Basu, K

Bose, S N

English

IACS Institutional Repository

38A NOTE ON DIRAC EQUATIONS AND THE ZEEMANEFFECTBy S. N. BOSEANDK: BASU( K c i r i v t ' L i  /(}} p u b l i c a t i o n ,  S c p L  2 :',ABSTRACT. A new Ircatniuit lias been g i \n i  for ])irru’\s e(jun(if)ns foi Jiytlro-atoms, niid llie radial funetions aie t‘\j)res.sed in teiins of a coinbinalion ol Soniiu spolynomials J  ^ f^ i),  ^ | fp) of only f\\o i‘onsecnlive d e c r e e s n —i; and llie i lemrntar\propeitiefi of such polvTioniials have enabled us to Uicklc Lire Zeeman t iK-et pnddem in ;^eneral (homogeneous fiidd) leading to the standard i|uadratic eijuatioii in enei^»v ba tlie etfcelT. With the helf) of the tvvo-clinionsionnl niatrices .•>//, v of Pauli, llie wave-equations of Dirac can he pul in the well-known matrix fonn:(1.1)I.hr li + Ko +Zc‘^ X  + D V - o , K - K o  +Zr=Y +1  )X =where D is the operatord.vI I+  .S„ \0vaa :If, similarly .S—v.v.f+ yv«, + r.-.s s, and .s= M-vsj.+ ;v.';,/d-r.S; ), (lienS. D .^  r ■ + L,  where I, “ + M(,.s« + Mj.'ie). drAnd the following coniniutation rules can he easily deduced;,s(L-1) H- (L - J).v o, 1)(I.-1) + (L-  j)D = o ; also s'  ^h: i .Hence multiplying the equations on the left hy *S we have(1 .2)ihrIv+!',(, + '^-2— j,SX + rr + Id rhrr  r  uh ^ P.o SY + r + 1r dvAssuming«i!t<2=  q>; Y  = g(j)Y + ( e - l ) ^ ’ - o ;X  +  ( L - i ) X = n .'I' where 'h, are similar one-“ 4;coluinii matrices and functions of <l>), the equations can be rewritten as302 S . N . Bose ami K . Basu( i . d h V '"E I hio + A<l> H- M/ + e ( ’ ) t)M'= O,(J r 1h r I r i dvJ rThe oqiialious become easily scpaiablc if the matrices ‘1* and are so chosen tlmt at first ( L —j)M^ — and = and as ^  i, it follows tlicrefore from commntaliou niles thatd> -  and (L"” i) W '— — /,We observe at once that I^(b — /-(A’ v^ ’herecrr!j9y sin'-Wyand therefore A- can Ik* eithei a [positive or a ne^*ative intei’er.Secondly, if (d>, 'h) are the matrices for positive A’ , then d>) are tlie matrices for ne[»ati\e A’, Also remeniberini^ tlmt the oi>eratoi'a v  aa )commnles with the erjuation-systeir., the angularmatiices can be expressed in terms of spherical harmonics of order A and A 1 in the following foiiii :' I ' -  ; , I-:A' 1 I '  /,•A / A’ + in ,, '  2A--“ r ^I . / /> -  »/ - 1 I ^i W o-i//.•4  m-t-T 2M -I(I:, a positive intenc*r). / 2 h + i  (/v’ - / d !  I( a  = c o s  0 }The functions are interchanged for negative values of />. Writing /(r) =  iF(r) so as to remove the imaginary from the radial equations, we gel(i-5)i/r.dr ' )(/F _ . F .(/ ( r1he + Ivo I-Iv ■ I'n tF - o ,( 1 - 0 .'I'o solve the e(|uatioiis we assumeU’ .i) F =  /o'-“ '''» '‘F , ( 0, C .-^ o c T ^ 'A 'r i fr ) ,  and N =  aK/(Fj| -\vherc Zt'^/hc — a.Note on Dirac Equations and Zeeman Effect 303Substiluting in (1.5), and (ij are easily seen lo satisly the followine equations:(e.2)drJF- A G  , + AF, + + F) = u .I h V^ ^ » - A F ,t A ( ', ,- ''- ^ / V ,- ^ .( K ,,  .K ) * ’ ' - o .r i\ rA he- / I 'lo - l 'U(2.3) where - ^ V o -F j.h r A -  -  -l |'-iand AI'.n-i l', ^F,, M-If F] lG i  —2x F G | - h ‘ i - - ’\^, eliniiiiiilion leads easily to the following equations:-(' t ' "  )*(2.4) r ^ ' / l   ^ (2/‘  ^ I -  .oG) _.A(N -  - o ,li r" ti ipiovidocl -  ,«2 =  n '-!(i .;2'-  i<;«j; l.:2 „ 2.Taking the first eiiuation it can he easily seen that it admits of iiolynoniial solutions if N - /i =  11 (an integer). Writing i> foi eA/, ami writing a Soiiiiie Iiolynoniial in the foiiiiif>) ^II Hn II ~ I . -  j ) ^ / +  -^1 J/<- —x) H 'l> -  J , l> H  ^ , ------------- 1>?.o)' (ir^““ /) //+ 2/^( ’ P 'with eoiKStaut ' so dioscii that it is nurinali/cd according toJ \  ^fr'' T^ I'ht>Yr^ l^ \i>)di>= Iu ’e g e t  I . {'I i r - 1 )} '^ .Hence it easily follows that x' ^^ d^ x " will have solutions as given below :iu: j;where the normalising factor  ^ is to be detennined presently.Our radial functions of cqs. ^2.1) stand thus^2.7) X '=s(^N/ r(2.8) r = f^  N + kU.q) f  =(2.10) Gi = s A / ( .  + |-tL2e  ^ p (p)X or mill i sat ii )ji i ciiuircsj  ( 1 ’^ H (_» ^ )  J  —  J ,. /  A i ;w l l l d l  ; . - J V L S  0 : - -  ' \ /  ^VVc ran vvrilc llir two solutions (A) and (B), currcspondiiiii to posilivr and jjcrativr values of Jc respuctivcly.304 S. N. Bose and K . Basu(A) Iw *fH)■LK2'-A- + /1 ^Iy ,(r )  , , / i  r ' a - i ' * A — V /;\ /  2/1 i I r2/ .- -  1: , y J r )1 r A - i ’ 2 h  4  I rk - f i  2 /.' +  II J t ) , , / i r//'■  +  /* A’ - ( r )  „ / i/,' +  /' f t 2 k +  I(~(r) +  1 r * V  2A — j r A -wild rill it is undrrstood that j -  and arc obtained by changiiij^ tlic sign of /v', in tlie expressions for F  and O in (1.5) above.When the atom is in an electromagnetic field defined by the vecior- poleiilial (A.tA,,A  J ,  the Diiac-eciuations becomeI.h<I.h<K-I I'o +z/-’Zc2X + 1)Y  I '■ AV =  u heY  1 D X ^ /  A X - oheW’h ere A --1 1A .r>' r + A \ „ "I- A .^v ^  ]In the case of a constaiil magnetic field H, in the direction of Z-axis(S’,0 A ^ i l l 7 O t— (sin 0oWe observe in jiassing that in the absence of A, the wave function X  is generally small compared with V —the radial component of F(r) has the factor* /  1 -^' ; ,  while 0(/) has a /  i ; so that the perturbation effect of\'^ h<« V‘ AX IS small as compared with effect due to the term A Y in e<i. (3.1). hi heNote on Dirac Equations and Zeeman Efcct 305Remembering~ 1 cin 0 ---- /(/>■’ ■“ ~/i-t i),-M . /(/c-t/i)(/o +/( + i). ,/<c siu ^  ^ ^   ^ + ,) ( ,/ ,_ ,)V '/ ‘  s i l l  ( 9 =  / ( !  +  / * + i)(l- +  / ' - l  ->) „  / ( / v 7 / < ) ( / . - / i - i )  , . / * + 1 ./'■’ ■ ' V  (2/.’ + i)(:2 ’^ + 3 ) \  (j A -  i )(2A-+i) 'we seek an approximate soluliotA of the equation by djoosiui; one set <»f aiiKuhu function (V^ .^  , and assnininy; the cxislenee of both the sets i\^! ,and (Y^  ^  ^ Tliis can lie clone bj’ suitably loinbiiiingllie (A) and (B) tyj)es, thusTake M j = C jh 'J I-Cl, M i)"!'";, ( / - i, 2, 3. |), vvlieic («i, a.j, u--,, 114) corresponds to A—+/; and (I'Yi "^ '2, v-li "ej) corrcspoiids to /,■ -—(/ 1 ij. Moie explicitly, llieir values arc'  2 /  " I  r, _  _  / <  “ / '  -  I. 1 * -v  2/-hi r ’ ' \ /  2/ + 1  r/+(l) y/'/ — 1-  / / - / r - l - i  , r/ '  /_(r)V -’ ; + 7  rLU) v/' * ' /I 1- n + '  2l t 31 H 1 y-(r) / 1 1 1-'  / ■ A .// H- It + 1 '  .;/ I 1X (v) .,A' . . / '(r) v/ ‘ ' ■^r~ ' / A’  ^ “/ / -  A-V  -’ / + 1 r 'Tlie constants A, N, At of the two tyi»cs of solutions arc dilTcrcnl and arc expressed by the following relations(3..1) A, = (K § - K ? ) - / h r ,  =  N . -aK ,/ | ]C ,1 -J ' ;^]%and N , =  lu -I- /i-. [for (A) -  type with /> < 1.(3.5) A_ =  (E^-E?) '^ 'hr),  (/ h i l 2 - a 2 =.-;u£, N =aE_/|JC;-i-I 'Hi^ and N _= )i- ( /‘ -  [ for (B)—type with — (/1-i)J.If iii and ii-d iffer slightly from one another, vve liave the following ai>proxi- inatc relations(3-6) -t-1, Ni ==N-, and i*By following the usual method of perturbation vve see easily Ibat the chaxac- teristic equation for detenniniug the Eigen value E  W'oitld be(3-7)(.heK - E +/’"a I+ /)ot 1 paj 5jlheE -  li- +  pa.j.jwhere l< — i l l  I.:hi, andd \306 S . N. Bose and K . Basu( , V « )M . = / X? /av. h-Y? ;ax ,  X^ *aV_ 4 \ t  ,ax_  X !  jaV_ +V *  /ax_ X t  rav^ I V* ,aX.dVdVi/Vand X f  etc. are Ininspo.sed conjiinates of X . (the a is jiiveii in (;v,t)) Carryin;.  ^ unt inlef^ialioiis over the an^le-vaiial)le-spaee we gel" 11" 2 2 ■ji.( |/<4-e) i f ,  ,zi-(4fi -i- j)il I- 1) '(( \-i f1 O — O J . V ( / - mM/J M+i)e/ 4-1 ,e I T /. e_)dr.To evaluate the integral { 1 }  or -{lU, we substitute the values,of tlie con esiionding /, g, ami rcuicnibering ; ~ jA, we obtainA, lu{ i r 2N , UK - -‘I/-/ \ <o- ( n , ;|“ - + 0 / e v ' ‘ ''/> J/t f  1 r i* ' (/>) <//'Tf'iio-) ^ )•//j.'riic iniu}.;nil.s au^  uvalualcd cjiiickl}  ^ by rupcalec] partial inlegratifjiis ; thus/ . - V + I 1 n \p) { - r(IP “  /V ?/ !l ’ ( fi H- ;:/A 4f  r J f ^ )   ^ i f ft 2 P I nV ft liftr. -ftI ' p __  1  ^ clJ  //! -b( i^‘I 1) dp= (;/ + l)(// 4- -h l) “  // (// + 2p) =  :iN+ H- 1. Similarlyuei/-  p 2p 4”e p « — i ' di>= >K  ^ -  j..'Vn easy substitution of the value of A+ in the foregoing leads to the following results :CjO(4 ■ I ) / r / . i 'nNote on Dirac Equations ami Zeeman E0ecl 307U-2) Jrl->The evaluation of the third iiilegral cannot, however, he exactly cxpievSscd in a neat form ; for the simple reason that A,, as \\ell as N^, luarc diflerent in the two solutions, as also /) =eA./ and r- H* howe\'er,\vc introduce approximations in the he c^dmiing, and lake(/^.3) A . ^ - A _ ™  |A':, U , - - ! - - . ,  4 i , n i , e ►vve see the third integral'^(rhlr . ]■I. J r  -e(I'riie two deiiiiite integrals can he easily evaluated in Hit' same way as before- We give tlie results ])eU>\N .( J f r  V ' ' t  ’ . - - ; -J  /  V a — I ! ?/ ! i (^ / ^ ^~ ii i~  > ) { » + i)/*'" + •••< i r ‘+ :;/< i -i 2it +  2)V(N -/)(N + ?+ i).(4-5^HenceCfjJ r  "  '■ (r )r t t  '\lr=2  V (N W ) ( N  -  / -  >).{ I I U =  - :N I'nObserving Ko' E-*], vve see{III>  =1 K(.N ,, +iS-' \ /  K\  N j21* 2 N ■hrKo2N== “(, and f'h / — ^ / V N -  / - 1hrl*-()titiiliiig tile values of tlie three integrals (and making ) we see that thedeterininaiilal equation (.l-/) takes the form(4.6)l i - l i f - - - ' -  w/'(iH 2I t- I./illll i)(H- m + i) '- /+!  i„, +1 )2/ + 1wherein we have put M + i — (magnetic quantum number),I C - K - -  * " m/hiH ;2 / + T308 S. A  ^ Bose and K, Basu1'hc result (<1.6) agrees completely with that quoted by Betlie (1933^ supposed to have been worked out by liini from Pauli’s equations. Condon and Shortley (1035) obtained similar determinant from principles of quantum mecha­nics Ijy applying two-fold perturbations (spin-orbit and magnetic) siimiltaiicously. It may be noted that Darwun (igaS) has iimiiy similar features with our mode of attack, and we can claim some elegance by our introduction of Sonine’s piopcrtics, which exhibit our solutions in a good perspective.nNivr:iisiT\\Kamma, Dacca.C a1.('T)TYA t ' \ l \  liliSlTV, f ’l'JM K C'mna ak Roao, CAM'r’nR K 1 ’ R V N C' S n , 1033, Jhiiidbiuh det 24, pe»mdnti, [•; r  ami Slmiiu-v Ti. II. 1035, The Thn}t v ol Atomic Sptu i}a fCaml)ri(li»i- rnivi'tsilv pDarwin C. Tnu /Ou' Six , l,omlon fA), II8 p. 1^3 1

A Note on Dirac Equations and the Zeeman Effect

http://arxiv.iacs.res.in:8080/jspui/bitstream/10821/1309/1/A%20note%20on%20dirac%20equations%20and%20the%20zeeman%20effect_Bose%20S%20N.pdf

A Note on Dirac Equations and the Zeeman Effect

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