Abstract. 1. Introduction. Paul Bracken Department of. University of. Mathematics, kinetic. The quantum. and the. magnetic field. version.

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J Mod Phs 58-6 doi:436/mp33 Publishd Onlin August (http://wwwscirporg/ournal/mp) A Modl for th Quantization of th all Rsistanc in th Quantum all Effct Paul Brackn Dpartmnt of Mathmatics Univrsit of T

conductanc A singl particl Schrödingr modl is introducd and coupld with a constraint quation formulatd from th anon pictur Th Schrödingr quation-constraint sstm can b convrtd to a singl nonlinar

1 J Mod Phs 58-6 doi:436/mp33 Publishd Onlin August ( A Modl for th Quantization of th all Rsistanc in th Quantum all Effct Paul Brackn Dpartmnt of Mathmatics Univrsit of T Edinburg USA brackn@panamdu Rcivdd Januar 5 ; rvisd Fbruar 6 ; accptdd March Abstract Som pcts of anon phsics ar rviwd with th intntionn of stablishing a modl for th quantization of th all conductanc A singl particl Schrödingr modl is introducd and coupld with a constraint quation formulatd from th anon pictur Th Schrödingr quation-constraint sstm can b convrtd to a singl nonlinar diffrntial quation and solutions for th modl can b producd Kwords: Anon all Rsistanc Conductanc Composit Particls Introduction Th quantum all ffct is a rcntl discovrd and now wll known phnomnon which appars in a two- dimnsional lctron sstm which hibits spctacular phnomna whn subctd to an intns transvrs magntic fild [] First ncountrd primntall [] th intgr quantum all ffct h rcivd much stud and w subsquntl followd b th fractional all ffct [34] Rfrring to ths two ffcts th quan- tum all ffct th all rsistanc is found primn- tall to hibit plataus at th quantizd valus h R () f whr f is ithr an intgr or a simpl rational frac- tion Thus () incorporats both of ths ffcts For th intgr c f taks on intgr valus f n 3 and som prominnt fractions for th fractional n c appar in squncs such f n n 4n Th two ffcts show rmarkabl similaritis dspit th diffrncs in origin In both ffcts th localization of lctrons and quiparticlss is blivd to b rsponsibl for th formation of th plataus in th all conductivit At th transitions btwn succssiv plataus in th in- tgr quantum all ffct scaling bhavior h bn obsrvd Thorticall th aim in undrstanding this is to solv th man-bod quantum mchanical problm dfind b th man bod amiltonian givn b p Ar ( ) m c U( r k ) () r k Th first trm onn th right-hand sid is th kintic nrg in th prsnc of a constant trnal magntic fild; th scond trmm is th Coulomb intractionn nrg; and th third trm iss a on-bod potntial du to a uni- to mov in th two-dimnsional -plan It is th intntion hr to st up and solv a simpl form positiv background Th lctrons ar constraind vrsion of () subct to a phsical constraint Such pic- lvls turs occur oftn in this ara for ampl Landau ar dtrmind b solving th Schrödingr quation with a harmonic oscillator potntial Thus a simpl phsical modl which mphizs gomtr in th problm is constructd for a all sstm and it is shown that solu- undr tions can b found A wavfunction is obtaind som spcific sumptions It will b sn that som phsical proprtis that ar vr rlvant can b stab- of lishd from th modl; in particular th quantization th all rsistanc () can b obtaind To bgin to st up th modl som mor phsical concpts nd to b introducd Lt us procd to this [ 56] Stting up th Modl-Composit Particls Lt Ψ( ) b th lctron fild An anon ma b thoughtt of a flu carring a boson or frmion quantum numbr A composit-particl fild ( ) an oprator ph transformation ( imθ( ) ) ) Th ph fild Θ( ) is dfind b Θ( ) d ( ) ( ) is dfind b (3) (4) Copright SciRs

2 whr in (3) is an intgr and ( ) in (4) is th angl mad btwn th vctor and th -ais; rprsnts anon dnsit Th ffct of th oprator ph transformation (3) is to attach m flu quanta to ach lctron Composit particls princ th ffctiv magntic fild B ( ff ) dscribd b th potntial Α ( ) whr Α ( ) dpnds on th trnal vctor t potntial A ( ) and a fild Ck ( ) which is an auiliar fild dtrmind soll b th dnsit ( ) t A ( ) A ( ) C( ) (5) Thrfor from (5) it follows that B ( ) A ( ) B m ( ) (6) ff i i D and so th ffctiv magntic flu is th sum of th ral magntic flu and a trm which can b rgardd a Chrn-Simons flu Now suppos that A ( ) in (4) satisfis th Coulomb gaug condition A ( ) (7) It is possibl to prss A ( ) in trms of a scalar fild A( ) A( ) kka( ) (8) This conclusion is onl possibl in a planar gomtr Substituting (8) into Bff ( ) th fild A( ) can b rgardd th scalar potntial of th ffctiv magntic fild Bff ( ) A( ) (9) This is bicall th tp of constraint w would lik to appl in ordr to solv (); that is b taking a particular ronabl form for Bff ( ) Th stat vctor Ψ is sumd to full or vr narl charactriz th lctronic stat of th sstm Th total fr charg is givn b Q d () s Th stad stat tim-indpndnt wavfunction is givn b iet / whr Ψ is tim-indpndnt and will hav to satisf th tim-indpndnt Schrödingr quation E () Lt us incorporat an additional sumption into th 59 construction of this modl hr Lt us suppos that w can writ Bff ( ) B( ) in th following form B ( ) k () whr k is rlatd to th total magntic flu through th surfac; that is th numbr of flu quanta of th magntic fild and othr constants Th magntic flu dnsit affcts th lctronic stats it modifis th amiltonian Of cours th amiltonian is modifid b th vctor potntial which in a simpl-connctd domain is givn b th usual formula A B For ampl suppos w writ and us () in th form B ( ) a (3) and a is a constant which satisfis Q Φ = Bd ( ) a s s da an (4) In (4) N is th numbr of rlvant currnt carring charg quanta Morovr lt M dnot th numbr of magntic flu quanta which mans th total flu can b writtn h Φ M (5) Whn th flu and charg ar quantizd ths rsults impl that a is a fraction which can b prssd in trms of th flu quantum M h a N (6) On a simpl connctd rgion th vctor potntial can b rprsntd a on-form givn in trms of a singl function which stands for A hr A d d (7) Using (7) th magntic fild can b calculatd and thn (3) ilds a constraint quation (8) a Φ 3 Solution of th Schrödingr Equation Th main obctiv hr is to solv th tim-indpndnt Schrödingr quation coupld with Equation (8) to obtain Ψ Of cours vctor potntial (7) appars in th Schrödingr quation can b clarl sn from () This procdur will lad to a nonlinar quation; howvr it will b found that solutions with th corrct phsical proprtis can b dtrmind in closd form Kping th first trm in () th lft hand sid without th ovrall multiplicativ constant appling (7) lads to i Copright SciRs

3 6 Thrfor () writtn out in full taks th form i me (9) Now th problm taks th form of finding solutions to (9) subct to th condition (8) This will not b don in a compltl gnral wa but with som sumptions which will lad to a phsicall rlvant rsult Suppos th lctron sstm dscribs a rctangular gomtr in th plan Morovr lt Ψ hav a plan wav dpndnc in th dirction so solutions which hav th structur is sought whr Ψ ik () is a ral function of Lt us tak th function in th vctor potntial to b indpndnt of ( ) () Th drivativs of Ψ can b calculatd bd on () and thn substitutd into (9) k k me () This taks th form of a scond ordr quation for in (7) but it is coupld to k me (3) If is sumd to hav th form () thn and (8) sums th simpl form a (4) Sinc th right-hand sid of (4) dpnds onl on (4) can b intgratd onc to obtain which appars in (3) in trms of Imposing a d (5) Substituting (5) into (3) this coupld sstm is rducd to th following nonlinar ignvalu problm me ( k a d ) (6) In Thrfor th dpndnt variabl in (6) is addition to (6) it is usful to writ down a dcoupld vrsion which is obtaind b introducing a nw variabl givn b k a d (7) Equation (6) can b writtn in th form of a pair of quations follows me a (8) Th all rsistanc for this two-dimnsional sstm can b calculatd bd on (8) in fact it can b writtn in trms of Th gomtr is that of a rctangular plat with dgs which ar paralll to th and - To b consistnt with () whr th -dpndnc in Φ is sumd to b a plan wav onl th dimnsion will b of significanc hr Th trminations for intgration localizd at fid -coordinats ar trmd th lft (L) and right (R) dgs of th gomtr Th all potntial is dfind th diffrnc of potntials btwn ths two dgs of th rctangl In fact th all potntial can b obtaind from (6) or bttr in trms of th solution for b mans of V R L m (9) whr R and L rfr to right and lft Onl th longitudinal or plan wav componnt of th currnt dnsit contributs RiA k a d m m (3) Th potntial V is transvrs to th currnt From (8) sinc can b rlatd to th currnt dnsit can b rprsntd ntirl in trms of th variabl h m am am Intgrating and using th dfinition of V givn in (9) I can b rlatd to V follows I d d R L V L L R am R am a (3) B mans of (6) th quantit a can b liminatd from (3) to produc th following rmarkabl formula N I V M h (3) Th rsult in (3) immdiatl implis th all rsistanc is quantizd according to V M h R (33) I N Copright SciRs

4 6 Finall it will b shown that a wavfunction can b dtrmind bd on th coupld sstm (8) In fact th coupld quations in (8) can b combind into a singl nonlinar diffrntial quation for th function from which can b dtrmind To bgin to do this diffrntiat th first quation in (8) and thn divid this b to obtain (34) Diffrntiating both sids of this thr follows (35) Squaring both sids of (34) an additional prssion for / is obtaind Substituting this into th right hand sid of (35) (36) 4 From th scond quation in (8) upon dividing b it follows that me (37) Substituting (37) into (36) a third ordr nonlinar quation in trms of th indpndnt variabl rsults whr w put E (38) me E (39) A gnral solution to (38) ma not b possibl howvr somthing can b don Not that upon omitting ( E ) from (38) th quation can b intgratd Thus w hav ln ln and intgrating givs This can b intgratd wll to giv c 3 c c3 A spcific 3c phsicall ralistic solution to th gnral form of (38) can b approachd follows Th first quation in (8) implis that th sign of is dtrmind b a thrfor whn dos not vanish must b a monotonic function Consquntl on wa in which a cls of solution can b obtaind is to considr th c in which is onl a function of (4) w In fact g can b dtrmind plicitl Diffrntiating both sids of (4) with rspct to w gt w w w w w (4) w w Substituting (4) and (4) into (38) givs ris to th following quation for w ww w E (4) Clarl (4) is nonlinar howvr thr is a wa to produc a solution which is phsicall ronabl Thr ists a quadratic polnomial solution for w which can b prssd in trms of w( ) Ths constants can b spcifid upon substitution in (4) and it will constitut a solution providd that and w E (43) Taking (43) and rplacing th rsult in (4) it is clar th rsulting quation can b sparatd to giv d c (44) E Th ngativ sign givs a tangnt function solution which will b pron to hav pols and can b writtn E tan c owvr th othr choic of sign in (44) givs ris to th rsult arctan( ) c E E This can b solvd plicitl for th function E C E tanh Ec E E C (45) B diffrntiating (45) an prssion for is obtaind Th function which w nd to writ th wavfunction () is found from th squar root of this naml E 4 E a E C C (46) Th wavfunction is thn dtrmind using (46) b mans of Copright SciRs

5 6 / Ψ iet ik This is a boundd function on an right half ais and squar intgrabl ovr th rctangular ara Thus thr ists a solution with th dsird phsical proprtis Thrfor it h bn sn how () mrgs and that phsical clss of solutions to () can b invstigatd Most importantl a link btwn th wavfunctions implid b th modl and th calculation of a corrsponding rsistnc for th modl h bn shown 4 Conclusions An lmntar modl for th quantum all ffct h bn dvlopd It is known in this fild that simpl modls bd on Schrödingr quations can b vr usful in studing th ffct For ampl th quation is solvd with th harmonic oscillator potntial to dscrib and obtain th nrgis of Landau lvls Th modl mphizs svral pcts of th gomtr of th sstm in obtaining th rsults (333) It is quit intrsting that a singl particl Schrödingr quation can b obtaind and solvd in closd form and which incorporats a significant amount of th phsics involvd 5 Rfrncs [] J K Jain Composit Frmions Prspctivs in Quantum all Effcts In: S D Sarma and A Pinczuk Eds J Wil and Sons Nw York 987 [] K von Klitzing G Dorda and M Pppr Nw Mthod for igh-accurac Dtrmination of th Fin Structur Constant Bd on Quantizd all Rsistanc Phsics Rviw Lttrs Vol 45 No 6 98 pp [3] R B Laughlin Quantizd all conductivit in Two dimnsions Phsical Rviw B Vol 3 No 98 pp [4] R E Prang and S M Girvin Th Quantum all Effct Springr Vrlag Nw York 987 [5] Z F Ezawa Th Quantum all Effcts Fild Thortical Approach and Rlatd Topics World Scintific Singapor [6] R B Laughlin Fractional Quantization Rviws of Modrn Phsics Vol 7 No Copright SciRs

Hydrogen Atom and One Electron Ions

Hydrogen Atom and One Electron Ions Hydrogn Atom and On Elctron Ions Th Schrödingr quation for this two-body problm starts out th sam as th gnral two-body Schrödingr quation. First w sparat out th motion of th cntr of mass. Th intrnal potntial