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
  1. 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 ( 2π ) 3 [ z2( e k μ ) ] χ 0 ( k,k',K,z ) ψ k 1 ( k ) d 3 k ( 2π ) 3 d 3 q ( 2π ) 3 v( q ) χ 0 ( kq,k',K,z ) ψ k 1 ( k ) = =[ z2( 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=He9q8qqQ8frFve9Fve9Ff0dmeaacaGacmGadaWaaiqacaabauaagaaakeaacqaHhpWydaWgaaWcbaGaaGimaaqabaGcdaqadaqaaiaahUgacaGGIaGaaiilaiaahUgacaGGNaGaaiilaiaahUeacaGGSaGaamOEaaGaayjkaiaawMcaaiabg2da9iabeI8a5naaBaaaleaacaWHRbGaai4jaaqabaGcdaqadaqaaiaahUgacaGGIaaacaGLOaGaayzkaaGaey4kaSYaa8qaaeaadaWcaaqaaiaadsgadaahaaWcbeqaaiaaiodaaaGccaWGRbWaaSbaaSqaaiaaigdaaeqaaaGcbaWaaeWaaeaacaaIYaGaeqiWdahacaGLOaGaayzkaaWaaWbaaSqabeaacaaIZaaaaaaakmaadmaabaWaaSaaaeaacaaIXaaabaGaaGOmaiaadwgadaWgaaWcbaGaaC4AamaaBaaameaacaaIXaaabeaaaSqabaGccqGHsislcaaIYaGaamyzamaaBaaaleaacaWHRbGaai4jaaqabaGccqGHsislcaWGPbGaeq4TdGgaaiabgUcaRmaalaaabaGaaGymaaqaaiaadQhacqGHsislcaaIYaGaamyzamaaBaaaleaacaWHRbaabeaakiabgUcaRiaaikdacqaH8oqBaaaacaGLBbGaayzxaaGaeqiYdK3aaSbaaSqaaiaahUgadaWgaaadbaGaaGymaaqabaaaleqaaOWaaeWaaeaacaWHRbGaaiOiaaGaayjkaiaawMcaaaWcbeqab0Gaey4kIipakiabgwSixpaaGaaabaGaamOzaaGaay5adaWaaWbaaSqabeaacqGHxiIkaaGcdaqadaqaaiaahUgadaWgaaWcbaGaaGymaaqabaGccaGGSaGaaC4AaaGaayjkaiaawMcaaaaa@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@
  2. 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 z2 e k +2μ d 3 q ( 2π ) 3 v( q )χ( kq,k',K,z ) = = ( 2π ) 3 δ( kk' )+ F ( + ) ( K,k )1 z2( 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 ( 2π ) 3 [ ( 2π ) 3 δ( kk' ) v( kq ) z2 e k +2μ ]χ( q,k',K,z ) = = d 3 k ( 2π ) 3 [ ( 2π ) 3 δ( kq ) v( kq ) z2 e k +2μ ] χ 0 ( k,k",K,z ) = = ( 2π ) 3 δ( qk" ) 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 2 a m [ 1 + O ( | k a | ) ] Γ 0 ( k , k ' , K , z ) Γ 0 ( 0,0,0,0 ) = 4 π 2 a m Γ ( k , k ' , K , z ) Γ ( 0,0,0,0 ) = 4 π 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 2 a / m MathType@MTEF@5@5@+=feaagCart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbdfgBPjMCPbqefmuyTjMCPfgarmqr1ngBPrgitLxBI9gBamrr1ngBPrwtHrhAYaqeguuDJXwAKbstHrhAGq1DVbWefv3ySLgznfgDOfdarCqr1ngBPrginfgDObYtUvgarqqr1ngBPrgifHhDYfgasaacH8qrps0lbbf9q8WrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0RYxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaacaGacmGadaWaaiqacaabauaagaaakeaacaWG2bWaaeWaaeaacaWHRbaacaGLOaGaayzkaaGaeyiKHWQaeu4KdC0aaeWaaeaacaaIWaGaaiilaiaaicdacaGGSaGaaGimaiaacYcacaaIWaaacaGLOaGaayzkaaGaeyypa0JaaGinaiabec8aWjabl+qiOnaaCaaaleqabaGaaGOmaaaakiaadggacaGGVaGaamyBaaaa@5E6D@ , v ( r ) = 4 π 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: 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. q = 2 m ( V 0 E ) MathType@MTEF@5@5@+=feaagCart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbdfgBPjMCPbqefmuyTjMCPfgarmqr1ngBPrgitLxBI9gBamrr1ngBPrwtHrhAYaqeguuDJXwAKbstHrhAGq1DVbWefv3ySLgznfgDOfdarCqr1ngBPrginfgDObYtUvgarqqr1ngBPrgifHhDYfgasaacH8qrps0lbbf9q8WrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0RYxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaacaGacmGadaWaaiqacaabauaagaaakeaacqWIpecAcaWGXbGaeyypa0ZaaOaaaeaacaaIYaGaamyBamaabmaabaGaamOvamaaBaaaleaacaaIWaaabeaakiabgkHiTiaadweaaiaawIcacaGLPaaaaSqabaaaaa@52F4@ κ = 2 m | E | MathType@MTEF@5@5@+=feaagCart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbdfgBPjMCPbqefmuyTjMCPfgarmqr1ngBPrgitLxBI9gBamrr1ngBPrwtHrhAYaqeguuDJXwAKbstHrhAGq1DVbWefv3ySLgznfgDOfdarCqr1ngBPrginfgDObYtUvgarqqr1ngBPrgifHhDYfgasaacH8qrps0lbbf9q8WrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0RYxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaacaGacmGadaWaaiqacaabauaagaaakeaacqWIpecAcqaH6oWAcqGH9aqpdaGcaaqaaiaaikdacaWGTbWaaqWaaeaacaWGfbaacaGLhWUaayjcSdaaleqaaaaa@5291@ q ctg ( q r 0 ) < 0 MathType@MTEF@5@5@+=feaagCart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbdfgBPjMCPbqefmuyTjMCPfgarmqr1ngBPrgitLxBI9gBamrr1ngBPrwtHrhAYaqeguuDJXwAKbstHrhAGq1DVbWefv3ySLgznfgDOfdarCqr1ngBPrginfgDObYtUvgarqqr1ngBPrgifHhDYfgasaacH8qrps0lbbf9q8WrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0RYxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaacaGacmGadaWaaiqacaabauaagaaakeaacaWGXbGaeyyXICTaci4yaiaacshacaGGNbWaaeWaaeaacaWGXbGaamOCamaaBaaaleaacaaIWaaabeaaaOGaayjkaiaawMcaaiabgYda8iaaicdaaaa@552B@ E B : = E MathType@MTEF@5@5@+=feaagCart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbdfgBPjMCPbqefmuyTjMCPfgarmqr1ngBPrgitLxBI9gBamrr1ngBPrwtHrhAYaqeguuDJXwAKbstHrhAGq1DVbWefv3ySLgznfgDOfdarCqr1ngBPrginfgDObYtUvgarqqr1ngBPrgifHhDYfgasaacH8qrps0lbbf9q8WrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0RYxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaacaGacmGadaWaaiqacaabauaagaaakeaacaWGfbWaaSbaaSqaaiaadkeaaeqaaOGaaiOoaiabg2da9iabgkHiTiaadweaaaa@4E3D@ . Épp akkor jelenik meg a kötött állapot, amikor q ctg ( q r ) = 0 MathType@MTEF@5@5@+=feaagCart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbdfgBPjMCPbqefmuyTjMCPfgarmqr1ngBPrgitLxBI9gBamrr1ngBPrwtHrhAYaqeguuDJXwAKbstHrhAGq1DVbWefv3ySLgznfgDOfdarCqr1ngBPrginfgDObYtUvgarqqr1ngBPrgifHhDYfgasaacH8qrps0lbbf9q8WrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0RYxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaacaGacmGadaWaaiqacaabauaagaaakeaacaWGXbWaaWbaaSqabeaacqGHxiIkaaGccqGHflY1ciGGJbGaaiiDaiaacEgadaqadaqaaiaadghadaahaaWcbeqaaiabgEHiQaaakiaadkhaaiaawIcacaGLPaaacqGH9aqpcaaIWaaaaa@5689@ E B = 0 MathType@MTEF@5@5@+=feaagCart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbdfgBPjMCPbqefmuyTjMCPfgarmqr1ngBPrgitLxBI9gBamrr1ngBPrwtHrhAYaqeguuDJXwAKbstHrhAGq1DVbWefv3ySLgznfgDOfdarCqr1ngBPrginfgDObYtUvgarqqr1ngBPrgifHhDYfgasaacH8qrps0lbbf9q8WrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0RYxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaacaGacmGadaWaaiqacaabauaagaaakeaacaWGfbWaaSbaaSqaaiaadkeaaeqaaOGaeyypa0JaaGimaaaa@4C82@ ctg q r 0 = 0 MathType@MTEF@5@5@+=feaagCart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbdfgBPjMCPbqefmuyTjMCPfgarmqr1ngBPrgitLxBI9gBamrr1ngBPrwtHrhAYaqeguuDJXwAKbstHrhAGq1DVbWefv3ySLgznfgDOfdarCqr1ngBPrginfgDObYtUvgarqqr1ngBPrgifHhDYfgasaacH8qrps0lbbf9q8WrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0RYxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaacaGacmGadaWaaiqacaabauaagaaakeaaciGGJbGaaiiDaiaacEgacaWGXbWaaWbaaSqabeaacqGHxiIkaaGccaWGYbWaaSbaaSqaaiaaicdaaeqaaOGaeyypa0JaaGimaaaa@518A@ q r 0 = π 2 MathType@MTEF@5@5@+=feaagCart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbdfgBPjMCPbqefmuyTjMCPfgarmqr1ngBPrgitLxBI9gBamrr1ngBPrwtHrhAYaqeguuDJXwAKbstHrhAGq1DVbWefv3ySLgznfgDOfdarCqr1ngBPrginfgDObYtUvgarqqr1ngBPrgifHhDYfgasaacH8qrps0lbbf9q8WrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0RYxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaacaGacmGadaWaaiqacaabauaagaaakeaacaWGXbWaaWbaaSqabeaacqGHxiIkaaGccaWGYbWaaSbaaSqaaiaaicdaaeqaaOGaeyypa0ZaaSaaaeaacqaHapaCaeaacaaIYaaaaaaa@508D@ q 2 r 0 2 = π 2 / 4 MathType@MTEF@5@5@+=feaagCart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbdfgBPjMCPbqefmuyTjMCPfgarmqr1ngBPrgitLxBI9gBamrr1ngBPrwtHrhAYaqeguuDJXwAKbstHrhAGq1DVbWefv3ySLgznfgDOfdarCqr1ngBPrginfgDObYtUvgarqqr1ngBPrgifHhDYfgasaacH8qrps0lbbf9q8WrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0RYxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaacaGacmGadaWaaiqacaabauaagaaakeaacaWGXbWaaWbaaSqabeaacqGHxiIkcaaIYaaaaOGaamOCamaaDaaaleaacaaIWaaabaGaaGOmaaaakiabg2da9iabec8aWnaaCaaaleqabaGaaGOmaaaakiaac+cacaaI0aaaaa@539E@ 2 m V 0 r 0 2 2 = π 2 4 MathType@MTEF@5@5@+=feaagCart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbdfgBPjMCPbqefmuyTjMCPfgarmqr1ngBPrgitLxBI9gBamrr1ngBPrwtHrhAYaqeguuDJXwAKbstHrhAGq1DVbWefv3ySLgznfgDOfdarCqr1ngBPrginfgDObYtUvgarqqr1ngBPrgifHhDYfgasaacH8qrps0lbbf9q8WrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0RYxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaacaGacmGadaWaaiqacaabauaagaaakeaadaWcaaqaaiaaikdacaWGTbGaamOvamaaDaaaleaacaaIWaaabaGaey4fIOcaaOGaamOCamaaDaaaleaacaaIWaaabaGaaGOmaaaaaOqaaiabl+qiOnaaCaaaleqabaGaaGOmaaaaaaGccqGH9aqpdaWcaaqaaiabec8aWnaaCaaaleqabaGaaGOmaaaaaOqaaiaaisdaaaaaaa@56B8@ így végül V 0 = π 2 2 8 m r 0 2 MathType@MTEF@5@5@+=feaagCart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbdfgBPjMCPbqefmuyTjMCPfgarmqr1ngBPrgitLxBI9gBamrr1ngBPrwtHrhAYaqeguuDJXwAKbstHrhAGq1DVbWefv3ySLgznfgDOfdarCqr1ngBPrginfgDObYtUvgarqqr1ngBPrgifHhDYfgasaacH8qrps0lbbf9q8WrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0RYxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaacaGacmGadaWaaiqacaabauaagaaakeaacaWGwbWaa0baaSqaaiaaicdaaeaacqGHxiIkaaGccqGH9aqpdaWcaaqaaiabec8aWnaaCaaaleqabaGaaGOmaaaakiabl+qiOnaaCaaaleqabaGaaGOmaaaaaOqaaiaaiIdacaWGTbGaamOCamaaDaaaleaacaaIWaaabaGaaGOmaaaaaaaaaa@55E6@ Ha V 0 V 0 MathType@MTEF@5@5@+=feaagCart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbdfgBPjMCPbqefmuyTjMCPfgarmqr1ngBPrgitLxBI9gBamrr1ngBPrwtHrhAYaqeguuDJXwAKbstHrhAGq1DVbWefv3ySLgznfgDOfdarCqr1ngBPrginfgDObYtUvgarqqr1ngBPrgifHhDYfgasaacH8qrps0lbbf9q8WrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0RYxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaacaGacmGadaWaaiqacaabauaagaaakeaacaWGwbWaaSbaaSqaaiaaicdaaeqaaOGaeyyzImRaamOvamaaDaaaleaacaaIWaaabaGaey4fIOcaaaaa@4F3D@ , akkor van kötött állapot.

Ha V 0 < V 0 MathType@MTEF@5@5@+=feaagCart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbdfgBPjMCPbqefmuyTjMCPfgarmqr1ngBPrgitLxBI9gBamrr1ngBPrwtHrhAYaqeguuDJXwAKbstHrhAGq1DVbWefv3ySLgznfgDOfdarCqr1ngBPrginfgDObYtUvgarqqr1ngBPrgifHhDYfgasaacH8qrps0lbbf9q8WrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0RYxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaacaGacmGadaWaaiqacaabauaagaaakeaacaWGwbWaaSbaaSqaaiaaicdaaeqaaOGaeyipaWJaamOvamaaDaaaleaacaaIWaaabaGaey4fIOcaaaaa@4E7B@ , akkor nincs kötött állapot.