaagCart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbdfgBPjMCPbqefmuyTjMCPfgarmqr1ngBPrgitLxBI9gBamrr1ngBPrwtHrhAYaqeguuDJXwAKbstHrhAGq1DVbWefv3ySLgznfgDOfdarCqr1ngBPrginfgDObYtUvgarqqr1ngBPrgifHhDYfgasaacH8qrps0lbbf9q8WrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0RYxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaacaGacmGadaWaaiqacaabauaagaaakeaacaWGXbGaeyyXICTaci4yaiaacshacaGGNbWaaeWaaeaacaWGXbGaamOCamaaBaaaleaacaaIWaaabeaaaOGaayjkaiaawMcaaiabgYda8iaaicdaaaa@552B@
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. Épp akkor jelenik meg a kötött állapot, amikor
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így végül
V
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Ha
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, akkor van kötött állapot.
Ha
V
0
<
V
0
∗
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, akkor nincs kötött állapot.