The Relativistic Stern-Gerlach Force C. Tschalär 1. Introduction

Transcription

1 Th Rlativistic Strn-Grlach Forc C. Tschalär. Introduction For ovr a dcad, various formulations of th Strn-Grlach (SG) forc acting on a particl with spin moving at a rlativistic vlocity in an lctromagntic fild hav bn put forward [] xprimnts proposd. To answr ths spculations, th SG intraction, including th ffct of th Thomas prcssion, hav bn drivd from th wll-stablishd non-rlativistic SG intraction nrgy using th Lagrangian formalism. For a particl of mass, charg, spin s, gyro-magntic factor g, th Lagrangian L in th rst fram (RF) of th particl is [] L c L SG whr th astrisk indicats RF quantitis, is th lctro-magntic scalar potntial in th RF, LSG s is th componnt of th RF Lagrangian (rfrrd to in th following for short as Lagrangian ) rprsnting th spin intraction with th lctro-magntic fild [4]. Hr, is th angular vlocity of th spin prcssion around th magntic fild B in th RF dscribd by [] ds / dt s whr g B.. Transformation to th Laboratory Fram Th RF is moving with vlocity c with rspct to th laboratory fram (LF). Th lctromchanical intraction Lagrangian L c obys th rquirmnt that th product γl E b invariant undr Lorntz transformations [] such that L L L whr / /, so that th lctro-mchanical Lagrangian L E in th LF is L E c / A whr A ar th scalar vctor potntials of th lctro-magntic fild in th LF. Th transformation of th spin motion ds / dt in th RF to th LF has bn drivd in rf. [] using th 4-vctor rprsntation S α of th spin s : S s; s s which, in th RF, bcoms S 0; s.

2 Th rsult is [] that th spin motion in th LF bcoms ds / dt s / s T s () whr T is th Thomas prcssion of th RF with rspct to th LF if th particl, with it, its RF, is acclratd prpndicularly to : T E / c whr E is th lctric fild in th RF. Thrfor, th spin prcssion vlocity in th LF is / g T B E / c. Th Lagrangian L of th spin intraction with th lctro-magntic fild in th LF is thrfor L s LSG / st which, xprssd in lctro-magntic filds E B in th LF, is g g g L se/ c sb sb. () L is th wll-know spin intraction Lagrangian for a particl in an lctro-magntic fild [4]. Whil th transformation of LSG to L is thus modifid by th Thomas prcssion of th RF, th lctro-mchanical intraction Lagrangian L E is not affctd by this rotation of th RF.. Canonical omntum Th canonical momntum P for th Lagrangian L is dfind as [5] L L cp c A c A cp. () Th notation / dnots th vctor ( / ; / ; / ). Th Strn-Grlach-Thomas part cp L / of cp is, according to q. (), P g s E/ c s E/ c sb c g s B s B B s. (4)

3 4. Strn-Grlach-Thomas Forc Th Lagrang quation of motion stats that From qs. () (5), w obtain whr F L is th Lorntz forc F th Strn-Grlach-Thomas forc dp / dt L L / x ; L / x ; L / x c d( )/ dt = FL F F Ec B, L. (5) (6) F L dp / dt. (7) Bfor w valuat q.(7) from qs. () (4), it is usful to commnt on a practical application of th forc. Bcaus this forc dpnds linarly on th particl spin s, it was proposd [] to us this forc to polariz an un-polarizd particl bam or altrnativly to masur th polarization of a bam (polarimtr). In ithr cas, th forc is applid or masurd by passing a stord particl bam through a st of radio-frquncy cavitis. Howvr if a particl travrss a localizd fild rgion (zro fild outsid th rgion), it can b sn from qs.(6) (7) that th chang in mchanical momntum c, i.. th intgral c dtcd / dt fild rgion is not affctd by th forc trm dp / dt bcaus it is a total diffrntial in tim P is zro outsid th fild rgion. In this cas, th only contributing forc is L whr / L E c B B G s G s G s xk xk xk xk g G. For compltnss, w calculat th total forc of q. (7). Sinc P, in analogy to th lctromagntic momntum P E A, is tratd as a function of x t only, w hav F L dp / dt H cp c P P / t H P / tc P whr H c P L is th -componnt of th Hamiltonian.

4 Sinc, according to q. () P, k L s s c s t t k t k t k k s s s s k t k t t k kt whr, from q. (), s / t s, s / k 0, w find P, k c s s s s t k k kt kt i.. th prcssion s / tof th spin dos not contribut to th tim variation P / t of P Assuming, for simplicity, to point in th x -dirction, such that, from qs. () (4), c g P,, G s,b sb, ; c g P sbg sb; H G sb G s B whr w hav dfind se / c, w find., /,, / c F H x P t P, g sb G,, x, x x, B B, B G s, s s s x x x, B sb,, F, H / t P / t g sb G sb x x B G s sb x whr th dots indicat /c tims th partial drivativ with rspct to tim se / c. 4

5 For highly rlativistic particls, th forc componnts rduc to sbsb ds,b sb,, F, G x, dt G L / x, dp, / dt d F, s B G sb G G sbsb / x dt G L / x dp / dt plus trms of ordr (/γ), whr d / dt / t/ x + trms of ordr /. Th lading trm of th x -componnt, proportional to γ, is a total tim drivativ th forc dos not contain any γ -trms. In th rst fram ( 0 ), th forc bcoms g g F sb se c 0 / c g g g P s E c L s E c sb 0 0 / ; / Rfrncs ). Cont t al., Strn-Grlach Forc on a Prcssing agntic omnt, Procdings of PAC07, ( ) J.D. Jackson; "Classical Elctrodynamics", Wily & Sons, 975, ch... ) Ibid. ch... 4) A.A. Pomranskii J. B. Khriplovich; Journal of Exprimntal Thortical Physics; Vol. 86, No. 5, ) P.. ors & H. Fshbach; thods of Thortical Physics, cgraw-hill, 95, pp