On the Hamiltonian of a Multi-Electron Atom
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1 On th Hamiltonian of a Multi-Elctron Atom Austn Gronr Drxl Univrsity Philadlphia, PA Octobr 29, Introduction In this papr, w will xhibit th procss of achiving th Hamiltonian for an lctron gas. Making th simplification that th lctron gas b in a static lctromagntic fild, as wll as th simplification that thir vlocitis b much smallr than th spd of light, w nd up with a multi-trm xprssion which, in th non-rlativistic limit, is a vry good approximation. Along th way to this final xprssion, xplanations will accompany intrmdiat stps. In addition to this, a physical dscription of ach trm will also b providd for th bnfit of th radr. 2 Procdur 2.1 Elctron-Fild Intractions A good plac to start will b th rlativistic nrgy of a singl lctron. E = (m c 2 ) 2 (pc) 2 (1) In ordr to xpand this nrgy into multipl trms w must slightly altr its currnt form. E = m c 1 2 pc ( m c 2 )2 (2) Making us of th binomial xpansion, w gt: [ E = m c ( ) 2 pc m c 2 1 ( ) 3 pc 8 m c 2.. (3) Lt s now look in mor dpth at th trms that cam from this xpansion. By multiplying th factor of (m c 2 ) through th brackt, th first trm which appars is th rst nrgy of th lctron. E 0 = m c 2 (4) Th scond trm which appars is th classical kintic nrgy of th lctron. p2 E 1 = 2m (5) Th nxt trm is th first ordr rlativistic corrction to th lctron kintic nrgy. Whr: p4 E 2 = 8m 3 c 2 (6) p 4 = ( p p) 2 (7) By th principl of minimal lctromagntic coupling to (5), w can xprss th Hamiltonian in trms of th magntic vctor potntial A. Th principl of lctromagntic coupling says that th quation of motion of a chargd particl in som xtrnal lctromagntic fild will b obtaind from th fr quation by th following substitution: p p q c A. Th nrgy E 1 will bcom: p2 E 1 = 2m 2 2m c 2 A A p A A p c 2m (8) Lt us now considr multipl lctrons, an lctron gas. So far, w hav only considrd th kintic nrgy of an lctron with th addition of prforming 1
2 th gaug transformation: p p q c A. Th Hamiltonian thn bcoms: p 2 N p 4 N 8m 3 c 2 p A(r ) A(r ) p c 2m 2 2mc 2 A(r ) A(r ) (9) Whr m is now undrstood to b m (th mass of th lctron) th indx rprsnts th th lctron, and r rprsnts th coordinat of th th lctron. th xtrnal magntic fild, w also obtain a contribution to th prcssion of th spin in th xtrnal lctric fild: 1 2 m 2 c 2 Ŝ ( E(r ) p ) (13) Whr th 1 2 is th Thomas Prcssion Factor, a kinmatic corrction includd du to th fact that th lctron is not in its rst-fram, it is acclrating. This trm taks car of th lctrons orbital angular momnta. A moving lctron ss: Point particls undr th influnc of th Lorntz Forc ( F = E xt (r ) c [ v H xt (r )), would giv us a Hamiltonian including ths trms: p 2 N 2m mc p A xt r 2 2mc 2 A xt (r ) 2 Φ xt (r ) (10) So with this intraction in mind, w add th last trm ( φ xt ) to our growing list of trms in th total Hamiltonian of our lctron gas. W must also considr th spin of th lctron intracting with th xtrnal fild. Th classical Hamiltonian of a singl lctron in an xtrnal magntic fild is: µ B xt. Quantum Mchanically, w can rplac this hamiltonian with: Again, for multipl lctrons: Ĥ = mcŝ B xt (r ) (11) mcŝ B xt (r ) (12) This rsult is of cours basd on th approximation that th magntic momnt of an lctron is 2 µ B, whr th Bohr Magnton µ B = h 2mc. It s rally: µ B 2(1 α 2π...), whr th fin structur constant α In addition to th trm involving th intraction btwn th lctrons dipol momnts and and B ff = γ( B xt p mc E xt ) (14) E ff = γ( E xt p mc B xt ) (15) Plugging th ffctiv magntic fild in for B in quation (12), w gt: γ mcŝ( B p mc E) (16) Aftr som multiplication, and in th limit of v << c γ 1: mcŝ B xt (r ) 2m 2 c 2 Ŝ [ E xt (r ) p (17) Now suppos th lctron is an xtndd particl - i.. that is has som probability dnsity ρ(r) cntrd around r 0, with th condition that: ρ(r)dv = 1. Aftr all, it must b found somwhr! If w xprss th last summation in quation (10) as: ρ(r)φ(r)dv (18) And, xpanding Φ(r) about r 0 w gt: ρ(r)[φ(r 0 ) r i Φ i (r 0 ) 1 2 r i r Φ i (r 0 )... (19) 2
3 By multiplication and latr intgration of quation (19), w achiv: Φ xt (r 0 ) 1 r Φ (20) Kp in mind som important stps: 1 : ρ(r)dv = 1 By symmtry: 3 : 2 : r i ρ(r)dv = 0 r i r ρ(r)dv = δ i 1 3 r2 Th sourc of th xtrnal potntial is all of th charg xtrnal to ach individual lctron. Th rlationship is: 2 Φ xt = 4πρ xt (21) Gnralizing for all of th lctrons: π h 2 Φ xt (r ) 2m 2 c 2 ρ xt(r ) (22) At this point w ar don with all singl lctronfild intraction trms which contribut to our total Hamiltonian. Hr is all of th work w hav don so far in constructing th Hamiltonian of a gas of lctrons: p 2 N 2 2mc 2 A xt (r ) 2 p 4 N 8m 3 c 2 Φ(r ) 2m 2 c 2 Ŝ [ E xt (r ) p mc p A xt (r ) mcŝ B xt (r ) π h 2 2m 2 c 2 ρ xt(r ) (23) 2.2 Elctron-Elctron Intractions Although w hav considrd all possibl intractions involving lctrons and th lctromagntic fild thy ar in, w havn t discussd any contributions to th Hamiltonian involving th magntic fild producd by ach lctron s magntic dipol momnt. In an attmpt to construct trms involving lctron-lctron intractions w must sparat out what w hav alrady don. What w will do is to r-xprss th filds as th following: Φ = Φ xt Φ int B = B xt B int E = E xt E int A = A xt A int Th intrnal or intrinsic contributions will b th ons du to lctron-lctron intractions. Th most obvious lctron-lctron intraction trm in our Hamiltonian will com from th Coulomb rpulsion of th lctrons. 2 r (24) To avoid doubl counting, w will sum ovr < k (this will b tru for almost all of th trms in this sction; k is th indx givn to th lctron that th th lctron will s). Th nxt intraction will b du to th orbital motion of th th lctron in th k th lctron s magntic fild. By applying A xt A xt A int to th third trm in (23), w will gt: mc p ( A xt A int ) (25) Now, th intrinsic magntic vctor potntial is du to th k th lctron. It is gottn by taking th curl of th magntic fild of th k th magntic dipol: H k (r) = c ( r r k) v k r r k 3 (26) 3
4 Th magntic vctor potntial of th k th lctron is: A k (r) = [ ( r rk )( r r k ) 1 2c r r k 3 v k (27) r r k Making th substitution of A k (r) in for A int in (25), w gt th first trm back along with this scond trm: 2 2m 2 c 2 p ( r r k )( r r k ) r 3 1 p k (28) r Nxt will b th intraction btwn th spin of th th lctron in th magntic fild producd by th k th lctron s orbital angular motion. By plugging in xprssion (26) into th 6th trm in (23) w will gt: k 2 1 m 2 c 2 r 3 Ŝ [( r k r ) p k (29) Nxt w will mak th substitution that E xt E xt E int into th quation involving th dot product of th spin of th th lctron with ( E xt p ): 2m 2 c 2 Ŝ [( E xt E int ) p (30) k By xpanding th cross product as wll as th dot product, w obtain th original rsult back along with anothr trm. First I will stat that at rst, th lctric fild of th k th lctron is xprssd as: E k = ( r r k) r r k 3 (31) Th additional trm in quation (30), aftr substitution of (31) is th following: k 2 1 2m 2 c 2 r 3 Ŝ [( r r k ) p (32) This trm can roughly b intrprtd as th intraction of th spin of th th lctron in th k th lctron s fild (similar in natur to th 7 th trm of (23), but causd by th k th lctron s fild, not by th xtrnal fild itslf). W must also tak into account th mutual intraction btwn spin magntic momnts of th th and k th lctrons, which ar not mutually pntrating. Th contribution to th Hamiltonian will b as follows: 2 m 2 c 2 Ŝ B Ŝk (33) Whr upon substituting th xprssion for th magntic fild B (th) into th prvious xprssion abov, it turns into: 2 3( r r k )( r r k ) m 2 c 2 Ŝ r 5 1 r 3 Ŝk (34) Th nxt trm is vry important in astronomy and astrophysics. Hyprfin structur is ssntially th diffrnc in nrgy lvls of atoms and molculs du to intrnal filds. Hyprfin splitting occurs bcaus th nuclar magntic dipol momnt is locatd in th magntic fild gnratd by th lctrons. A nutral Hydrogn atom for instanc has on lctron, which upon transition of spin from ( ) will produc a chang in nrgy, which w must tak into account. It is of astrophysical importanc bcaus it is associatd with intrstllar rgions of HI, and it is this magntic dipol transition which producs radiation with a charactristic wavlngth of about 21cm (radio). Th contribution to th Hamiltonian is: 8π 2 3m 2 c 2 δ( r r k )Ŝ Ŝk (35) Th last trm in our Hamiltonian will b obtaind by rplacing Φ xt Φ xt Φ int into (20) (rmmbr - this particular rsult cam about by assuming th lctron has a distribution in its location): (1 r2 6 2 )(Φ xt Φ int ) (36) Th intrinsic or intrnal potntial w ar spaking of is th potntial producd by th k th lctron: Φ int (r) = k r r k (37) 4
5 Aftr substitution into (36), w obtain th prvious rsult plus anothr trm: Which simplifis to: r2 2 (38) 6 r π 2 h 2 m 2 c 2 δ( r r k ) (39) Lt s rcapitulat. It it now tim to xprss th full hamiltonian in all of its glory: p 2 N p 4 N 8m 3 c 2 mc p A xt (r ) Rfrncs (1) T. Itoh 1965 Rv. Mod. Phys. 37, (2) R. Gilmor 2010, Lcturs on Quantum Mchanics - Drxl Univrsity (3) H. Hoogland 1978 J. Phys. A: Math. Gn (4) D. Griffiths, Introduction to Quantum Mchanics, 2nd d.: Prntic Hall (2005) 2 2mc 2 A xt (r ) 2 Φ(r ) 2m 2 c 2 Ŝ [ E xt (r ) p mcŝ B xt (r ) π h 2 2m 2 c 2 ρ xt(r ) 2 2 r 2m 2 c 2 p ( r r k )( r r k ) r 3 1 p k r k 2 1 m 2 c 2 r 3 Ŝ [( r k r ) p k 2 1 2m 2 c 2 r 3 Ŝ [( r r k ) p k 2 3( r r k )( r r k ) m 2 c 2 Ŝ r 5 1 r 3 Ŝk 8π 2 3m 2 c 2 δ( r r k )Ŝ Ŝk π 2 h 2 m 2 c 2 δ( r r k ) (40) 5
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