Quantum Correlations and Temperature Fluctuations in Nanoscale Systems

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(1)Quantum Correlations and Temperature Fluctuations in Nanoscale Systems Brange, Fredrik. 2019. Document Version: Publisher's PDF, also known as Version of record Link to publication. Citation for published version (APA): Brange, F. (2019). Quantum Correlations and Temperature Fluctuations in Nanoscale Systems. Lund University, Faculty of Science.. Total number of authors: 1. General rights Unless other specific re-use rights are stated the following general rights apply: Copyright and moral rights for the publications made accessible in the public portal are retained by the authors and/or other copyright owners and it is a condition of accessing publications that users recognise and abide by the legal requirements associated with these rights. • Users may download and print one copy of any publication from the public portal for the purpose of private study or research. • You may not further distribute the material or use it for any profit-making activity or commercial gain • You may freely distribute the URL identifying the publication in the public portal Read more about Creative commons licenses: https://creativecommons.org/licenses/ Take down policy If you believe that this document breaches copyright please contact us providing details, and we will remove access to the work immediately and investigate your claim.. L UNDUNI VERS I TY PO Box117 22100L und +46462220000.

(2) FREDRIK BRANGE . Quantum Correlations and Temperature Fluctuations in Nanoscale Systems 2019. Departement of Physics Faculty of Science Lund University ISBN 978-91-7895-046-1. 9 789178 950461. In this thesis, we investigate two different topics related to the physics of nanoscale systems. The first topic concerns quantum correlations, in particular how to generate and detect entangled electrons. The second topic concerns temperature fluctuations, especially how to utilize them for nanoscale calorimetry. The first part of the thesis includes an introduction to the research field and an overview of the main findings of the papers. In the second part, the five papers of the thesis are found in their entirety.. Quantum Correlations and Temperature Fluctuations in Nanoscale Systems FREDRIK BRANGE FACULTY OF SCIENCE | LUND UNIVERSITY.

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(28) 3. .... 2 3 2. 1 1. (a). (b). (c). (d). 'JHVSF B " HFOFSJD OBOPTDBMF TZTUFN ZFMMPX DPVQMFE UP B OVNCFS PG UFSNJOBMT HSBZ IFSF UISFF WJB DPVQMJOH TUSFOHUIT γ γ BOE γ SFTQFDUJWFMZ 5IF UFSNJOBMT DPOTJTU PG QBSUJDMF SFTFSWPJST XJUI XFMMEFmOFE UFNQFSBUVSFT 5 5 BOE 5 BOE DIFNJDBM QPUFOUJBMT BOE 5IF BJN PG RVBOUVN USBOTQPSU JT UP DIBSBDUFSJ[F UIF USBOTGFS PG QBSUJDMFT IFBU P PUIFS QIZTJDBM RVBOUJUJFT UISPVHI UIF TZTUFN C " TJOHMF SFTPOBODF MFWFM GFSNJPOJD PS CPTPOJD DPVQMFE XJUI B DPVQMJOH TUSFOHUI γ UP B SFTFSWPJS XJUI UFNQFSBUVSF 5 BOE NFBO PDDVQBUJPO OVNCFS ¯ O = (exp[ε/(L# 5)] ± )− XJUI + − GPS GFSNJPOT CPTPOT BOE ε UIF SFTPOBODF FOFSHZ 5IJT TZTUFN JT VTFE BT BO FYBNQMF TZTUFN UISPVHIPVU UIF UIFTJT UP JMMVTUSBUF WBSJPVT DPODFQUT 5ZQJDBM FYBNQMFT PG .POUF $BSMPTJNVMBUFE UJNF USBDFT PG UIF QBSUJDMF FNJTTJPO TUBUJTUJDT GPS D GFSNJPOJD QBSUJDMFT BOE E CPTPOJD QBSUJDMFT GPS ¯ O = . &WFSZ QBSUJDMF FNJTTJPO FWFOU JT JOEJDBUFE CZ B TQJLF. B OVNCFS PG EJĊFSFOU DPODFQUT UP DIBSBDUFSJ[F USBOTQPSU TUBUJTUJDT BOE TIPX IPX UIFTF NBZ CF FYUSBDUFE GSPN UIF -JOECMBE FRVBUJPO 4FD ñò *O QBSUJDVMBS XF DPOTJEFS GVMM DPVOUJOH TUBUJTUJDT JO UIF MPOHUJNF MJNJU XIFSF [FSPGSFRVFODZ OPJTF NFBTVSFNFOUT GPSN BO JOEJTQFOTBCMF UPPM GPS JOWFTUJHBUJOH CPUI RVBOUVN DPSSFMBUJPOT $I ò BOE UFNQFSBUVSF nVDUVBUJPOT $I ó JO OBOPTDBMF TZTUFNT 4VDI NFBTVSFNFOUT DPOTUJUVUF BO JOUFHSBM QBSU PG BMM mWF QBQFST PG UIJT UIFTJT 8F XJMM UISPVHIPVU UIF DIBQUFS DPOTJEFS B HFOFSJD OBOPTDBMF TZTUFN <TFF 'JH ñð B > DPVQMFE UP POF PS TFWFSBM SFTFSWPJST 8IJMF UIF PWFSBMM HPBM JT UP JOUSPEVDF UPPMT UP EFTDSJCF UIF FNJTTJPO BCTPSQUJPO PS OFU DVSSFOUT PG QBSUJDMFT PS IFBU nPXJOH CFUXFFO UIF TZTUFN BOE POF PS TFWFSBM PG UIF SFTFSWPJST UIF NBJO GPDVT XJMM CF PO QBSUJDMF FNJTTJPO .PTU DPODFQUT BOE SFTVMUT NBZ FBTJMZ CF HFOFSBMJ[FE UP PUIFS USBOTQPSU RVBOUJUJFT 5P QSPWJEF BT TJNQMF BOE DPODSFUF FYBNQMFT BT QPTTJCMF PG WBSJPVT DPODFQUT XF XJMM DPOTJEFS QBSUJDMF FNJTTJPOT GSPN BO FYBNQMF TZTUFN DPOTJTUJOH PG B TJOHMF OPOEFHFOFSBUF SFTPOBODF MFWFM FJUIFS GFSNJPOJD PS CPTPOJD XFBLMZ DPVQMFE XJUI B SBUF γ UP B TJOHMF SFTFSWPJS <TFF 'JH ñð C o E > *O UIF GFSNJPOJD DBTF UIJT DPVME CF B RVBOUVN EPU PQFSBUJOH JO UIF TUSPOH $PVMPNC CMPDLBEF SFHJNF XIJMF JO UIF CPTPOJD DBTF JU DPVME CF B NJDSPXBWF QIPUPO DBWJUZ JEFOUJDBM UP UIF POF JO 1BQFS 7 #FTJEFT JMMVTUSBUJOH TPNF PG UIF LFZ DPODFQUT PG UIJT DIBQUFS BOE PG 1BQFS 7. UIF FYBNQMF TZTUFN BMTP BMMPXT VT UP IJHIMJHIU TPNF PG UIF GVOEBNFOUBM EJĊFSFODFT CFUXFFO CPTPOJD BOE GFSNJPOJD USBOTQPSU. ñð 4UBUFT BOE PCTFSWBCMFT " LFZ DPODFQU OPU POMZ PG USBOTQPSU UIFPSZ CVU BMTP PG DMBTTJDBM QIZTJDT UIFSNPEZOBNJDT BOE RVBOUVN NFDIBOJDT JT UIBU PG B TUBUF <óö> *O B CSPBE TFOTF B TUBUF BJNT BU EFTDSJCJOH UIF QIZTJDBM SFBMJUZ PG B TZTUFN JF UIF QIZTJDBM RVBOUJUJFT XIJDI IBWF QSFEFUFSNJOFE BOE. ðï.

(29) XFMMEFmOFE WBMVFT XIFUIFS XF NFBTVSF UIFN PS OPU .PSF QSFDJTFMZ UIF TUBUF PG B TZT UFN BMMPXT VT BU MFBTU UP TPNF FYUFOU UP QSFEJDU UIF PVUDPNF PG NFBTVSFNFOUT PG DFSUBJO QIZTJDBM RVBOUJUJFT o PCTFSWBCMFT o PG UIBU TZTUFN )PXFWFS BMUIPVHI UIFZ QSPWJEF B DPN QMFUF EFTDSJQUJPO PG UIF QIZTJDBM SFBMJUZ RVBOUVN TUBUFT BSF IJHIMZ SFTUSJDUFE DPNQBSFE UP UIFJS DMBTTJDBM BOBMPHT BT UIFZ BSF FWFO JO QSJODJQMF JODBQBCMF PG QSFEJDUJOH UIF PVUDPNF PG FWFSZ QPTTJCMF NFBTVSFNFOU XJUI DFSUBJOUZ "DDPSEJOH UP RVBOUVN NFDIBOJDT UIJT JT OPU B SFTVMU PG UIF UIFPSZ CFJOH JODPNQMFUF CVU SBUIFS BO JOIFSFOU QSPQFSUZ PG PVS XPSME JO XIJDI JU JT JNQPTTJCMF GPS BMM QIZTJDBM PCTFSWBCMFT UP GPSN QBSU PG POF BOE UIF TBNF SFBMJUZ TJNVMUBOFPVTMZ <ó÷> ɨF SFNBJOEFS PG UIJT TFDUJPO GPDVTFT PO JOUSPEVDJOH UXP JNQPSUBOU LJOET PG TUBUFT ɨF mSTU LJOE PG TUBUFT QVSF TUBUFT QSPWJEFT B DPNQMFUF RVBOUVN NFDIBOJDBM EFTDSJQUJPO PG BMM UIF NJDSPTDPQJD EFHSFFT PG GSFFEPN PG B QIZTJDBM TZTUFN BOE DPSSFTQPOET UP B NJDSPTUBUF JO TUBUJTUJDBM QIZTJDT <ð> )PXFWFS BT XF XJMM TFF TVDI TUBUFT BSF UZQJDBMMZ JOTVċDJFOU UP EFTDSJCF PQFO RVBOUVN TZTUFNT ɨJT NPUJWBUFT VT UP JOUSPEVDF UIF TFDPOE LJOE PG TUBUFT o NJYFE TUBUFT "OBMPHPVT UP NBDSPTUBUFT JO TUBUJTUJDBM QIZTJDT <ð> UIFZ EFTDSJCF B TZTUFN TUBUJTUJDBMMZ JO UFSNT PG FOTFNCMFT PG QVSF TUBUFT. ñðð 1VSF TUBUFT NJDSPTUBUFT. *O UIF 4DISÚEJOHFS QJDUVSF PG RVBOUVN NFDIBOJDT QVSF TUBUFT BSF SFQSFTFOUFE CZ OPSNBM J[FE UJNFEFQFOEFOU TUBUF WFDUPST |Ψ(U) XJUI Ψ(U)|Ψ(U) = UIBU CFMPOH UP B )JMCFSU ˆ BDUJOH PO UIFTF TUBUFT GPS TQBDF <óø> 0CTFSWBCMFT BSF SFQSFTFOUFE CZ IFSNJUJBO PQFSBUPST " TJNQMJDJUZ XF IFSF BTTVNF UIBU UIFZ EP OPU IBWF BOZ FYQMJDJU UJNF EFQFOEFODF 8IJMF B QVSF TUBUF TVQQPTFEMZ QSPWJEFT B DPNQMFUF EFTDSJQUJPO PG UIF SFBMJUZ PG B RVBOUVN TZTUFN JU NBZ POMZ QSFEJDU XJUI DFSUBJOUZ UIF PVUDPNF BΨ PG B TJOHMF NFBTVSFNFOU JG |Ψ(U) JT BO ˆ ˆ XJUI "|Ψ(U) ˆ FJHFOTUBUF PG " = BΨ |Ψ(U) 'PS BOZ PUIFS TUBUF GPS XIJDI XF TBZ UIBU " MBDLT SFBMJUZ UIF NFBTVSFNFOU PVUDPNFT nVDUVBUF BOE RVBOUVN NFDIBOJDT POMZ QSPWJEFT UIF BWFSBHF WBMVF PG TVDI NFBTVSFNFOUT UIF FYQFDUBUJPO WBMVF ˆ ˆ ˆ U = Ψ(U)|"|Ψ(U) = US{"|Ψ(U)Ψ(U)}. " ɨF NBHOJUVEF PG UIF nVDUVBUJPOT o UIF RVBOUVN VODFSUBJOUJFT o JT ˆ ˆ U − " ˆ U , ˆ≡" ˆ − "U . σ" (U) = (ǈ") U = " ǈ". ñð. ññ. ˆ BOE # ˆ #] ˆ #− ˆ = ˆ GPS XIJDI [", ˆ ≡" ˆ # ˆ" *NQPSUBOUMZ GPS TPDBMMFE JODPNQBUJCMF PCTFSWBCMFT " UIFSF BSF OP DPNNPO FJHFOTUBUFT UIFTF PCTFSWBCMFT UIVT BMXBZT MBDL TJNVMUBOFPVT SFBMJUZ ɨJT JT UIF FTTFODF PG UIF )FJTFOCFSH VODFSUBJOUZ QSJODJQMF <óø> ˆ ˆ σ" (U)σ# (U) ≥ |[", #]U |, ðð. ñò.

(30) XIJDI MJFT BU UIF IFBSU PG RVBOUVN NFDIBOJDT ɨF QSJNF FYBNQMFT PG JODPNQBUJCMF PCTFSW BCMFT BSF UIF NPNFOUVN BOE QPTJUJPO PG B QBSUJDMF PS UIF QSPKFDUJPO PG B TQJO BMPOH EJĊFSFOU BYFT ɨF RVFTUJPO XIFUIFS UIF )FJTFOCFSH VODFSUBJOUZ QSJODJQMF JT BO JOIFSFOU QSPQFSUZ PG PVS QIZTJDBM SFBMJUZ PS B SFTVMU PG RVBOUVN NFDIBOJDT CFJOH BO JODPNQMFUF UIFPSZ XJMM CF GVSUIFS EJTDVTTFE JO $I ò #Z mOEJOH UIF MBSHFTU TFU PG DPNQBUJCMF PCTFSWBCMFT B TPDBMMFE DPNQMFUF TFU POF DBO DPO TUSVDU BO PSUIPOPSNBM CBTJT PG FJHFOTUBUFT |ΦJ UIBU GVMMZ QSFEJDU UIF PVUDPNFT PG UIF MBSHFTU QPTTJCMF TFU PG NFBTVSFNFOUT )FSF ΦJ SFQSFTFOUT UIF XIPMF TFU PG UIF DPSSFTQPOEJOH FJHFO WBMVFT UIFTF BSF LOPXO BT RVBOUVN OVNCFST "OZ PUIFS TUBUF NBZ CF XSJUUFO BT |Ψ(U) = DJ (U)|ΦJ , DJ (U) = ΦJ |Ψ(U), ñó. J. XIFSF QJ (U) = |DJ (U)| JT UIF QSPCBCJMJUZ PG PCUBJOJOH UIF NFBTVSFNFOU PVUDPNF BΦJ XIFO ˆ JT NFBTVSFE GPS XIJDI "|Φ ˆ J = BΦJ |ΦJ UIF PCTFSWBCMF " %VF UP UIF JOEJTUJOHVJTIBCJMJUZ PG JEFOUJDBM QBSUJDMFT JO RVBOUVN NFDIBOJDT B HSFBU EFBM PG DBSF IBT UP CF UBLFO JO NBOZQBSUJDMF TZTUFNT 0OF XBZ PG EPJOH UIJT JT UP VTF UIF PDDVQBUJPO OVNCFS SFQSFTFOUBUJPO <ôï> XJUI B CBTJT DPOTJTUJOH PG UIF TPDBMMFE 'PDL TUBUFT |ΦJ = |O O ...ON ,. ñô. XIFSF POMZ UIF OVNCFS PG QBSUJDMFT OJ JO FWFSZ BDDFTTJCMF TJOHMFQBSUJDMF TUBUF |φJ JT TQFDJmFE JF OP SFGFSFODF UP UIF JEFOUJUJFT PG TJOHMF QBSUJDMFT NBEF .PSF GPSNBMMZ UIFTF JT CFJOH⊗O 4 H UIBU JT B EJSFDU TVN PWFS BMM NBOZQBSUJDMF TUBUFT CFMPOH UP B 'PDL TQBDF F = ∞ O= ± UFOTPS QPXFST PG UIF TJOHMFQBSUJDMF )JMCFSU TQBDF H <ôï> )FSF 4+ JT UIF BOUJTZNNFUSJ[JOH PQFSBUPS GPS GFSNJPOT BOE 4− UIF TZNNFUSJ[JOH PQFSBUPS GPS CPTPOT 5P FWFSZ TJOHMFQBSUJDMF TUBUF |φJ XF JOUSPEVDF DSFBUJPO PQFSBUPST ˆB†J BOE BOOJIJMBUJPO PQ FSBUPST ˆBJ XIJDI JODSFBTF BOE EFDSFBTF SFTQFDUJWFMZ UIF QBSUJDMF OVNCFS JO UIBU TUBUF <óø> √ √ ˆB†J |O ...OJ ...ON = OJ +|O ...OJ +...ON , ˆBJ |O ...OJ ...ON = OJ |O ...OJ −...ON . ñõ. 8F OPUF UIBU O ˆJ ≡ ˆB†J ˆBJ NBZ CF JOUFSQSFUFE BT OVNCFS PQFSBUPST XIPTF FJHFOWBMVFT BSF OJ UIFTF PQFSBUPST GPSN UIF DPNQMFUF TFU PG DPNQBUJCMF FJHFOPQFSBUPST PG UIF 'PDL TUBUFT ɨF TZNNFUSZ QSPQFSUJFT PG GFSNJPOJD BOE CPTPOJD TUBUFT BSF OBUVSBMMZ JODPSQPSBUFE JOUP UIF BOUJ DPNNVUBUJPO QSPQFSUJFT PG UIF DSFBUJPO BOE BOOJIJMBUJPO PQFSBUPST [ˆBJ , ˆB†K ]± = δJK ,. [ˆBJ , ˆBK ]± = [ˆB†J , ˆB†K ]± = ,. ñö. ˆ# ˆ EFOPUFT UIF BOUJDPNNVUBUPS + GPS GFSNJPOT BOE UIF DPN ˆ #] ˆ±# ˆ" ˆ± ≡ " XIFSF [", NVUBUPS − GPS CPTPOT 'VSUIFSNPSF δJK EFOPUFT UIF ,SPOFDLFS EFMUB *NQPSUBOUMZ GPS GFSNJPOT XF PCUBJO UIF 1BVMJ QSJODJQMF <ôð> ˆB†J ˆB†J = ˆBJˆBJ = . ðñ. ñ÷.

(31) 'PS UIF GFSNJPOJD WFSTJPO PG PVS FYBNQMF TZTUFN UIJT NFBOT UIBU UIF SFTPOBODF MFWFM DBO CF PDDVQJFE CZ BU NPTU POF QBSUJDMF BU B UJNF *O UIBU DBTF VTJOH UIF TUBUFT | BOE | XJUI [FSP BOE POF QBSUJDMFT BT B CBTJT UIF DSFBUJPO BOE BOOJIJMBUJPO PQFSBUPST NBZ CF SFQSFTFOUFE JO UFSNT PG UIF IFSNJUJBO 1BVMJ NBUSJDFT XJUI FJHFOWBMVFT ±. −J , σ ˆZ = , σ ˆ[ = , ñø. σ ˆY = J − BT ˆB† =. σZ ≡ σ σ ˆY + Jˆ ˆ+ , . ˆB =. BOE. σZ ≡ σ σ ˆY − Jˆ ˆ− . . ñðï. .PTU JNQPSUBOUMZ BOZ PUIFS PQFSBUPS NBZ CF FYQSFTTFE JO UFSNT PG UIF BOOJIJMBUJPO BOE DSFBUJPO PQFSBUPST 'PS FYBNQMF UIF )BNJMUPOJBO EFTDSJCJOH UIF FOFSHZ PG B TZTUFN NBZ GPS PVS FYBNQMF TZTUFN CF FYQSFTTFE BT VQ UP BO BSCJUSBSZ DPOTUBOU. † ˆ = εˆB†ˆB + ñðð. εJˆCJ ˆCJ + UJˆB†ˆCJ + U∗J ˆC†J ˆB , ) J. J. XIFSF ˆB BDUT PO UIF TJOHMF NPEF PG UIF TZTUFN CPTPOJD PS GFSNJPOJD ˆCJ BDUT PO NPEF J JO UIF SFTFSWPJS BOE UJ JT UIF UVOOFMJOH BNQMJUVEF GSPN NPEF J JO UIF SFTFSWPJS UP UIF TJOHMF ˆ†ˆ ˆ& = ˆ 4 = εˆB†ˆB JT UIF )BNJMUPOJBO GPS UIF TZTUFN ) NPEF JO UIF TZTUFN )FSF ) J εJ CJ CJ. †ˆ † ˆ JOU = JT UIF )BNJMUPOJBO PG UIF FOWJSPONFOU BOE ) UJˆB CJ + U∗ˆC ˆB JT UIF JOUFSBDUJPO J. J J. )BNJMUPOJBO UP CF VTFE MBUFS *O UIF CPTPOJD DBTF UIJT )BNJMUPOJBO JT FYBDUMZ UIF TBNF BT UIF POF VTFE JO 1BQFS 7 UP EFTDSJCF B NJDSPXBWF DBWJUZ *O UIF GFSNJPOJD DBTF UIF )BNJMUPOJBO UP TPNF FYUFOU SFTFNCMFT UIF POF JO 1BQFS * IPXFWFS JO UIF MBUUFS DBTF XF IBWF TJY EJĊFSFOU OFBSFTUOFJHICPVS DPVQMFE RVBOUVN EPUT FBDI POF BMTP DPVQMFE UP B SFTFSWPJS XIJDI QSPEVDFT B TPNFXIBU NPSF DPNQMJDBUFE )BNJMUPOJBO UP CF EJTDVTTFE JO $I ò "T XF XJMM TFF JO UIF OFYU TFDUJPO UIF GVMM )BNJMUPOJBO QSPWJEFT UIF UJNF FWPMVUJPO PG UIF TZTUFN BOE JUT FOWJSPONFOU )PXFWFS TJODF UIF FOWJSPONFOU NBZ DPOUBJO JOmOJUFMZ NBOZ EFHSFFT PG GSFFEPN JU JT EFTJSBCMF UP mOE BO FĊFDUJWF EFTDSJQUJPO PG UIF EZOBNJDT PG UIF TZTUFN BMPOF CZ JOUFHSBUJOH PVU UIF EFHSFFT PG GSFFEPN CFMPOHJOH UP UIF FOWJSPONFOU ɨJT JT POF PG UIF BJNT PG UIF UIFPSZ PG PQFO RVBOUVN TZTUFNT. ñðñ .JYFE TUBUFT NBDSPTUBUFT. 8IFO EFBMJOH XJUI PQFO RVBOUVN TZTUFNT UIF TUBUF WFDUPS EFTDSJQUJPO JOUSPEVDFE BCPWF UVSOT PVU UP CF JOTVċDJFOU 5P TFF UIJT XF mSTU OPUF CBTFE PO &R ñð UIBU GPS BOZ TUBUF WFDUPS UIFSF JT B EFOTJUZ NBUSJY ρˆ(U) = |Ψ(U)Ψ(U)| EFTDSJCJOH UIF TBNF TUBUF CZ B QSPKFDUJPO PQFSBUPS /PX TVQQPTJOH UIBU ρˆ(U) EFTDSJCFT B DPNQPTJUF TZTUFN DPOTJTUJOH PG ðò.

(32) BO PQFO TZTUFN 4 BOE JUT FOWJSPONFOU & XF NBZ BMTP XBOU UP DPOTJEFS UIF EFOTJUZ NBUSJY ρˆ4 (U) PG UIF PQFO TZTUFN 4 BMPOF 5P FYUSBDU UIJT EFOTJUZ NBUSJY XF JOUFHSBUF PVU UIF EFHSFFT PG GSFFEPN CFMPOHJOH UP UIF FOWJSPONFOU UIVT PCUBJOJOH ρ(U)} = ϕJ |Ψ(U)Ψ(U)|ϕJ , ñðñ. ρˆ4 (U) = US& {ˆ J. XIFSF US& EFOPUFT UIF QBSUJBM USBDF PWFS UIF FOWJSPONFOUBM EFHSFFT PG GSFFEPN BOE |ϕJ JT BOZ PSUIPOPSNBM CBTJT GPS UIF )JMCFSU TQBDF PG UIF FOWJSPONFOU .PTU JNQPSUBOUMZ ρˆ4 (U) DBOOPU JO HFOFSBM CF FYQSFTTFE BT B QSPKFDUJPO PQFSBUPS 8F UIVT DPODMVEF UIBU BO PQFO TZTUFN NBZ OPU BMXBZT CF EFTDSJCFE CZ B QVSF TUBUF *OTUFBE XF JOUSPEVDF B NPSF HFOFSBM DMBTT PG TUBUFT NJYFE TUBUFT DPOTJTUJOH PG B TUBUJTUJDBM NJYUVSF PG QVSF TUBUFT |Ψ(U) XJUI XFJHIUT RJ ≥ ɨFZ NBZ CF SFQSFTFOUFE <ñ> CZ EFOTJUZ NBUSJDFT PG UIF GPSN RJ |ΨJ (U)ΨJ (U)|, RJ = . ñðò. ρˆ(U) = J. J. " TUSBJHIUGPSXBSE HFOFSBMJ[BUJPO PG UIF FYQSFTTJPO JO &R ñð ZJFMET UIF FYQFDUBUJPO WBMVFT ˆ U= ". . ˆ J (U) = US{"ˆ ˆ ρ(U)}. RJ ΨJ (U)|"|Ψ. ñðó. J. 'PS B DFSUBJO CBTJT UIF EJBHPOBM FMFNFOUT PG UIF EFOTJUZ NBUSJY BSF DBMMFE QPQVMBUJPOT BT UIFZ EFTDSJCF UIF QSPCBCJMJUJFT PG mOEJOH UIF TZTUFN JO B DFSUBJO CBTJT TUBUF ɨF PĊEJBHPOBM FMFNFOUT BSF DBMMFE DPIFSFODFT BOE EFTDSJCF TVQFSQPTJUJPOT UIFTF BSF QBSUJDVMBSMZ JNQPSUBOU GPS FOUBOHMFE TUBUFT &RVBUJPO ñðò CSJOHT UXP EFmOJOH QSPQFSUJFT PG EFOTJUZ NBUSJDFT US {ˆ ρ(U)} = BOE ρˆ(U) ≥ JF ρˆ(U) JT B QPTJUJWF PQFSBUPS ɨFTF DPOEJUJPOT FOTVSF UIBU JO BOZ CBTJT UIF QPQVMBUJPOT BSF OPOOFHBUJWF BOE TVN VQ UP " DPODSFUF BOE JNQPSUBOU FYBNQMF PG B EFOTJUZ NBUSJY JT UIBU PG B RVCJU <ñ> (I + ζ(U)a(U) · σ) , ñðô. XIFSF I JT UIF JEFOUJUZ PQFSBUPS a(U) JT B VOJU WFDUPS XIJDI NBZ DIBOHF PWFS UJNF σ = ˆZ , σ ˆ[ ) JT UIF 1BVMJ WFDUPS BOE ≤ ζ(U) ≤ JT B QBSBNFUFS EFUFSNJOJOH UIF NJYFEOFTT (ˆ σY , σ PG UIF RVCJU TUBUF 8F XJMM TFF JO $I ò UIBU UIJT LJOE PG TUSVDUVSF BMTP BQQFBST XIFO XF DPOTJEFS FOUBOHMFNFOU XJUOFTTFT GPS OBOPTDBMF TZTUFNT "OPUIFS JNQPSUBOU DMBTT PG TUBUFT BSF TUFBEZ TUBUFT GPS XIJDI EˆρEU(U) = JF XF NBZ XSJUF UIFN BT ρˆ(U) = ρˆT 5P mOE UIF TUFBEZ TUBUF PG B TZTUFN XF IBWF UP DPOTJEFS JUT UJNF FWPMVUJPO XIJDI XF EJTDVTT JO UIF OFYU TFDUJPO )PXFWFS XF BMSFBEZ OPX OPUF UIBU GPS PVS FYBNQMF TZTUFN UIF TUFBEZ TUBUF TIPVME JO BDDPSEBODF XJUI TUBUJTUJDBM QIZTJDT CF UIF UIFSNBM TUBUF ρˆ(U) =. ˆ. ρˆT =. †. F−β()4 −ˆB ˆB) US{F−β()ˆ 4 −ˆB ˆB) } †. ðó. ,. ñðõ.

(33) XJUI β = /(L# 5) UIF JOWFSTF UFNQFSBUVSF BOE UIF DIFNJDBM QPUFOUJBM PG UIF TZTUFN DPNQBSFE UP UIF SFTFSWPJS ɨF BWFSBHF PDDVQBUJPO OVNCFS JT UIFO O ¯ = US{ˆB†ˆBρˆT } =. , Fβ(ε−) ± . ñðö. ZJFMEJOH UIF 'FSNJo%JSBD EJTUSJCVUJPO GPS GFSNJPOT + BOE UIF #PTFo&JOTUFJO EJTUSJCVUJPO GPS CPTPOT − XJUI = GPS OPODPOTFSWFE QBSUJDMFT MJLF QIPUPOT /PUF UIBU ≤ O ¯≤ GPS GFSNJPOT BOE ≤ O ¯ < ∞ GPS CPTPOT 'SPN OPX PO XF XJMM VTF UIF TJHO DPOWFOUJPO UIBU UIF VQQFS TJHOT JO ± BOE ∓ SFTQFDUJWFMZ BMXBZT SFGFS UP UIF GFSNJPOJD DBTF BOE UIF MPXFS TJHOT SFGFS UP UIF CPTPOJD DBTF VOMFTT PUIFSXJTF TUBUFE .JYFEOFTT o 1VSJUZ BOE FOUSPQZ 8F IBWF BMSFBEZ OPUFE UIBU BOZ QVSF TUBUF NBZ CF EFTDSJCFE CZ B EFOTJUZ NBUSJY CVU UIF PQQPTJUF EPFT OPU IPME USVF .PSF QSFDJTFMZ B EFOTJUZ NBUSJY SFQSFTFOUT B QVSF TUBUF JG BOE POMZ JG JU IBT POMZ POF TJOHMF OPO[FSP FJHFOWBMVF UIBU JT FRVBM UP )PX NJYFE UIF TUBUF SFQSFTFOUFE CZ B EFOTJUZ NBUSJY JT NBZ CF RVBOUJmFE CZ JUT QVSJUZ <ñ> Q = US{ˆ ρ(U) },. ñð÷. XJUI Q = GPS QVSF TUBUFT BOE Q < GPS NJYFE TUBUFT 'PS B mOJUF TZTUFN PG EJNFOTJPO E UIF NBYJNBMMZ NJYFE TUBUF ρˆ = I/E HJWFT UIF NJOJNBM QVSJUZ Q = E 'PS UIF RVCJU TUBUF JO &R ñðô XF IBWF Q =. +|ζ| . "T BO BMUFSOBUJWF POF NBZ DIBSBDUFSJ[F UIF NJYFEOFTT CZ UIF WPO /FVNBOO FOUSPQZ <ñ> 4(U) = −US{ˆ ρ(U) log ρˆ(U)},. ñðø. XIJDI JT UIF RVBOUVN BOBMPH PG UIF #PMU[NBOO FOUSPQZ 4 = −L# O QO log QO XJUI QO UIF QSPCBCJMJUZ PG CFJOH JO B NJDSPTUBUF O ɨF FOUSPQZ JT B NFBTVSF PG UIF BWFSBHF JOGPSNBUJPO PCUBJOFE XIFO POF NFBTVSFT UIF FYBDU NJDSPTUBUF PG B NBDSPTUBUF 'PS B QVSF TUBUF UIF FOUSPQZ JT [FSP ɨF NPSF NJYFE UIF TUBUF JT UIF MBSHFS FOUSPQZ "T XF XJMM TFF JO $I ò UIF WPO /FVNBOO FOUSPQZ NBZ CF VTFE UP JEFOUJGZ XIFUIFS B QVSF TUBUF JT FOUBOHMFE PS OPU JG UIF PWFSBMM TUBUF IBT [FSP FOUSPQZ CVU UIF SFEVDFE EFOTJUZ NBUSJDFT IBWF OPO[FSP FOUSPQZ UIF TUBUF JT FOUBOHMFE. ññ 5JNF FWPMVUJPO 8F OPX NPWF PO UP DPOTJEFS UIF UJNF FWPMVUJPO PG RVBOUVN TZTUFNT 'PS B DMPTFE TZTUFN ˆ 4 (U) XIFSF BO FYQMJDJU UJNF EFQFOEFODF NBZ EFTDSJCF XPSL QFSGPSNFE XJUI )BNJMUPOJBO ) ðô.

(34) WJB FYUFSOBMMZ DPOUSPMMBCMF EFHSFFT PG GSFFEPN UIF UJNF FWPMVUJPO JT PCUBJOFE GSPN UIF 4DISÚEJOHFS FRVBUJPO <óø> E ˆ 4 (U)|Ψ(U), J |Ψ(U) = ) EU. ññï. PS JO UFSNT PG EFOTJUZ NBUSJDFT CZ UIF -JPVWJMMFoWPO /FVNBOO FRVBUJPO J ˆ Eˆ ρ4 (U) ˆ4 (U)]. = − [) 4 (U), ρ EU . ññð. "MUFSOBUJWFMZ UIF UJNF FWPMVUJPO NBZ CF SFQSFTFOUFE CZ UIF UJNF FWPMVUJPO PQFSBUPS J. ˆ U ) = T← F− 6(U,. U U. ˆ 4 (U )EU ). ,. ñññ. ˆ 4 (U) XIFO JU EPFT XIFSF T← JT UIF UJNFPSEFSJOH PQFSBUPS FOTVSJOH B DPSSFDU PSEFSJOH PG ) ˆ U )|Ψ(U ) OPU DPNNVUF XJUI JUTFMG BU EJĊFSFOU UJNFT ɨJT BMMPXT VT UP XSJUF |Ψ(U) = 6(U, † ˆ U )ˆ ˆ (U, U ) ≡ ΦU,U ρˆ(U ) XIFSF ΦU,U JT B TPDBMMFE RVBOUVN PQFSB ρ4 (U )6 PS ρˆ4 (U) = 6(U, UJPO PS RVBOUVN EZOBNJDBM NBQ EFmOJOH UIF UJNF FWPMVUJPO PG UIF EFOTJUZ NBUSJDFT <ôñ> ɨF RVBOUVN PQFSBUJPO PSJHJOBUJOH GSPN UIF 4DISÚEJOHFS FRVBUJPO EJTQMBZT TFWFSBM JNQPSU BOU QSPQFSUJFT FOTVSJOH UIBU JU NBQT QIZTJDBM TUBUFT EFOTJUZ NBUSJDFT POUP PUIFS QIZTJDBM TUBUFT BT UJNF HPFT CZ JU JT J MJOFBS JJ USBDF QSFTFSWJOH BOE JJJ DPNQMFUFMZ QPTJUJWF ɨF MBUUFS DPOEJUJPO NFBOT UIBU JG UIF DMPTFE TZTUFN XJUI )JMCFSU TQBDF H4 JT FYUFOEFE XJUI B HFOFSJD OMFWFM TZTUFN XJUI )JMCFSU TQBDF $O UIF UJNF FWPMVUJPO ΦU,U ⊗ *O BDUJOH PO UIF DPNCJOFE )JMCFSU TQBDF H4 ⊗ $O TUJMM QSFTFSWFT UIF QPTJUJWJUZ PG UIF TUBUF 8F XJMM UBLF UIF UISFF QSPQFSUJFT J o JJJ BT B EFmOJUJPO PG B RVBOUVN PQFSBUJPO *O BEEJUJPO UP UIFTF DSJUFSJB UIF 4DISÚEJOHFS RVBOUVN PQFSBUJPO BMTP EJTQMBZT TFWFSBM QSPQ FSUJFT UIBU BSF DIBSBDUFSJTUJD GPS DMPTFE TZTUFNT 'PS JOTUBODF UIF UJNF FWPMVUJPO JT VOJUBSZ BOE QSFTFSWFT UIF WPO /FVNBOO FOUSPQZ PG UIF TZTUFN ɨJT JT DPOOFDUFE UP UIF GBDU UIBU UIFSF JT OP IFBU FYDIBOHF XJUI UIF FOWJSPONFOU GPS B DMPTFE TZTUFN 5P BDDPVOU GPS IFBU BOE QBSUJDMF FYDIBOHF XF JOTUFBE OFFE UP DPOTJEFS UIF UJNF FWPMVUJPO PG PQFO TZTUFNT. ññð 2VBOUVN PQFSBUJPOT 'PS BO PQFO RVBOUVN TZTUFN UIF FRVBUJPO PG NPUJPO NBZ CFDPNF NVDI NPSF DPNQMJD BUFE UIBO UIF -JPVWJMMFoWPO /FVNBOO FRVBUJPO <&R ññð > )PXFWFS VOEFS UXP RVJUF HFOFSBM BTTVNQUJPOT XF NBZ TUJMM EFTDSJCF UIF UJNF FWPMVUJPO XJUIJO UIF GSBNFXPSL PG UIF RVBOUVN PQFSBUJPOT EFmOFE JO UIF QSFWJPVT TVCTFDUJPO 5P UIJT FOE XF DPOTJEFS B HFOFSJD TZTUFN DPVQMFE UP BO FOWJSPONFOU BT TIPXO JO 'JH ññ B ɨF mSTU BTTVNQUJPO JT UIBU UIF TZTUFN UPHFUIFS XJUI JUT FOWJSPONFOU GPSNT B DMPTFE TZTUFN JF UIF UJNF FWPMVUJPO PG UIF DPNCJOFE TZTUFN JT TUJMM EFTDSJCFE CZ UIF -JPWJMMFoWPO /FVNBOO FRVBUJPO ɨF FWPMVUJPO. ðõ.

(35) (a). (b). ˆ 4 BOE EFOTJUZ NBUSJY ρˆ4 (U) DPVQMFE UP BO FOWJSPONFOU ZFMMPX XJUI 'JHVSF B "O PQFO TZTUFN CMVF XJUI )BNJMUPOJBO ) ˆ & BOE EFOTJUZ NBUSJY ρˆ& (U) UIBU UPHFUIFS DPOTUJUVUF B DMPTFE TZTUFN EFTDSJCFE CZ UIF UPUBM )BNJMUPOJBO )BNJMUPOJBO ) ˆ BOE EFOTJUZ NBUSJY ρ(U) ˆ JOU EFTDSJCFT UIF JOUFSBDUJPO CFUXFFO UIF TZTUFN BOE JUT FOWJSPONFOU ) ˆ 5IF )BNJMUPOJBO ) C #Z DPOTJEFSJOH UIF EZOBNJDT PG UIF DPNQPTJUF DMPTFE TZTUFN BOE UIFO USBDJOH PVU UIF FOWJSPONFOUBM EFHSFFT PG GSFFEPN POF NBZ EFSJWF B RVBOUVN EZOBNJDBM NBQ ΦU,U UIBU EFTDSJCFT UIF UJNF FWPMVUJPO PG UIF SFEVDFE EFOTJUZ NBUSJY ρˆ4 (U) POMZ. PG UIF TZTUFN EFOTJUZ NBUSJY ρˆ4 (U) BMPOF JT UIFO PCUBJOFE CZ USBDJOH PVU <DG &R ñðñ > UIF FOWJSPONFOUBM EFHSFFT PG GSFFEPN <ôñ> ZJFMEJOH ˆ U )ˆ ˆ † (U, U )} ≡ ΦU,U ρˆ4 (U ), ρ(U)} = US& {6(U, ρ4 (U ) ⊗ |Ψ& Ψ& |6 ρˆ4 (U) = US& {ˆ. ññò. XIFSF XF IBWF NBEF VTF PG UIF TFDPOE BTTVNQUJPO UIBU UIFSF BSF OP DPSSFMBUJPOT CFUXFFO UIF TZTUFN BOE UIF FOWJSPONFOU BU UIF JOJUJBM UJNF JF ρˆ(U ) = ρˆ4 (U ) ⊗ |Ψ& Ψ& | 8F ˆ † (U, U ) OPX JT UIF UJNF FWPMVUJPO PQFSBUPS BDUJOH PO UIF DPNCJOFE DMPTFE OPUF IFSF UIBU 6 TZTUFN ɨF NBQ ΦU,U EFmOFE CZ &R ññò EFTDSJCFT UIF FĊFDUJWF UJNF FWPMVUJPO PG UIF TZTUFN GSPN UJNF U UP U <TFF 'JH ññò C > BOE GVMmMMT BMM UIF QSPQFSUJFT PG B RVBOUVN PQFSBUJPO <ôñ> 8F UIVT DPODMVEF UIBU VOEFS WFSZ HFOFSBM BTTVNQUJPOT UIF UJNF FWPMVUJPO PG BO PQFO TZTUFN JT EFTDSJCFE CZ RVBOUVN PQFSBUJPOT ˆ )|Ψ& XIFSF |ϕJ JT BO PSUIPOPSNBM *OUSPEVDJOH UIF ,SBVT PQFSBUPST ,J (U) ≡ ϕJ |6(U, CBTJT PG UIF FOWJSPONFOU )JMCFSU TQBDF XF OPUF UIBU BOZ RVBOUVN PQFSBUJPO NBZ CF FY QSFTTFE BT † ,J (U)ˆ ρ4 (U ),†J (U), ,J (U),J (U) = . ññó. ρˆ4 (U) = J. J. )PXFWFS UIF RVBOUVN PQFSBUJPOT PS FRVJWBMFOUMZ UIF ,SBVT SFQSFTFOUBUJPOT DBO JO HFO FSBM CF WFSZ DPNQMJDBUFE UIFZ NBZ EFQFOE PO UIF TZTUFN UIF FOWJSPONFOU BOE UIFJS NV UVBM DPVQMJOH *O HFOFSBM UIFZ BMTP EFQFOE PO UIF JOJUJBM UJNF 'PS QSBDUJDBM QVSQPTFT UIFZ BSF UIVT OPU BMXBZT UIF NPTU DPOWFOJFOU UPPMT UP EFTDSJCF UIF UJNF FWPMVUJPO PG PQFO TZTUFNT 8F XJMM UIFSFGPSF NBLF BOPUIFS GFX BTTVNQUJPOT UP BSSJWF BU B TJNQMFS DMBTT PG RVBOUVN PQFSBUJPOT VTFE JO 1BQFST * *** BOE 7 DBMMFE NBTUFS FRVBUJPOT. ñññ .BTUFS FRVBUJPOT ɨF JEFB CFIJOE NBTUFS FRVBUJPOT JT UP DBTU CBDL UIF RVBOUVN PQFSBUJPOT JOUP UJNFMPDBM. mSTUPSEFS EJĊFSFOUJBM FRVBUJPOT MJLF UIF -JPVWJMMFoWPO /FVNBOO FRVBUJPO *O GBDU BMM. ðö.

(36) EJWJTJCMF RVBOUVN PQFSBUJPOT GPS XIJDI UIF JOWFSTF Φ− U,U FYJTUT GPS BMM UJNFT U ≥ U MFBE UP UJNFMPDBM NBTUFS FRVBUJPOT <ôñ> PG UIF GPSN Eˆ ρ4 (U) = LU ρˆ4 (U), EU. ññô. XIFSF LU JT UIF HFOFSBUPS PG UIF NBTUFS FRVBUJPO "O JNQPSUBOU DMBTT PG EJWJTJCMF RVBOUVN PQFSBUJPOT BSF UIPTF GVMmMMJOH UIF TFNJHSPVQ QSPQFSUZ ΦU ,U ΦU ,U = ΦU +U ,U GPS XIJDI UIF HFOFSBUPS JT UJNFJOEFQFOEFOU LU = L "DDPSEJOH UP UIF (PSJOJo,PTTBLPXTLJo 4VEBSTIBOo-JOECMBE UIFPSFN <ôò ôó> UIFTF HFOFSBUPST IBWF UP UBLF B WFSZ QBSUJDVMBS GPSN BU MFBTU GPS mOJUFEJNFOTJPOBM )JMCFSU TQBDFT EFTDSJCFE CZ UIF -JOECMBE FRVBUJPO.

(37) Eˆ ρ4 (U) J ˆ ˆ† ˆ † ˆ ˆ ΓJ -J ρˆ4 (U)-J − {-J -J , ρˆ4 (U)} . ññõ. = Lˆ ρ4 (U) = − [), ρˆ4 (U)] + EU J )FSF UIF mSTU UFSN PO UIF SJHIUIBOE TJEF SFTFNCMFT UIF -JPVWJMMFoWPO /FVNBOO FRVBUJPO ˆ JT OPU OFDFTTBSJMZ BOE EFTDSJCFT UIF DPIFSFOU UJNF FWPMVUJPO PG UIF TZTUFN )PXFWFS ) ˆ 4 PG UIF TZTUFN BT JU NBZ BMTP DPOUBJO B -BNC TIJGU UIF TBNF BT UIF TZTUFN )BNJMUPOJBO ) EVF UP UIF FĊFDU PG UIF FOWJSPONFOU PO UIF FOFSHJFT PG UIF TZTUFN 'VSUIFSNPSF UIF ˆJ EFTDSJCF JODPIFSFOU JOUFSBDUJPOT XJUI SBUFT ΓJ CFUXFFO UIF TPDBMMFE KVNQ PQFSBUPST TZTUFN USBOTGFS PG IFBU BOE QBSUJDMFT ɨF PQFSBUPST DJ ρˆ4 (U) = BOE UIF FOWJSPONFOU FH † ˆ† ˆ ˆ ˆ ΓJ -J ρˆ4 (U)-J − {-J -J , ρˆ4 (U)} BSF TPNFUJNFT DBMMFE EJTTJQBUPST 5JNF FWPMVUJPO VOEFS . UIF -JOECMBE FRVBUJPO NBZ CF OPOVOJUBSZ BOE MFBE UP B DIBOHF PG UIF WPO /FVNBOO FOUSPQZ PG UIF TZTUFN B DIBSBDUFSJTUJD QSPQFSUZ PG PQFO TZTUFNT ɨF -JOECMBE FRVBUJPO o PS UIF DMBTTJDBM 1BVMJ NBTUFS FRVBUJPO XIFSF OP DPIFSFODFT BSF QSFTFOU o JT VTFE FYUFOTJWFMZ JO CPUI 1BQFST * *** BOE 7 BOE GPSNT B DPSOFSTUPOF GPS BMM USBOTQPSU UIFPSZ EJTDVTTFE JO UIF SFNBJOEFS PG UIJT DIBQUFS *NQPSUBOUMZ UIJT FRVBUJPO NBZ CF NJDSPTDPQJDBMMZ EFSJWFE GPS FYBNQMF VOEFS UIF TPDBMMFE #PSOo.BSLPW BQQSPYJNBUJPO XIFSF UIF DPVQMJOH CFUXFFO UIF TZTUFN BOE JUT FOWJSPONFOU JT USFBUFE QFSUVSCBUJWFMZ UP TFDPOE PSEFS BOE UIF DPSSFMBUJPO UJNFT PG UIF FOWJSPONFOU BSF BTTVNFE UP CF NVDI TIPSUFS UIBO UIF UZQJDBM UJNF TDBMFT PG UIF TZTUFN EZOBNJDT TFF "QQ " GPS GVSUIFS EFUBJMT PG UIF NJDSPTDPQJD EFSJWBUJPO PG NBTUFS FRVBUJPOT ɨF GPSNBM TPMVUJPO UP UIF -JOECMBE FRVBUJPO NBZ CF XSJUUFO BT ρˆ4 (U) = FLU ρˆ4 (),. ññö. XIFSF ρˆ4 () JT UIF JOJUJBM EFOTJUZ NBUSJY ɨF FJHFOWBMVFT λα XIJDI NVTU CF OPOQPTJUJWF UP ZJFME B QIZTJDBM TPMVUJPO PG UIF NBUSJY SFQSFTFOUBUJPO PG UIF -JOECMBEJBO L EFUFSNJOFT UIF UJNF TDBMFT PO XIJDI WBSJPVT USBOTJFOUT EJF PVU ɨF USBDF QSFTFSWBUJPO PG UIF -JOECMBE FRVBUJPO FOTVSFT UIBU UIFSF JT BMXBZT BU MFBTU POF TUFBEZTUBUF TPMVUJPO ρˆT PCUBJOFE GSPN Lˆ ρT = . ð÷. ññ÷.

(38) 1SPWJEFE UIBU UIFSF JT B VOJRVF TUFBEZTUBUF UIF TZTUFN XJMM BQQSPBDI UIJT TUBUF BGUFS B TVċDJFOUMZ MPOH UJNF 8IFO XF DPOTJEFS USBOTQPSU TUBUJTUJDT XF XJMM UZQJDBMMZ BTTVNF UIBU UIF TZTUFN JT TUBSUJOH JO JF IBT BMSFBEZ SFBDIFE JUT TUFBEZ TUBUF. ññò +VNQ PQFSBUPST BOE DVSSFOU PQFSBUPST 5P VTF UIF -JOECMBE FRVBUJPO GPS QSBDUJDBM QVSQPTFT POF OFFET UP JEFOUJGZ UIF KVNQ PQ ˆJ EFTDSJCJOH UIF JOUFSBDUJPO PG UIF TZTUFN XJUI JUT FOWJSPONFOU ɨFTF PQFSBUPST FSBUPST EFQFOE PO UIF NJDSPTDPQJD QIZTJDT PG UIF TZTUFN BOE JUT DPVQMJOH UP UIF FOWJSPONFOU *O UIJT UIFTJT XF EFBM XJUI UXP LJOET PG KVNQ PQFSBUPST ɨF mSTU POF VTFE JO 1BQFS * JT ˆ(E) B†ˆB, J =ˆ. ΓJ = Γ(E) ,. ññø. BOE SFQSFTFOUT QVSF EFDPIFSFODF XJUI B EFDPIFSFODF UJNF /Γ(E) *U TVQQSFTTFT BOZ DPIFS FODFT PG UIF EFOTJUZ NBUSJY PO UIF UJNF TDBMF TFU CZ UIF EFDPIFSFODF UJNF VOMFTT UIFSF JT BOPUIFS NFDIBOJTN DPOUJOVPVTMZ DSFBUJOH OFX DPIFSFODFT *O 1BQFST * BOE 7 XF BMTP VTF KVNQ PQFSBUPST UP EFTDSJCF FNJTTJPO BOE BCTPSQUJPO PG QBSUJDMFT UP B SFTFSWPJS XJUI BWFSBHF PDDVQBUJPO OVNCFS O ¯ GPS XIJDI UIF KVNQ PQFSBUPST BSF (F). ˆ(F) = ˆB, -. ΓJ = Γ( ± O ¯). ˆ(B) = ˆB† , -. ΓJ. (B). = Γ¯ O.. ñòïB. ñòïC. *O UIJT DBTF XF OPUF UIBU XF NBZ JOUFSQSFU JF ρˆ4 (U) = γ( ± O ¯)ˆBρˆ4 (U)ˆB† ,. JB ρˆ4 (U) = γ¯ OˆB† ρˆ4 (U)ˆB,. ñòð. BT FNJTTJPO BOE BCTPSQUJPO DVSSFOU TVQFSPQFSBUPST SFTQFDUJWFMZ 'VSUIFSNPSF XF OPUF UIBU UIF FNJTTJPO BOE BCTPSQUJPO SBUFT GVMmMM EFUBJMFE CBMBODF (F). (B). ΓJ = Fβε ΓJ ,. ñòñ. XIFSF ε JT UIF FOFSHZ PG UIF RVBOUB UIBU JT CFJOH USBOTGFSSFE "T B DPODSFUF FYBNQMF PG B -JOEMCBE FRVBUJPO XF OPUF UIBU GPS PVS FYBNQMF TZTUFN XF PCUBJO

(39).

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