The study of interaction and second quantization of electromagnetic fields
for non-zero mass system has boon undertaken in angular momentum
basis and the selection rules for omission and absorption of a massive
particle are derived. Reducton of real transverse electromagnetic
vector potential for non-zero mass system has been derived in terms of
irreducible representations of proper, ortho-chronous, inhomogeneous
Lorentz group in angular momentum basi

Parkash, Om

Rajput, B S

Singh, B

English

IACS Institutional Repository

Indian J. Phy8, 48» 609-519 (1974)Second quantization and interaction of electromagnetic fields for non zero mass system in angular momentum basis : IIOm Parkash, B. Singh and B. S. RajfutDepartment of Kurukfihetra University, Knrtilcshetra(Received 14 June 1973, revised 24 September 1973)The $tudy of interaction and second quantization of electromagnotic fields for non-zero mass system has boon imdortaken in angular momentum basis and the selection rules for omission and absorption of a massive particle are derived. Reducton of real transverse electromagnetic vector potentia] for non-zero mass system has been dearived in terms of irreducible representations of prope r^, ortho-chronous, inhomogeneous Lorentz group in angular momentum basis1, I n tr o d u c tio nIn the present paper we have undertaken tlu^  study of interaction of electro­magnetic fields for non-zero mass system in angular momentum basis. For ll\is purpose the transverse reduced expansion of real electromagnetic ve(?tor potential for non-zero mass system in linear momentum basis derived by Parkash ot al (1974) (henceforth to be referred to as T) has been transformed to that in angular momentum basis in terms of vector spherical harmonics. Replacing the amplitude corresponding to particle wave-function and their complex conjugate in this reduced expansion in angular momentum basis by annihilation and creation operators respectively, the second quantization of real electromagnetic vector potential has been carried out in angular momentum basis and the commutation rules for these operators have been derived. Using the second quantized reduced expansion of real electromagnetic vector potential, the expression for field- Hamiltonian is derived in terms of particle number operators quanta in angular momentum basis, which correspond to circularly polarized field states.Second quantized relativistic reduced expansions of purely transverse real electromagnetic vector potential operator in angular momentum basis has been used to derive tho transition probability for emission and absorption of a massive particle in the inter6w?tion of massive electromagnetic fields with an atomic system and it has been shown that the probability of spontaneous emission in this ca«e also, like that in linear momentum basis for non-zero mass system given in I and that in linear and angular momentum basis for zero mass system (Rajput i970,1971) is non-vanishing. Moreover it is shown that the probability of transi­tion in both the oases (emission and absorption) is proportional to the magnitude509510 Om Parkash, B. Singh and B. . Rajputof linear momentiun of the field. The selection rules for the emission and ab­sorption of a particle in this interaction are similar to those derived in the inter­action of electromagnetic field for zero-mass system with an atom (Rajput 1971).2. R educkd E x p a n s io n  o f  E le ctro m a g n etic  V ector  P o t e n t ia l  IN L in e a r  M o m entum  B asisReduced expansion of real transverse electromagnetic vector potential derived in I for non-zero mass system in linear momentum basis may also be written as follows for the general caseA ( x )  = ( 1 )where=  4 ^ 7 * j ; ^ [ 6 i / ( p . l ) + « * / ( p - 2 ) + e J ( p , 3 ) l e x p [ i { p . x - a . ( 4 > ) t } ] .  .. .  (2)where Cj, Cg* the unit vectors along first, second, and third axes respective-Iv. Let us define now the functions/(P . 7)  =  i ) + ^ r / ( p . 2)1where 7 "= ± 1 and/ ( p . 3 ) = / ( p .  0),and the corresponding unit vectors 64.^ , e_j and Cg as(3)(4)-te.‘•+1 y/2-  _  +~  V 2* 0  =  *» •Substituting eqs. (3), (4) and (6) in eq. (2) we getI ;^ r « ^ i / (P ’ + l)+ «-x /(P . -l)+ «o /(p .O )]Xexp[i{p.*—o»(j>)«}].(8)... (6)3. Rbdfotion in Angular Momentum BasisIn angular momentum basis a wave function for non-zero mass system depends on the energy E, total angular momentum quantum number j, quantum number m of j  (2—component of angular momentum operator), and on the variable a (the absolute value a of which is identified with the spin of the particle). Thus the wave-function in this case is represented as F(E ,j, m, a) which is related to the components of corresponding wave-function in linear m<^montum basis by tlxe following eq. (Moses 1967b)Second quantization fo r  non-zero m ass system  611p \ S / a „ _ i  y.i<1>)F{EJ, m, a). ... (7)whore y =  -(-1, 0, —1, designates the components of the wavefunction, and 0) and 0) are the generalized spherical harmonics of 0 and <j>which are the polar angles of vector p and vary from 0 to n and from 0 to 2tt res­pectively, p — p(sin S cos 0, sin 0 sin 0, cos 0), p =  (E^—/jfi)K The well knownexpansion for exp (ip.x) in this basis is given byexp(ip.x*) =  47t S Sit-iy  ( e j l  ... (8)where r =  \x\, t/jt[(JS^ —/^ ®)*r] is the spherical Bessel’s function of order k, and and (f> are the polar angles which describe the direction of x*Substituting eqs. (8) and (7) in eq. (6), we get2 1 oo j ao kA '(x ,y )^  „ Y. Y Y Y Y {i)k-m+y+2- 2.X idif, i  d e fin e  i  dE Jk[(E^-p^MYjt^\^(e, (!>)0 0X y iV *(» ,0 )  Yk^\^*(0,<f>)X F { E y j j m , o c . ) e x p { - - i E t ) .  ••• (0)This reduced expansion in angular momentum basis is used for calculating the field Hamiltonian in the next section. However, for the compact form of reduced expansion which may be conveniently used for the study of interaction, the fol­lowing properties of generalized spherical harmonics are used (Moses 1967);/ 3 2A?-f 1 \47t 2 «/+ 1 /i: -r = | jfc - i| \X(fc,m', l . y l i ,  1, J, m '+y)(k, o,\,a.\k,\, J ,a)(IQ)which may be readily derived from the well-known relation ~ / et(p—p').x)dx =  (^p—p').and( 4  i n  EBinddSdil>-¥ l i m S  ... (22)which is aogalar momentum analogue of the similar approximationf dp = lim S in linear momentum baeis.(2nfJ pSimilarly the volume integral of second term of B+(x) given by eq. (19) is obtained as514 Om Parkash, B- Singh and B. S. RajputS s  S I ^  (£;*-;»*)*4 t ffl at J aX F*(E, h, m, a) F{E, k, m, a). ... (23)where in addition to relations (2 1 ) and (2 2) we have used the following well known relations also (Edmonds 1957)1:4-1 \ */ d k~  [ - ( s t T r )  i z r - f )*"-^Vt[(E«-/t*)*r]4-(^?*-y»*)V*_i[(^-/t«)*r]and/d f/d d sin ^ r*.* ' >m0> <5) =0 0 flliS(24)(25)(26)7ic,ic>m("> 0) the usual vector spherical haxmonios. The volume integration of third term in H+(x) is== - i s  S S J ^  «*. « ) W  «)• -  (27)Second gpm ntization fo r  non-zero m ass ayatem 516Combining relations (20), (23) and (27) we getS S j  k, m, a)F(E, k, m, a). ... (28)Tn a similar maimer one can calculate the contribution of quantized operator A+ *(x, y) to the field Hamiltonian.Thus^ = i  S S L dE{F*(E, k, m, ol)F{E, k, m,a) ^k m a /t*)»-\-F{E, k, m, a)F*{E, k, m, a)]. (29)U.sing the commutation relation given by eq. (16) and dropping the zero-point energy, we getE... (30)H = S S S I F*{E, k, m, a)F(E, k, m, a).l/)t us now define new annihilation and creation operators as b(8, a) =  [ll{E^ -fi^ Y'*]F{s, a)b*{s, a) =  (jg2_r^ 2ji74 “ )>w)iere s collectively denotes the variables E, k, m. These operators satisfy the following commutation relations(31)[6(5, a), 6(5'. a')] =  [6*(5, a), 6*(5', a')] =  0, [6(5, a), b*{a', a')] = 5(5—s')^,,',(32)where5(5—s') =  5(2?—E')Sic,ie’Stntm' •In terms of those operators the field Hamiltonian becomes #  =  S S S / <2.® Eb*{s, a)b(s, a)t  m Cl= S SfdE E[ (^s, - fl)+ 2^ (5, -l)+ .^ (a, 0)],k mwhore.^ (5, + 1 ), ^(s, - 1 ) and i^(s,0 ) we three different particle number operatm-s given byE(s, a) =  b*(s, a) b(s, a)1(33)(E » -^ ‘ )*F*(s, a)F(a, a), (34)516 O m  P a rk a s h . B . S in g h  a n d  B . S. R a jp u twhich commute with each other and have the independent positive integer eigen values »(«, a). Thiis the energy of the field is15 = S S J EdE[n(8, +l)+n(«, —l)+n(«, 0)].k m(35)Let us assume that all the quanta of this real field are aligned and move along the third axes (i.e., z-direction). p =  (0, 0, p). Then /(p, 0) =  0 due to condi­tion*** (2) and hence {^s, 0) = 0ThenH =  S S J dE E [N {8 ,-l)+N {s, + 1 )] k m=  S S  S i  dEEb*(s,a)b(s,a)k in «-±l=  S S S i dE EI}{8, a).k m « -± i(36)where N[s, +1) and N{a, —1) are the nttmber operators for the particles with helicity + 1  and —I respectively. The quanta of particle number operatorsN{s, a) correspond to two circularly polarized field states (Part I).The base vector which spans the Hilbert space can be chosen to contain these circularly polarized particles of mass /i and spin 1. Let us define the state which contains n-particles (each of mass /t) with variables s^ , s ,^. , .  ,s„ and helicities aj..... a„, as|n > =  n  6*(s«,Oi)j0 > (37)where |0> designates the vacuum state. If all these n- particles correspond to a weU defined quantum state i.e.,«i =  «« = «„ = «, = a* = = a, =  athenn >  — I«(«, 1 )>  =  7=-Vn{8, a) 6*»(«,a)10>.... (38)Aimihilation and creation operators act upon these basis vectors in the following mannerb*(a, a) 1«(«. a)> = [»(«, a)4-l]*|«(«t «)+l >  b{8, «) I n{a, a) > =  [»(«, a)]» | n(a, a)— 1 >... (39) ... (40)* Using leduoed expansion (12) one oon prove that div A is zero in angnlar momentum basis also.In terns of operators 6(s, a) and h*{s, a), the reduced expansion for the operator A+(x), transforms to.!+ (* ) =  J: By A+(x, y)7-±lS I  2 *)XMjr* «= ±i | , _ iSecond quafdizcU ion fo r  non-zero m ass system  517X ^exp(— 6(«, a) &y M  2(2Jb+l) ]2(2Jb+l)m-7»o K ^X (* + l ,m —7, l ,y \ k + l , l ,k ,m )  Jjc+Ji(E^—fi )^h'\YAr+1~ 4 2 (2)H I) ]* 1 > Vl *>—1 » *'> /t*)‘r]m-yiO  ^ A. 1X T (&, ----- j s  a.{k, m—y, 1, y j i ,  1, fc, m)Jk[{E^—fi^ )^ r'\ifc-l V  ^« l-7 > 0  A ^ 1 1x 7  {&,</>).]].Substituting the vector spherical harmonicswhereW ( i  g o tm-7,0 >vF»,t',in((9, )^ == 2  (t)% r (0, ^ )(k', m—y, 1 , y 1 1 , k, m)y Jt'k' ~ k—ly k, k-\-1 ,i* (x )=  \ 2 S I  (i)*-™;  a)11= ± 1  fc- 1  m—Jkx {  i [2(2] ^ )-*  [ 2| S t ) 1‘(41)(42)(43)Reduced expansion of operator A(x) may be readily derived by using the relationJi(*) =  A+(»)+A+*(*)« ••• (44),5. IntbbaotionIn the angular momentum base, the transition probabiUties of emission and absorption in the interaction of electromagnetic fields for non-zero mass ease with an atomic system may be calculated in the manner similar to that for linear mo­mentum basis given in I. For emission, the initial state before interaoti<m ocux- sists of field state containing n {a, a) particles and atomic state with quantum numbers k , m and n for angular momentum, z-component of angular momentum and parity. The final state in emission consists of field state with (n(«, a)-|-l] particles and atomic state with qusmtum numbers kf, mj, n/. The matrix element of interaction Hamiltonian for emission is thus given below0) .»|^7 > =  <^f\ <SI.4+*(jc,0) .o l V> 1 1518  O m  P a rk a s h , B . S in g h  a n d  B . S . B a jp u t(2n)i a )-f 1]*4  - ' (X <  ft\  >  -i-t ("2^1^, )X <^/\  ij>).\v \^t >  -a J * (F *X <  v^ /1 4>).v\fi> ], (46)where wo have used eqs. (39) and (44) from which it is clear that only 0)part of A{x, 0) contributes to the interaction Hamiltonian for emission, while the other part i.e., A+(x, 0) contributes to the Hamiltonian for absorption. The matrix element given by eq. (48) consists of the terms like<  f f  \ Y*k,k'<m{9, 4>)‘ V\rlr>, (fc' =  k+l, k)which can be written in terms of quantum numbers of the initial and final states as follows<  k/mfjtf 1 F%,!,'„(<?, ^ ) . i; I hitHm > , (46)where Y*ic,k',m(9,0). v is an irreducible tensor of rank k. Applying Winger-Echart theorem, it follows that only those matrix element like (49) are non-vanishing for which, the following selection rules are satisfied.k^  — kf'^k) ••• I ,=  mf+m.(47)(48)The probability far the emission x>er unit time in transition from to is pro- porticmal to the square of matrix element (48). Hence it is proportional to [{E^  —/t*)*n(«, a)+ l], which is non-vanishing oven for n(e,tt.) =  0 .In the abBorpti(m the initial state is the same as that of emission while the final field state consists of (n(«, a)—1] particle. The matrix element of interest in this case is <  i/rf>A+(x, 0). v^t>1Second quantization  fa r  non-zero m ass system  519(2n)iX [ » ( 4>) • O'! f i >I t t  )X <  <i>) ■v\t/rf>  —aJ*[(jS*—X < ^ f\  Y*je,ic,m( ,^ <f>) •v\^i >]•Fn tho similar maimer as discussed for emission we get the following selection rules for absorption,kf =  k t+ k ,k i+ lc --l ... \ki--k\ ... (49)nif ™ ... (50)Tlie probability for abosrption is proportional to the number of particles of a givoii kind in the initial state i.e., [{E^—‘/i^ )^ n(s, a)]. The ratio of the probability of omission to that of absorption is proportional to [w(«, a)+l]/n(5, a).The selection rules for emission and absorption of a particle in this case are similar to those obtained in the interaction of electromagnetic fields for zero mass system with an atom. Transitions can take place between those atomic states (luantum number of which satisfy the conditions (50) and (51) for omission and (52) and (53) for absorption. The interaction consists of transition of the atomic system from one quantum state to another. This transition results into an omission or absorption of a particle (which is assumed to have the same mass /^ , oiicjgy E, angular momentiun number fc, ^-component o f quantum number m Uolicity a).A c k n o w l e d g m e n tOne of the authors (O.P.) is thankful to the C.S.I.R. for providing the finau(?ial assistance in the form of research fellowship.R eferencesKumumls, A. R. 1967 Angular Momentum in Quantum Meohanios (Princeton, New Jersey),Chapter 5.Mohos h . E. 1967a J. Math, Phya. 8, 1134.Mos(J8 H. E. 1967b Ann. Phya. 41, 166.Mo808 H. E. 1966 J. Math. Phya. 6, 1244.Parkash Om & Rajput B. S. 1974 Ind. J. Phya. 48, 369. liajpiifc B. 8. 1970 Ind. J. Phya. 44, 609.I^ ajput B. S. 1971 Nuovo Cimento 2B, 46.

Second Quantization and Interaction of Electromagnetic Fields for Non Zero Mass System in Angular Momentum Basis : II

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Second Quantization and Interaction of Electromagnetic Fields for Non Zero Mass System in Angular Momentum Basis : II

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