Lecture 14 (Oct. 30, 2017)

Transcription

1 Ltur Quantum Thory I, Fall Ltur 14 (Ot. 30, 017) 14.1 Magnti Monopols Last tim, w onsidrd a magnti fild with a magnti monopol onfiguration, and bgan to approah dsribing th quantum mhanis of a hargd partil moving in suh a magnti fild. Th prsn of th monopol violats th ondition B = 0, whih mans that w annot globally dfin a vtor potntial A suh that B = A. W bgan by onsidring an isolatd magnti monopol of strngth g loatd at th origin, whih givs a magnti fild g B = ê r. (14.1) r Considr a sphr of radius r, and a losd irular ontour C at θ on th sphr. Th magnti flux through th uppr ap Σ + boundd by C is If w hoos thn w hav C ˆ A dl = πa = Φ C = πg(1 os θ). (14.) A = A ê, (14.3) Σ+ B ds = πg(1 os θ), (14.4) whih givs us g(1 os θ) A =. (14.5) Whr is this potntial wll-dfind? W s that it divrgs at r = 0, but w xptd this, baus th magnti fild is divrgnt at r = 0. As θ 0, th osin in th numrator approahs 1 mor quikly than th sin in th dnominator approahs 0, so A is wll-dfind at θ = 0. Howvr, it blows up at θ = π. Lt us rnam this vtor potntial, as A + g(1 os θ) =. (14.6) In alulating this vtor potntial, w mad th arbitrary hoi to onsidr th magnti flux through th uppr ap boundd by C instad of th lowr ap. If w instad onsidr th flux through th lowr ap, w would find a vtor potntial A = A ê, whr g(1 + os θ) A =. (14.7) This vtor potntial is not wll-dfind at θ = 0, but is wll-dfind at θ = π. Thus, w annot us A + at th south pol, and w annot us A at th north pol, but vrywhr ls w ould qually wll hoos ithr xprssion for th vtor potntial. Baus both of ths vtor potntials dsrib th sam ltromagnti filds in ths rgions, thy must diffr by th gradint of som funtion. Indd, w s that A + A = ( ) A + A ê = g ê = g. (14.8) Thus, as xptd, ths two vtor potntials diffr from on anothr by a gaug transformation.

2 Ltur Quantum Thory I, Fall W hav found that, in ordr to dsrib th magnti monopol fild in trms of vtor potntials, w an writ B = A, whr { A + g(1 os θ) = A = ê, for 0 θ < π + ɛ, A g(1+os θ) = ê, for π. (14.9) ɛ < θ π, In th rgion whr A + and A qual A, thy ar wll-dfind. In th ovrlap rgion, th diffr by a gaug transformation QM of a Chargd Partil Moving in a Magnti Monopol Fild Considr a partil of ltri harg. For 0 θ < π + ɛ, lt th wavfuntion b ψ+ (r, θ, ), and for π ɛ < θ π, lt th wavfuntion b ψ (r, θ, ). Assum that w hav alrady dtrmind ths by solving th Shrödingr quation. In th ovrlap rgion, ψ + and ψ must b rlatd by a gaug transformation. W saw in a prvious ltur that if w mak a gaug transformation thn th wavfuntion must hang by A + = A + g, (14.10) ψ + = ψ ig/. (14.11) Now, w not that th wavfuntions ψ + and ψ must b singl-valud whn + π. This rquirs that g = n Z, (14.1) i.., nh g =, n Z. (14.13) π This is th Dira quantization ondition: th magnti monopol fild strngth must b quantizd, with th quantum a funtion of th ltri harg. W an gain intuition about this rsult with a vagu lassial analogu. Not, howvr, that this quantization is a purly quantum fft, and so annot b fully xplaind lassially. Lt s onsidr th hargd partil moving in th fild of an magnti monopol from a lassial point of viw. Considr an ltri monopol of strngth and a magnti monopol of strngth g, both stati, and displad from on anothr by distan d along th z-axis. W an thn ask about th total angular momntum stord in th ltromagnti fild; this information is ontaind in th Poynting vtor, whih is proportional to E B. Purly by symmtry, w onlud that th angular momntum must b dirtd along th z-axis. If w arry out th alulation, w find that th total angular momntum is indpndnt of th distan d, and is proportional to g. In quantum mhanis, w know that angular momntum is quantizd. If w rquir th total angular momntum w found to b an intgr multipl of, thn w rovr th Dira quantization ondition. Inidntally, this givs us a modl of what is spinning in a spin- 1 partil: a bound stat of a bosoni magnti monopol and a bosoni ltri monopol has spin- 1. Th naïv statistis of this bound stat would b bosoni, baus both th ltri and magnti monopols ar bosoni; howvr, th intrations btwn th two hargs lad to a hang in th statistis.

3 Ltur Quantum Thory I, Fall Chargd Partil in a Uniform Magnti Fild Considr a partil of ltri harg moving in a uniform magnti fild B = Bẑ in thr dimnsions. Th Hamiltonian is ( p H = A). (14.14) Only B z 0, and so w an always hoos A z = 0 and A x, A y indpndnt of z. Th Hamiltonian thn boms p z H = + Π x + Π y, (14.15) whr Π x,y = p x,y A x,y (14.16) ar th kinmati momnta in th x- and y-dirtions. Not that [p z, H] = 0, so w an labl th ignstats by p z. W an thn writ th Hamiltonian in th form p H = z + H d, (14.17) with Π x + Π y H d =, (14.18) and w only hav to dtrmin th sptrum of H d. Th trik is to noti that Π x and Π y hav a simpl ommutation rlation, [ [Π x, Π y ] = i x A x, i y ] A y = i. (14.19) Thus, ths two kinmati momnta (appropriatly rsald) ar anonially onjugat variabls, and th Hamiltonian H d looks lik th sum of squars of anonially onjugat variabls, whih is th Hamiltonian of th simpl harmoni osillator in on dimnsion. Mor prisly, lt Thn, [X, P ] = i, and Π x X =, P = Π y, (14.0) P H d = + 1 ( ) X = P + 1 mω X, (14.1) with ω =. (14.) m This is th on-dimnsional SHO Hamiltonian, with frquny ω, known as th ylotron frquny. As an asid, th lassial motion of a hargd partil a uniform magnti fild is dsribd by irular orbits in a plan orthogonal to th magnti fild. Mathing th ntrifugal for with th for from th magnti fild, w hav whih givs a radius of mv R = vb, (14.3) mv R =, (14.4)

4 Ltur Quantum Thory I, Fall known as th ylotron radius. Th tim priod of th orbit is πr T = v = π, (14.5) ω with ω th ylotron frquny. This is th lassial origin of th ylotron frquny. Now, w hav th Hamiltonian P H d = + 1 mω X (14.6) with [X, P ] = i. W an immdiatly onlud that th nrgy lvls ar ( ) E n (d) 1 = ω n +. (14.7) Th thr-dimnsional nrgy lvls of th full Hamiltonian H ar thn E (3d) p ( n (p z ) = z + ω n + 1 ). (14.8) Howvr, w ar not don, baus w do not know th dgnrais of ths nrgy lvls. W will find that th sptrum is highly dgnrat Dgnray Why is thr dgnray in th sptrum? On way to undrstand th dgnray is to noti that w an dfin nw oordinats in th problm, R x := x + Π y, R y := y Π x. (14.9) Not that [R x, R y ] = i. Thus, R x and R y ar anonially onjugat up to a multipliativ fator. Furthrmor, w not that [R x, Π x ] = [x, Π x ] + [Π y, Π x ] = i ( ) i (14.30) = 0. Similarly, w find that [R x, Π y ] = 0, and mor gnrally, Thus, [R i, Π j ] = 0. (14.31) [R i, H] = 0. (14.3) W hav two oprators that ah ommut with th Hamiltonian, but thy do not ommut with on anothr. Rall from a prvious homwork that whn this is th as, th Hamiltonian must b dgnrat. What is th physial maning of ths oordinats R i? Rall that lassially, th partil undrgos irular motion in th prsn of th uniform magnti fild. Classially, th vtor R = (R x, R y ) is th ntr of th ylotron orbit: if (x, y) ar th tim-dpndnt oordinats

5 Ltur Quantum Thory I, Fall of th partil moving in its irular orbit, and w tak Π i = mv i, thn (R x, R y ) ar th timindpndnt oordinats of th ntr of th orbit. This point is alld th guiding ntr. W s that in quantum mhanis, th oordinats of th guiding ntr do not ommut with on anothr in th prsn of th magnti fild. Th siz of th ylotron orbit will b fixd suh that th magnti flux through th orbit yilds on of th quantizd nrgis, but th loation of th orbit is not fixd, whih lads to th dgnray. Ths dgnrat nrgy lvls ar alld Landau lvls.