Nonlinear Boundaries in Quantum Mechanics

1 Non linear Boundaries in Quantum Mechanics A rt hur Davids on ECE D ep ar tm en t Ca r neg ie M ello n U n iv er sit y , P i tt sburgh, PA 15213 art dav@e ce. c m u.edu Abstract: B as e d on em piri cal evi den ce , 1 quantum syste m s appear t o be str i ct ly linear and gauge invariant. Th is wor k uses co ncise mat hem at ics to show t hat quant um eigenvalue equat i o ns on a one dimens i ona l r in g can e i t her be gauge invar i ant o r hav e a linear boundar y co nd it io n, bu t no t bot h. Fu rt her ana l ys is s ho ws t hat no n- line ar bo u nda r ie s fo r t he r i ng r est o r e gauge invariance but l ead u nexpect edly t o eigenfunct i ons wi t h a cont inuous eigenvalue spect rum, a d i s cr eet subset of which for m s a H il ber t space w ith en erg y bands. T his Hilbert space main ta ins the pr in ciple o f superpos i t i o n of ei genfunct ions despite t he n onlinear i t y . The m o mentu m op erator rem ains Hermi t ian. I f phy s ical reali t y requires gauge invar iance, i t woul d appear t hat quan t um mechanics s houl d inco rpor ate these nonli near boundary co nd i t i ons. Int roduc tion T o st art , t he t heo r y wil l be r e v ie wed s ho w ing t hat mo me nt u m in a n in f in i t e 1 d ime n s io na l syste m (not a ring) wi th boundaries at infini t y is fu lly linear and gaug e invari ant . T he m o mentu m e igenvalue equat ion will be wr i t t en expli ci t ly in gaug e dependent for m , and i t w ill be sh ow n t hat i dent ical eigenvalues resu l t, in dependent o f gauge. The lineari t y o f t his infini t e o ne d ime ns io na l s ys t e m w it h e f fe ct ive l y no bo u nd a r y c o nd i t io ns w il l a ls o be ap p a r e nt . T hese r e su lt s, o f cour se, are ex pected from the linear i t y and gauge invariance o f standard quant um mechanics. Next, a fini te 1 d imens i o nal r in g will be cons idered, aga in wi t h a mo mentu m eigenvalue e q u a ti on th at c an b e w r i tte n i n g a u g e de p en d en t f orm . B ut n ow th e wav e f un c ti on i s fi ni te all th e way around t he ring, and the wa y it in ter act s w i t h i t self at t he b o undary must be d escribed. I t will be shown that linear boundar y co ndit i o ns lead to dif ferent e i genvalues f or d if ferent gauges. It will t hen be shown that g ivi ng up linear i t y at the boundar y re st ores gauge invar i ance. The gauge invar iant e i genvalues w ill for m a co ntinuous spectr u m o f real numbers. Ho wever, a subset o f t he ei ge nfu nct i ons will be allowed by t he n onlinear boundary co ndi t i ons to form a Hilbert space a nd be superpo sable. This su bset t ur ns out to be w i t hi n a p hase fact or of the co m p let e o rthonor m al set all o wed by li near bo undaries, if t h e i ssue o f gauge i nvar iance is ignored. Thus the ad vantage o f nonlinear boundar y co ndi t i o ns f o r t he one d imensi o nal r ing is t hat essen t i a ls o f Hilbert space and the superpo siti o n pr in cip le are mainta ined al o ng w i t h gauge invariance and Her mi t ici t y o f the m o m entu m operato r. The eigenfunct i ons as soc iated w i th t he continuous spect rum are gener ally not o rthonorm a l or superpo sable. Thus it ca nnot be simply assumed that a ll eigenstat es of a syste m are superposable: i t may be t h at Schrodinger’s cat i s never f ou nd in super po sit i on b ecau se the t wo st ates of the cat ar e not ort honorm al a nd not super po sable . L ikewise, superpo sa bili t y ca nnot be assumed f or all eigenstat es or qubi t s o f a quantum co m puter . 2 The Infini te One Dime nsional Cas e Consider one coo r din ate d im ens i o n extending to ward p l us a nd minus infini t y . The gauge de p e nd e nt mo m e nt u m o p e r at o r c a n be w r it t en: ݌ ௢௣ = െ݅ డ డ௫ െ ݇ . Eq. 1 Un it s h a ve bee n c ho se n so t ha t ƫ = 1 , k is a n ar bi tr ary rea l constant d i ffere nt f o r eac h gauge and x repr esen t s the coordin ate. The gauge depen de nt ei ge nfu nct i on ca n b e wr i t t en ߰ ( ݔ ) = ܣ݁ ௜ ( ௡ା௞ ) ௫ Eq. 2 where A is a constant det ermi n ed by normalizat i o n. As is well kno wn in sta ndard quantu m mechanics, an oper ato r applied to one of i ts e i genfu nct i ons s houl d yield a rea l constant eigenv alu e mul t iplyi ng the same eigenfunction: Thus ቀെ݅ డ డ௫ െ ݇ቁ ൫ ܣ݁ ௜ ( ௡ା௞ ) ௫ ൯ = ݊ ൫ ܣ݁ ௜ ( ௡ା௞ ) ௫ ൯ Eq . 3 so that the real nu mber n is t he gauge independent e i genvalue. The number k, which is gauge depende nt, cance ls o ut of t he ei genva lue. Not e that bo undary cond i t i o ns play no role in this one d ime ns io nal p r o b le m e xt end ing far f ro m t he o rig i n. As u su a l in st and a r d q u ant u m me c ha nic s, t he eigenf unct i o ns ext end o ver the who le do m ain, a nd n f or m s a co ntin uous spect rum o f rea l values. S ince E q. 3 is linear and ho m o geneo us in th e eigenfu nct i on, a nd there are no e ffect ive boundary cond i t i o ns ot h er t han no r m alizat i o n, t hi s s ys t em, inc luding the op erato r and eigenfunct ion together, i s linear. All eigenfunct i o ns are part of t he Hi l bert space. Th i s sy s tem ha s a ga ug e d e pen den t m om en tum op e ra tor , a n d ga u g e de pen d en t eigenfunct i ons. Ho wever, w hen the o perator and an e igenf unct ion are put to gether t he resul t is a gauge indepe ndent e igenvalue. S ince measurement t heor y ho lds t hat t he eigenva l ue s hou ld co rrespond to possible measured m o ment um values, t he overall quantu m syste m here is pro perl y gauge inv ar i ant. T her e fo r e , it ha s be e n c o nf ir me d t hat t his in f in it e o ne d ime n s i o na l s ys t e m is l in ea r a nd gauge invariant . All t he eigenfu nct i o ns form t he usual ort honormal Hilbert space, and all ca n be superpo sed wi t h one anot h er in any linear co mbi nat ion. The Finite One Dimens io nal R ing wi th Linear Bound ary Condi tions Equat i ons 1, 2, and 3 sh ou l d appl y t o t he f i nite o ne d im ensi o nal ring as well , e xcept t hat t he ei ge nfunct i ons s hou l d n ow a l so sat isfy a defini te boundar y co ndi t io n. Standard quan t u m mechanics and mo st physicists ’ star t in g assu mpt i on wo u l d be t hat t hese boundary co ndi t i o ns are linear. T hat is, even t hough the coo r din ate may have a d isco ntin uit y so m ew here due t o the n atu r e of th e ri ng ge om etry , th e w av e fun c ti on sh oul d j o i n on to i tse l f sm ooth ly ev ery wh er e , including at the boundary in d icat ed by t he coo r din ate d i scont in uit y. Thus, if t h e coo rdinate i s c h osen su ch th at x ex t en ds f ro m – ʌ aro und the r ing to ʌ t hen t h e eigenfu nct i on in Eq. 2 m ust have (n + k) = m , where m is a n in teger . T his is needed f or t he wave funct i on t o b e period i c a nd to j o in s m oo t hly a nd linearly wit h i t s elf. The eigenvalue s pectr u m is discreet but gauge dependent . T hat i s, the e igenvalue n co rr esponding to equat i o n 3 will be n = m – k. T his eigenvalue depends explici t ly on k, an d so i s dependent o n the gauge. 3 There are o nl y two differences betwee n the infini t e line a nd the r in g: t he r i n g has a f ini t e c o ord in a te , a n d h a s a def i ni te b oun dary con di ti on. T h e f i ni te ri ng dom ai n i s th e sy s tem un de r invest i gat i on, so t he sali ent t hi ng to chan ge to resto re gauge invari an ce is t he boundar y c o n d it io n . The Finite One Dimens io nal Ring w ith Nonlinear Bou ndary Condi tions What boundar y co ndit i o n wou ld restore gauge invariance o f the m o mentu m ei genva lue? Th e ob vi ous thi n g to try is to d em an d sm ooth n ess a t th e c oo rd i n ate di sc on ti n ui ty n ot of < th e wa ve fu nct io n it se l f, bu t o f t he pr o bab i l it y de ns it y ȥ ȥ and the probabi li t y curre nt densi t y ȥȥ ׏ ) ) , where ׏ ) i s t he grad i ent o f the gauge dependent wave f u nct i on phase. Asi de f ro m being dist in ct l y no nlinear, t hese funct i ons o f the wave fu nct i on wou ld all ow t he phase ) to be di sc on ti n u ou s a t the b ou n da r y , wh er e th e c oordi n a te i s al s o di s c on t i n uou s. I t s h oul d b e n ote d t hat these pro posed nonlinear boundar y condit i o ns w ill not change t he Hermi t ian c haract er o f the m o mentu m operato r. The change f ro m linear co ndit i ons can be acco mm od at ed by phase rot ations o f the m o m entu m operat o r i n oppo site d i rect i o ns at t he b o undary , w hich pre serves Her mi t ici t y . T h e char act er o f the mo mentum o perator as a mat rix w i t h no nlinear boundaries i s devel o ped in A pp e ndix A . S in ce each e i ge nfunct i o n in Eq. 2 has a const ant am p li tude a nd consta nt phase grad ient, t he nonli near boundary condit i o ns are met for a ll r ea l n a nd k, e ven though ) i t se lf i s n ot gauge invariant. Thus the set of eigenfunct ions selected by t he boundary is the same cont inuous spect rum as f o r the i nfin i t e li ne discussed a bove. This means that all r ea l v alues are all ow ed for t he ei ge nvalue n, and thus gauge invar iance is resto red, j ust as f or the infi nite line. Of course, wi th i n te g ra ti on ov e r th e fi ni te d om ai n of th e ri ng co or d i n a te , th e s et of ei gen fun c ti ons wi th a co nt inuo us spect rum l o ses its o rt hono r m al it y. I n o t her w o r d s, i t c a nno t be as su med t hat fo r t he r ing a ll e ig e nfu nct ions w i t h r ea l eigenvalues w ill for m a Hilbert space. It i s kno wn, for examp le, t hat when two or m o r e eigenfunct ions wit h the for m of Eq. 2, wi th di ff er ent val ues o f n are put in s uperpo sit i on, t he r es u l t ing p r o ba b i l it y de ns it y w i l l ha ve p er io d i c var ia t io n, a nd no t a ll va lu e s o f n w ill p e r mit matching the per iod to that o f t he coordin ate r i n g. To ma k e t his expl ici t , suppo se one eigenfunct ion is c hosen as t he ini t i al st at e of the r ing, say wit h arbitrar y rea l m o m entu m ei gen val ue q. Th en we can sy stem ati cal ly go th rou gh all th e o th er ei gen f un cti on s , a d d th em all to t hat ini t ial st at e, and ask w hic h subset o f them w ill resu l t in p er i od ici t y o f t he probabi li t y dens i t y th at m a tc h es th e ri n g. T h e s up er p os i t i on wi l l l ook li ke ߰ ( ݔ ) = ܣ݁ ௜௞௫ ݁ ௜௤௫ ෍ܽ ௡ ݁ ௜௡௫ ௡ୀା ஶ ௡ୀି ஶ Eq. 4 whe r e k is st ill t he g a u g e v a r i a b le, a nd n is st ill t he se t o f re a ls a prior i . q is selected f ro m t he rea l co nt i n uous spect r um o f m o ment u m e igenvalues , a nd t he set a n is co mposed o f co m p le x co efficient s. W r i te o ut the superpo sed eigenfu nct i ons st art ing w i th lowe st o r der ter m , n=0, w hich was t he a ssu med in it ia l st at e o f t he p r o ble m: 4 ߰ ( ݔ ) = ܣ݁ ௜௞௫ ݁ ௜௤௫ ( ܽ ଴ + ܽ ଵ ݁ ௜௡ భ ௫ + ܽ ଶ ݁ ௜௡ మ ௫ + ڮ ) Eq . 5 where A is a rea l no r malizat ion constant, and t he a j are co m p le x coefficie nts. Th e n j are t aken a priori to be an arbi t r ar y sequence o f rea l numbers in increas ing order. W e are l oo king f or a subset o f n j ’s t hat will a ll ow ȥ ȥ to b e pe ri odi c on th e i n terv a l from – ʌ [ ʌ The complex conjugat e o f the above superpos i t i o n i s ߰ כ ( ݔ ) = ܣ݁ ି௜௞௫ ݁ ି௜௤ ௫ ( ܽ ଴ כ + ܽ ଵ כ ݁ ି௜ ௡ భ ௫ + ܽ ଶ כ ݁ ି௜௡ మ ௫ + ڮ ) Eq. 6 Co nsider the pro duct cross ter m s of Eq. 5 and Eq. 6 that contain terms wit h the zero -th orde r c o ef fi ci ents a 0 and a 0 *: ߰ כ ߰ = ܣ ଶ ( ܽ ଴ ܽ ଵ כ ݁ ି௡ భ ௫ + ܽ ଴ כ ܽ ଵ ݁ ௜௡ భ ௫ ) + ( ܽ ଴ ܽ ଶ כ ݁ ି௡ మ ௫ + ܽ ଴ כ ܽ ଶ ݁ ௜௡ మ ௫ ) + … Eq. 7 Each of these zero order terms can be writt en as periodic real expressi o ns suc h as: ( ܽ ଴ ܽ ଵ כ ݁ ି௡ భ ௫ + ܽ ଴ כ ܽ ଵ ݁ ௜௡ భ ௫ ) = 2 | ܽ ଴ || ܽ ଵ | cos ( ݊ ଵ ݔ + ߔ ଴ଵ ) Eq. 8 where ) 01 i s t he p hase angle o f the co m plex number ( a 0 a 1 *). Each ter m will ha ve the n eeded pe ri odi ci t y onl y if th e n j are in t egers. Mo reover, since the hi gher o rder cross t erms will be funct ions o f di ffere nces between t hese integers, t h ose t er m s w ill a lso be prop er ly per i od ic. Therefor e, t he gen era l wave- funct i on w i t h superpo sed e i ge nfu nct i ons ca n be wr i t ten as Eq. 4, with n rest r i ct ed to t he set o f in tegers, while k a nd q r e main arbi t rar y reals. Also, t he indiv idual t erm s o f eq. 4, w i t h n rest ricted to in tegers f o rm an ort h onormal s et a nd a co nventiona l quantu m Hilbert space. I n f act , sinc e t h e measured k inet i c energ y o f this s ys t em is pro port i onal t o t he square o f (q + n), wi t h q conti nuous and n discrete, the Hilbert space w ill be t hat of a f ree part icle in a sys t e m o f quadrat ic en erg y bands. Compar ison to th e We inberg n onl ine a rity . The well known Weinberg nonlinear i t y 2 i s th ou g h t to c on f l i ct wi th ca us ali t y . E s sen ti all y , t hi s is because t hat class of nonlinear i t y m ixes e i genvect or s and changes their direct i o ns in Hilbert space. T he nonlinear boundary co ndi t io ns discuss ed here w ill not have t his pr o perty. Cro ss terms will be gener at ed at the boundar y f ro m the ȥ ȥ nonlinear i t y, but t hese onl y co ntribute to ei ge nfunct i o n select ion, and do n ot feed back in to t he eige nfunct ions t hems elve s. The dynamic vari able r e mains t he co m ple x w ave funct i o n e vo lvi ng acco r din g to t he still perfect l y linear Schrö d in ger equat ion. 5 Summary Because gauge invar iance appare nt ly requi r es no nlinear bo undaries in the o ne d ime ns io na l r ing, t he r e is a co nt inu o us in fi n it y o f no n o rt ho no r ma l mo m e nt u m e ig e n fu nct i o ns with a continuous spectr u m o f real gauge independe nt e i ge nvalues. A di s cret e subset o f t hes e eigenfunct i ons i s all o wed i n o r t honormal superp o si t ion by the n o nlinear b oundar y condit i o ns. This su bset is w hat co nvent i o nal qua ntu m mechanics wo u l d select as t h e co mplete s et , but without gauge inva r iance a nd only a d iscreet spe ct rum . T he subset re m ains superpo sa ble suc h t hat a ray in Hi lbert space, once set up, i s unpert urb ed by the boundary no nlinear i t y . Th e i m pac t of thi s w ork i s l eas t on th e p h y si cs of th e on e di m en si o n al p e ri odi c sy s tem f or wh ic h e xpe r ime nt a l d iffe r e nc e s ma y b e d if f i c u l t t o d ist ing u ish. H o weve r , t h is ve r y s tra i ghtf orwa rd m ath em a t i c al pres enta t i on ope ns qua n tum mech an i cs to th e n onl in ea r r e al i t y t h at per m eates Newto ni an mechanic s and genera l r e lativi t y . The necessit y o f nonli near boundaries f o r thi s st raightf o rward s y ste m suggest s that nonlinearit y may be import ant f or t he desc r ipti on o f a quantu m syste m coup l ed to i t s environment, possibly alo ng the line of Fer mi 3 Ko s t in 4 N. Gisin 5 , D a vi ds on 6 , and others . Cert ainly the Josep hs on ju nct i ons used in s o me important quantum co m put er experiments 7 need to be re-examined, since t h ese de vices can be mod e l ed as a part icle o n a ring suc h as co nsidered her e . It ma y be no t e d t ha t Jos e p hso n ju nct io ns in t he qu a nt u m li m it h a ve b e e n a s su me d t o have d i scret e energ y levels. 8 This wo r k predicts t hat th e se junct ions will be fou nd to hav e e nerg y ba nd s inst ea d . Refe r enc es 1 J.J. Bo llinger , D . J. H e inze n, Wa yne M. I t a no, S . L. G ilbert , D.J. W ine land, P hys. Re v. Let t. 63, 10, 1031 (1989). 2 Steven W e inberg, Phys. Rev . Lett. 62, 485 (1989 ). 3 E. Fermi , Re nd. Lincei . 5, 795 ( 1927) 4 M. D. K ostin , J. Ch em. Phys , 57,3589 ( 1972), and M .D. Kostin , J. Stat. Phy s, 12, 145, (1975). 5 N. Gisin , J. Phys . A: M ath. Gen. 14 (1981) 2259-2267 6 A . Davidso n, Phy s. Rev . A, 41, 6, p . 3395 (1990 ) 7 N. Grøn bech-Jensen an d M. Ci rill o, Phys.Rev.L et t.95, 067001 (2005). 8 John M. Martini s, S. Nam , J. Aumentado, and C. Urbi na, Phys . Rev. Lett. 89, 117901 (2002). 6 Appendix A Nonlinear Hermiti an Bound ary Cond i tion for the Mome n tum Operator I t has be e n s ho w n in t h is a r t i c le t hat lin e ar bo und a r y co nd it i o n s fo r t he mo m e nt u m o perator on a 1D ring pro duce gauge depe ndent e i ge nvalues, w hich m ust be u nphysical. There f or e e i t her t he r ing is not a val id quantu m do m a in or t he li near boundar y condi t i o n s must be w r o ng. The a p p e nd ix la ys o ut so me de t a ils o f t he alt e r na t ive no nl in ea r a nd in ho mo g e neo us boundaries. The art icle showed t hat gauge inv ar iance was r est o red by cont i n u i t y at t he boundary o f t he wave f unct i o n a m plitude a nd of t he gradie nt o f t he phase. T hese nonli near boundar y co ndi t i ons added a co nt inuous spectr um o f eigenvalues to a m o d ified but still ort honormal and superpo sable Hilbert space. The appe ndix w ill sho w t hat t he m o m entu m o perat o r wi t h nonlinear boundaries st ill wo rk s in the us ual operato r ei genva l ue equat i o n f or m o m entu m . Th e pl an h er e i s to s tar t by p res en t i n g th e m a tr i x f orm of th e m o m en tum ope ra tor i n th e spatia l r epre se ntat i o n f or li near per iod i c boundar y condit i o ns. By “spat ial represe ntat ion” is meant t hat t h e syste m vect or is repres e nted by values o f t h e co m p lex wa ve-fu nct i on at d if ferent spatia l l o c at i ons aro und t he ring. In addi t i o n to lineari t y , st andard qua ntum mecha nics requ i res He r mi t ic it y, w hic h me a ns t he ma t r ix mu st be sq ua r e , w i t h r e a l d ia g o na l e le me nt s, a nd ot her elements suc h that t ranspos i t ion of rows and colu mns are co mplex conjugat es. The vari o us m atri x el em en ts c a n th en b e ch os en to c o r re s p on d i n th e l i m i t of sm all coo rd i n a te di f f ere n ce s to th e u s ual di ff eren t i al m o m en tum ope r a tor . Next, it will be considered how t his part i cu lar mat rix mo mentu m o perato r woul d change because of t he proposed nonli near boundary co nd i t i o ns. It will be argued t hat the only change necessar y is a ro t ation of the oper ato r in two par ticular ce lls suc h that Her mi t ici t y is pres erved. Th e de gr e e of ro ta ti on i n th os e tw o c el l s of th e ope r a tor d e p e n ds on th e ph a se of th e wav e- f unc t i on . Th i s de pen den c e o f th e tw o c el l s on th e wa v e f un cti on m a ke s th e op e ra tor n onl i n e a r in pr in cip le, but pr eserves the essent i al str uct ure of Hil bert space. F ina ll y, a c o mmer c ia l ma t rix so lver a was used t o fi nd t he eigenvalues a s the nonlinear boundary co ndit i on was varied in a 20 by 20 mo mentu m oper ator m at r i x . The predicted co ntin uous a nd discret e par t s of the eigenva lue spectr u m emerged, d irect ly co nf irming co ns ist enc y in t he u s e o f no nl in e ar bo u nd ar ie s. The linear period ic mome ntum opera tor The mat rix f or m o f t h e linear m o mentu m o perator i n t he s pat i a l represe ntat i o n wi t h l inear per i od ic b o undar y condi t i o ns is shown b elow in Eq . a1. Th e gauge t erm is s et to zero. This oper ato r is Her mi t ian. I t s size, 7 by 7, re flect s a div isi o n o f t he ring into 7 seg ments to be used as different i al e lements o f the oper a to r. The in creme nt repr esented by o n e segment i s dx . Ne xt neare st neighbor s are used t o cal culat e t he der ivative so t hat t he diago nal meets t he requ i re ment to be real , and t he matr i x as a who le to be Hermi t ian. Not i ce t he i and –i in t he (1 ,7) and (7, 1) co rners, which gives t he o perat or linear per iod i c bo undary co ndit i ons. T he op erato r is bot h l i near and h o m o geneous through t he wh o le do m a in, including t he b oundar y . A 7x7 m atr i x 7 was chose n here simply for co nv enience o f displ a y . Lat er, calculat i o n s wil l be s hown fro m a simil ar 20x20 matrix wi t h n o nlinear boundar i es. ܲ ௢௣ = ଵ ଶ ௗ௫ × Change to n onlinear boundary cond itions Ho w does t hi s matr ix oper ato r change for t he propo sed nonlinear bo undary cond i t i o ns ? Fi r st, express the der ivat ive o f an ar bi t rar y co m p lex funct i o n ȥ in t er m s o f i t s a mp l i t ud e A an d phase D : ݀Ȳ ݀ݔ = ( ܣ ௫ + ݅ܣ D ୶ ) e ୧ D where the x subscr ipt deno t es different iat i on by t he coo r din ate var iable. Ther e f o re, if t he no n l in e a r bo und ar y co nd it io n is t hat A, A x , an d D R x are s m oot h, then what i s requ i red is a way t o d if ferent i at e acro ss the boundar y w here t here m ay be a d i scont inui t y in D . T he li near m at r ix ope ra tor ܲ ௢௣ in Eq. a1 different ia t es acro ss the boundar y twice, o nce in t he fi r st row, and once in th e l as t row , s o th e s al i en t thi n g to d o a t th e b oun da ry o f th e n onl i nea r ope ra to r i s to r ota te th e (1, 7) an d (7,1) ce ll s in oppo si te d irecti ons by Ȱ = οȽ , a s sho wn be l o w in Eq . a3 :  ܲ ௢௣ ௡௟ = ଵ ଶ ௗ௫ × Here Ȱ = οȽ is the ph ase d iscont inui t y o f D at the boundary . So if the eigenfunct ion being o perated on ha s ph ase q x, where q i s real and x goes from – S to S , the n ) = 2 S q. 0- i 0 0 0 0 i i 0 - i 0 000 0 i 0 - i 000 00 i 0 - i 00 000 i 0 - i 0 0000 i 0 - i - i 0000 i 0 0 - i 0000 ie i ) i 0 - i 000 0 0i0 - i 0 0 0 00 i 0 - i 0 0 000 i 0 - i0 0000 i 0- i - ie - i ) 0000 i 0 Eq. a 1 Eq. a 3 Eq. a 2 8 W av e Func tion Phase Disconti nuity ) = 2 S q Momentum Eigenval ue O = (n + q ) - q - 1 q - 1 O = 1 + q 1 - q q - q T h i s ma k e s t h e ma t r i x o p e r a t o r fo r ܲ ௢௣ ௡௟ in E q. a3 i nhom o geneous in the wav e f u nct i o n, and since t hose t wo cells no w depend on the p hase o f the wave- f u nct i on, nonl inear as well. T he de r iva t ive o f t he w a ve - fu nct ion c a lc u lat ed t his w a y w i l l no net he le ss be s mo o t h and c o nt inu o us everywhere e xcept for the d iscont in u i t y in D , whi ch is exact l y what is needed to preserve gauge invariance. Eig enva lue sol uti on s A mat rix of the f o r m of Eq. a3, b ut 20 by 20 in stead o f 7 by 7, was diagonalized by the “e ig ” fu nct io n in Ma t la b a , fo r va r io us va lu es o f ) bet ween zero and S . Figure a1 be l ow p l ot s t he numerical value o f a r a nge o f t he “eig” so lut i ons for t he eigenvalues vert i ca lly against ) on th e ho r izo nt a l ax is. If O is t he real eigenvalue o f th e op era tor , th e s o l uti on f or th e nonlinear boundaries is O = (q+n ), wh e re q is an el em en t of th e cont in uo us a nd real e igenvalue spectr um , and n is an integer. Thi s is prec ise ly what e merged f ro m t he M a tl ab di ag on al i za ti on . In co nt r ast , line ar bo und ar y co nd it io ns would all o w ) = 0 only, a nd give o nly the quant ized m o mentu m v al ues on the l ef t verti cal axi s. F ig u re a1. T he l i nes re p re se nt t he v a ria t io n o f the eig e nvalu es of th e mo me ntum o per ato r abo ve, as c alcu late d by th e M atlab “e ig” f unct io n. a The magnitude o f th e phase disc ontinu ity I of th e w av e functio n is p lo tte d o n the hor izo ntal ax is , f rom 0 to S . The v ertic a l axis is t h e mo me ntum e ige nv alue no r mali zed to ƫ . Whe n I   on th e lef t axis, t h e e ige n val ues are dis cre e t w ith the v alue s -1 , 0 a n d + 1 s h o w n . A s I inc rease s, eac h ei ge n valu e s pli ts i nto tw o mome n tum b ands . An a r bit rary state will h ave a si ngle rea l v alue o f q, b ut mu l tip le n s ta te s i n supe rpo sitio n. Appen dix Endn ot e a Matr ix com put at i ons done wi t h co mm ercia l package Matlab 2008a, The MathWorks, Natick, MA (2008).