Derivation of Electron-Electron Interaction Terms in the Multi-Electron Hamiltonian

Transcription

1 Drivation of Elctron-Elctron Intraction Trms in th Multi-Elctron Hamiltonian Erica Smith Octobr 1, Introduction Th Hamiltonian for a multi-lctron atom with n lctrons is drivd by Itoh (1965) by accounting for both th trms that involv only a singl lctron, which w will rfrnc as th jth lctron, and th trms that involv lctron-lctron intractions. Th trms in th Hamiltonian com from four spac-tim filds: th scalar potntial Φ(x, t), th vctor potntial A(x, t), th lctric fild E(x, t), and th magntic fild B(x, t). Each of ths filds has two parts: xtrinsic and intrinsic. Th xtrinsic parts of ach fild is takn xclusivly from th jth lctron and dos not tak into account th othr lctrons. Th intrinsic parts, howvr, com from th intraction of th othr n - 1 lctrons. W also do not tak th lctrons to b stationary; thus thy xprinc an ffctiv lctric and magntic filds du to motion. Th ffctiv lctric fild is and th ffctiv magntic fild is E ff = γ(e + v c B) B ff = γ(b v c E). Howvr, sinc w know that th ratio of th spd of th lctron to th spd of light is small, w can tak γ 1. For th xtrinsic parts of th filds w will lav th quations as statd; howvr, for th intrinsic parts of th filds w must rmmbr that th vlocitis usd ar actually th vlocity diffrncs btwn th kth and jth lctrons. 1

2 Singl Elctron Trms W bgin with th singl lctron trms of th Hamiltonian: H singl = j j j 1 m p j (1) 1 8m 3 c p4 j () mc p j A x (r j ) (3) mc A x(r j ) (4) φ x (r j ) (5) mc s j B x (r j ) (6) m c s j E x (r j ) p j (7) π h m c ρ x(r j ) (8) Th trms (1) and () ar drivd from th usual rlativistic quation for nrgy E = (mc ) + (pc) = mc 1 + ( pc mc ). Taking a binomial xpansion of th squar root givs E = mc + p m p 8m c, nglcting highr ordr trms. Hr w s that (1) is th scond trm of th prvious quation, summd ovr all lctrons, and () is th third trm. Th trms (3) and (4) com from th principl of minimum lctromagntic coupling, which causs us to rdfin momntum as follows: p p q c A.

3 Plugging this into (1) givs back (1) as wll as (3) and (4). W gt (5) from th rlationship btwn potntial and potntial nrgy, which ar rlatd by a factor of charg. W can s that th trms (1), (3), (4), and (5) mak up th usual Hamiltonian for a singl lctron undr th Lorntz forc 1 E x (r j ) c v j B x (r j ). In ordr to driv (6) and (7), w must tak into account th intraction btwn th spin magntic momnt of th lctron with th magntic fild 1. Th nrgy of this intraction for th jth lctron is givn by µ s B ff whr µ s is th spin magntic momnt. This is rlatd to th spin by µ s = g s µ B s j h. whr g S = is th g-factor for th lctron, and µ B is th Bohr magnton. Th Bohr magnton is givn by µ B = h mc. Plugging this and g S into th quation for th spin magntic momnt w gt µ s = h s j mc h = mc s j and plugging this back into our original quation for th nrgy du to th spin magntic momnt in a magntic fild, w gt mc s j B ff. Finally, plugging in for our B ff, w gt two trms: mc s j B x ( v mc s j x) c E. Th first trm, summd up ovr all j lctrons, givs us (6), which is th intraction of th spin magntic momnt of th lctron with th xtrnal 3

4 magntic fild 1. W would lik to put th scond trm in trms of momntum instad of vlocity, which givs us m c s j (p E x ). Multiplying by th Thomas prcssion factor of 1 which coms from making two Lorntz transformations and summing ovr all j givs us (7), which is th trm that causs th prcssion of th spin in th xtrnal lctric fild 1. To driv (8), w will tak th lctron to b an xtndd particl with probability dnsity ρ(r) cntrd at x 0 with sphrical symmtry around this point. Th nrgy for this lctron would b E = ρ(x)φ(x) dx. W thn want to xpand Φ about x 0 and nglct trms highr than scond ordr in x, which givs us E = ρ(x)φ(x 0 ) + x i Φ i (x 0 ) + 1 x i x j Φ ij (x 0 ) dx. Th intgral in th first trm of this quation intgrats to 1 and th scond trm is zro by symmtry. Th third trm bcoms: ( ) ( ) 1 1 Φ ij (x 0 )δ ij 3 r whr r = x + y + z. Th nrgy thn bcoms ( ) ( ) 1 r E = Φ x (Φ ij δ ij ) 3 ( ) r = Φ x (Φ ii ) 6 Th first trm is just (5). Sinc Φ ii = Φ x = 4πρ x whr ρ x is th charg dnsity, w can tak th scond trm to b ( ) r (4πρ x ) 6 4

5 W can now plug in for th man squar radius, which is r = 3 4 which givs us ( π h ) ρ m c x (r j ) which, summd ovr all j, givs us (8). 3 Elctron-Elctron Trms W now turn our attntion to th trms in th Hamiltonian that com from lctron-lctron intractions, which ar as follows: ( h mc ), H multipl = (9) r jk m c p (r j r k )(r j r k ) j + 1 p 3 k (10) r jk j k j k 1 s m c 3 j (r k r j ) p k (11) 1 m c r 3 jk m c s j s j (r j r k ) p j (1) 3(r j r k )(r j r k ) s k (13) 8π 3m c δ(r j r k ) s j s k (14) π h m c δ(r j r k ) (15) In ordr to driv any of ths trms, w must first construct quations for th intrinsic portions of th scalar potntial, vctor potntial, lctric fild, and magntic fild. Th scalar potntial producd by th kth lctron on th jth lctron is Φ k (r) = (16) r j r k 5

6 whr j k. This is simply th Coulomb potntial. Th vctor potntial producd by th kth lctron on th jth lctron is 1 A k (r) = (rj r k )(r j r k ) + (r j r k ) p m r j r k 3 k (17) but this is not th only componnt of th vctor potntial. Thr is also a portion of th ovrall potntial that is producd by th spin of th kth lctron, which is A k,spin (r) = µ (r j r k ) r j r k 3 (18) Th lctric fild du to th kth lctron on th jth lctron is givn by E k (r) = (r j r k ) r j r k 3. (19) Th magntic fild from th kth lctron on th jth lctron has two componnts. On is producd by th orbital motion of th kth lctron. According to th Biot-Savart law, this is givn by 1 ( B k (r) = (r r k ) mc) p k r r k. (0) 3 Th othr componnt is producd by th spin of th kth lctron. This fild is givn by B k,spin (r) = A k,spin = µ (r j r k ) r j r k 3 = 3µ (r j r k )(r j r k ) µ(r j r k ) (r j r k ) r j r k 5 (1) W now mov on to driving th lctron-lctron trms using ths filds. If w considr ach fild in th singl lctron trms not to b just th xtrinsic fild, but th total fild which includs both th xtrinsic and intrinsic trms, th singl lctron trms will b rturnd and w will also gt th lctronlctron trms. Sinc th total fild will also rturn th prviously drivd quations in most cass, w will just rplac th xtrinsic trms with th intrinsic trms. Th drivation of (10) is an xcption to this cas. By inspction, w s that (9) is simply th nrgy givn by th Coulomb potntial; howvr, to driv this w can plug (16) into (5), which will yild (9). 6

7 If w tak th A x (r) trms in (3) and (4) and rplac thm with (A x (r)+ A k (r)), w gt for (3) mc p j A x (r) m c p (r j r k )(r j r k ) j + 1 p 3 k r jk and for (4) w gt mc (A x(r) + A x (r)a k (r) + A k (r) ). Howvr, w want to nglct trms that ar quadratic in A k (r) as wll as th cross trm btwn A x (r) and A k (r) 1. Without ths trms, w s that th rplacmnt in (4) rturns th sam quation. From th rplacmnt mad in (3) w s that th sam quation is rturnd as wll as an xtra trm. This trm, summd ovr j < k, givs us (10). Using (0) in plac of B x (r) in (6) givs us mc s 1 j (r j r k ) p k mc r 3 jk 1 s m c 3 j (r k r j ) p k which, summd ovr j k is (11), which is th intraction of th spin of th jth lctron with th magntic fild productd by th orbital motion of th othr k lctrons 1. Similarly, using (19) in plac of E x (r) in (7) givs us m c s j (r j r k ) p 3 j 1 s m c j (r j r k ) p j r 3 jk which, summd ovr j k is (1), which is th trm for th intraction of th spins of th jth lctron with th lctric fild producd by th othr klctrons 1. 7

8 Taking (1) and putting µ s in trms of th spin s k givs us B k,spin (r) = 3s k (r j r k )(r j r k ) s k (r j r k ) (r j r k ) mc 5. Plugging this in for B x (r) in (6) givs us mc s 3s k (r j r k )(r j r k ) s k (r j r k ) (r j r k ) j mc 5 m c s 3(r j r k )(r j r k ) (r j r k ) (r j r k ) j s 5 k m c s 3(r j r k )(r j r k ) j 1 s 5 3 k which, summd ovr j < k, givs us (13). This trm is th intraction btwn th spin magntic momnts that ar not mutually pntrating 1. Th trm (14) is put togthr by construction. This trm taks mutual pntration into account, which rquirs that r j = r k. This is th rason for th Dirac dlta function; w can s that if r j = r k thn th (13) gos to infinity, and if r j r k thn thr is no mutual pntration, causing th trm to b zro. Th dot product btwn th spins rmain th sam. Howvr, (13) is also multipld by a corrction factor of 8π. 3 Th trm (15) is similar to (8) in that it is du to trating th lctron as an xtndd particl. To driv this trm w tak (16) and plug this into th drivation for (8) which givs ( ) r ( Φ k ) 6 ( ) ( ) ( ) 1 3 h 4π δ(r j r k ) 6 4 mc π h m c δ(r j r k ) Multiplying by to tak into account both lctrons, and summing ovr j < k, givs us (15). 8

9 Rfrncs 1 Itoh, Takashi Drivation of Nonrlativistic Hamiltonian for Elctrons from Quantum Elctrodynamics. Rviws of Modrn Physics vol 37: pgs