Charged Partle n a Magnet Feld Mhael Fowler 1/16/08 Introduton Classall, the fore on a harged partle n eletr and magnet felds s gven b the Lorentz fore law: v B F = q E+ Ths velot-dependent fore s qute dfferent from the onservatve fores from potentals that we have dealt wth so far, and the repe for gong from lassal to quantum mehans replang momenta wth the approprate dervatve operators has to be arred out wth more are We begn b demonstratng how the Lorentz fore law arses lassall n the Lagrangan and Hamltonan formulatons Laws of Classal Mehans Reall frst (or look t up n Shankar, Chapter ) that the Prnple of Least Aton leads to the Euler-Lagrange equatons for the Lagrangan L: d Lq (, q ) Lq (, q) = 0, q, q beng oordnates and velotes dt q q The anonal momentum p s defned b the equaton p L = q and the Hamltonan s defned b performng a Legendre transformaton of the Lagrangan: Hq (, p) = pq Lq (, q ) It s straghtforward to hek that the equatons of moton an be wrtten: q =, p = p q These are known as Hamlton s Equatons Note that f the Hamltonan s ndependent of a partular oordnate q, the orrespondng momentum p remans onstant (Suh a oordnate s termed l, beause the most ommon eample s an angular oordnate n a spherall smmetr Hamltonan, where angular momentum remans onstant)
For the onservatve fores we have been onsderng so far, L = T V, H = T + V, wth T the knet energ, V the potental energ Posson Brakets An dnamal varable f n the sstem s some funton of the q s and p s and (assumng t does not depend epltl on tme) ts development s gven b: d f f f f f ( q, p) = q + p = = { f, H} dt q p q p p q The url brakets are alled Posson Brakets, and are defned for an dnamal varables as: A B A B { AB, } = q p p q We have shown from Hamlton s equatons that for an varable f = { f, H} It s eas to hek that for the oordnates and anonal momenta, { q, q } = 0 = { p, p }, { q, p } = δ j j j j Ths was the lassal mathematal struture that led Dra to lnk up lassal and quantum mehans: he realzed that the Posson brakets were the lassal verson of the ommutators, so a lassal anonal momentum must orrespond to the quantum dfferental operator n the orrespondng oordnate Partle n a Magnet Feld The Lorentz fore s velot dependent, so annot be just the gradent of some potental Nevertheless, the lassal partle path s stll gven b the Prnple of Least Aton The eletr and magnet felds an be wrtten n terms of a salar and a vetor potental: 1 A B= A, E = ϕ t The rght Lagrangan turns out to be: q L = mv q + v A 1 ϕ (Note: f ou re famlar wth Relatvt, the nteraton term here looks less arbtrar: the relatvst verson would have the relatvstall nvarant ( q/ ) A μ dμ added to the aton A A, ϕ d d, d, d, dt Ths s the smplest ntegral, where the four-potental μ ( ) = and ( ) μ = 1 3 possble nvarant nteraton between the eletromagnet feld and the partle s four-velot A μ d q v A/ ϕ dt ) Then n the nonrelatvst lmt, ( q/ ) μ just beomes ( )
3 The dervaton of the Lorentz fore from ths Lagrangan s gven b Shankar on page 84 We gve the (equvalent) dervaton from the Hamlton equatons below Note that for zero vetor potental, the Lagrangan has the usual T V form For ths one-partle problem, the general oordnates q are just the Cartesan o-ordnates = ( 1,, 3 ), the poston of the partle, and the q are the three omponents = v of the partle s velot The mportant new pont s that the anonal momentum p L L q = = = + mv q A s no longer mass velot there s an etra term! The Hamltonan s Hq (, p) = pq Lq (, q ) q q = + + mv qϕ 1 = + 1 mv A v mv qϕ v A Reassurngl, the Hamltonan just has the famlar form of knet energ plus potental energ However, to get Hamlton s equatons of moton, the Hamltonan has to be epressed solel n terms of the oordnates and anonal momenta That s, H ( p qa(, t)/ ) = + qϕ (, t) m where we have noted epltl that the potentals mean those at the poston of the partle at tme t Let us now onsder Hamlton s equatons =, p = p It s eas to see how the frst equaton omes out, bearng n mnd that q q p = mv + A = m + A
4 The seond equaton elds the Lorentz fore law, but s a lttle more trk The frst pont to bear n mnd s that dp/dt s not the aeleraton, the A term also vares n tme, and n a qute omplated wa, sne t s the feld at a pont movng wth the partle That s, q q A p = m + A = m + + v A t j j The rght-hand sde of the seond Hamlton equaton p = s H ( p qa(, t)/ ) q A ϕ(, t) = q m q = v j A j q ϕ Puttng the two sdes together, the Hamlton equaton reads: q A q m = + v j ja + v j Aj q ϕ t Usng v ( A) = ( v A) ( v ) A, B = A, and the epressons for the eletr and magnet felds n terms of the potentals, the Lorentz fore law emerges: v B m = q E + Quantum Mehans of a Partle n a Magnet Feld We make the standard substtuton: p =, so that [, p ] = δ as usual: but now p mv j j Ths leads to the novel stuaton that the velotes n dfferent dretons do not ommute From t s eas to hek that mv = qa / q [ v, v ] = B m To atuall solve Shrödnger s equaton for an eletron onfned to a plane n a unform
5 perpendular magnet feld, t s onvenent to use the Landau gauge, Az (,, ) = ( B,0,0) gvng a onstant feld B n the z dreton The equaton s p Hψ p E m m 1 (, ) = ( + / ) + ψ(, ) = ψ (, ) Note that does not appear n ths Hamltonan, so t s a l oordnate, and p s onserved In other words, ths H ommutes wth p, so H and p have a ommon set of egenstates We p / know the egenstates of p are just the plane waves e, so the ommon egenstates must have the form: ψ / (, ) p = e χ( ) Operatng on ths wavefunton wth the Hamltonan, the operator p appearng n H smpl gves ts egenvalue That s, the p n H just beomes a number! Therefore, wrtng p = d / d, the -omponent χ of the wavefunton satsfes: ( ) Where d 1 χ( ) m ( 0) χ( ) Eχ ( ) + = md m 0 = p / We now see that the onserved anonal momentum p n the -dreton s atuall the oordnate of the enter of a smple harmon osllator potental n the -dreton! Ths smple harmon osllator has frequen ω = q B/m, so the allowed values of energ for a partle n a plane n a perpendular magnet feld are: E = ( n+ ) ω = ( n+ ) q B/ m 1 1 The frequen s of ourse the lotron frequen that of the lassal eletron n a rular orbt n the feld (gven b mv / r = qvb /, ω = v / r = / m) 1 Let us onfne our attenton to states orrespondng to the lowest osllator state, E = ω How man suh states are there? Consder a square of ondutor, area A = L L, and, for smplt, take perod boundar ondtons The enter of the osllator wave funton 0 must le between 0 and L But remember that 0 = p /, and wth perod boundar ondtons p L / e = 1, so p = nπ / L = nh/ L Ths means that 0 takes a seres of evenl-spaed
6 dsrete values, separated b So the total number of states N = L / Δ0, Δ 0 = h / L where Φ 0 LL B N = = A, h Φ0 s alled the flu quantum So the total number of states n the lowest energ level 1 E = ω (usuall referred to as the lowest Landau level, or LLL) s eatl equal to the total number of flu quanta makng up the feld B penetratng the area A It s nstrutve to fnd 0 from a purel lassal analss q Wrtng mv = v B n omponents, m =, m = These equatons ntegrate trvall to gve: m = ( 0), m = ( 0 ) Here ( 0, 0 ) are the oordnates of the enter of the lassal rular moton (the velot vetor r = (, ) s alwas perpendular to ( r r ) ), and 0 r s gven b 0 = mv / = p / 0 = + mv / = + p / 0 (Reall that we are usng the gauge Az (,, ) = ( B,0,0), and L q p = = mv + A, et) Just as 0 s a onserved quantt, so s 0 : t ommutes wth the Hamltonan sne + p /, p + / = 0
7 However, 0 and 0 do not ommute wth eah other: [ ], = / 0 0 Ths s wh, when we hose a gauge n whh 0 was sharpl defned, 0 was spread over the sample If we attempt to loalze the pont (0, 0) as well as possble, t s fuzzed out over an area essentall that ouped b one flu quantum The natural length sale of the problem s therefore the magnet length defned b l = Referenes: the lassal mehans at the begnnng s smlar to Shankar s presentaton, the quantum mehans s loser to that n Landau