Van der Waals Forces Between Atoms

Transcription

1 Van dr Waals Forcs twn tos Michal Fowlr /8/7 Introduction Th prfct gas quation of stat PV = NkT is anifstly incapabl of dscribing actual gass at low tpraturs, sinc thy undrgo a discontinuous chang of volu and bco liquids. In th 87 s, th Dutch physicist Van dr Waals ca up with an iprovnt: a gas law that rcognid th olculs intractd with ach othr. H put in two paratrs to iic this intraction. Th first, an attractiv introlcular forc at long distancs, hlps draw th gas togthr and thrfor rducs th ncssary outsid prssur to contain th gas in a givn volu th gas is a littl thinnr nar th walls. Th attractiv long rang forc can b rprsntd by a ngativ potntial an/v on going away fro th walls th olculs nar th walls ar attractd inwards, thos in th bulk ar attractd qually in all dirctions, so ffctivly th long rang attraction is quivalnt to a potntial wll xtnding throughout th volu, nding clos to th walls. Consquntly, th gas dnsity N/V nar th walls is dcrasd / kt an / VkT by a factor = an / VkT. Thrfor, th prssur asurd at th containing P N / V kt P = N / V an / VkT kt, or wall is fro slightly dilutd gas, so = bcos P + a( N / V ) V = NkT. Th scond paratr van dr Waals addd was to tak account of th finit olcular volu. ral gas cannot b coprssd indfinitly it bcos a liquid, for all practical purposs incoprssibl. H rprsntd this by rplacing th volu V with V Nb, Nb is rfrrd to as th xcludd volu, roughly spaking th volu of th olculs, to giv his faous quation N P + a ( V Nb) = NkT V This rathr crud approxiation dos in fact giv sts of isothrs rprsnting th basic physics of a phas transition quit wll. (For furthr dtails, and an nlightning discussion, s for xapl ppndix D of Thral Physics, by. airlin.) Ground Stat Hydrogn tos Our intrst hr is in undrstanding th van dr Waals long rang attractiv forc btwn lctrically nutral atos and olculs in quantu chanical trs. W bgin with th siplst possibl xapl, two hydrogn atos, both in th ground stat:

2 r r W labl th atos and, th vctors fro th protons to th lctron position ar dnotd by r and r rspctivly, and is th vctor fro proton to proton. Thn th Hailtonian H = H + V, whr H ħ = ( + ) r r and th lctrostatic intraction btwn th two atos V = +. + r r + r r Th ground stat of H is just th product of th ground stats of th atos,, that is, =. ssuing now that th distanc btwn th two atos is uch gratr than thir si, w can xpand th intraction V in th sall paratrs r /, r /. s on ight suspct fro th diagra abov, th lading ordr trs in th lctrostatic nrgy ar just thos of a dipoldipol intraction: V = ( r )( r ) = ( r )( r ) r r 5 Taking now th axis in th dirction, this intraction nrgy is V = x x + y y

3 Now th first ordr corrction to th ground stat nrgy of th two ato syst fro this intraction is = n H n, whr hr H = V and n =. ginning with th first tr xx in V n ( )( xx )( ) = ( x )( x ) is clarly ro sinc th ground stats ar sphrically sytric. Siilarly, th othr trs in V ar ro to first ordr. call that th scond ordr nrgy corrction is That is, n = H n. n n ( ) nl n l V =. n, l, n n n, l, typical tr hr is ( nl n l )( xx )( ) ( nl x )( n l x ) =, so th singl ato atrix lnts ar xactly thos w discussd for th Stark ffct (as w would xpct this is an lctrostatic intraction!). s bfor, only l =, l = contribut. To ( ) ak a rough stiat of th si of, w can us th sa trick usd for th quadratic Stark ffct: rplac th dnoinators by th constant (th othr trs ar a lot sallr for th bound stats, and continuu stats hav sall ovrlap trs in th nurator). Th su ovr intrdiat stats n, l,, n, l, can thn b takn to b copltly unrstrictd, including vn th ground stat, giving ( )( ) nl n l nl n l = I, n, l, n, l, th idntity oprator. In this approxiation, thn, just as for th Stark ffct, x x y y 4 ( ) ( )( + ) ( ) whr = yd., so this is a lowring of nrgy.

4 4 In ultiplying out ( x x y y ) +, th cross trs will hav xpctation valus of ro. Th ground stat wav function is sytrical, so all w nd is th ohr radius. This givs 4 4 a a 5 x = a, whr a is using = / a. ar in ind that this is an approxiation, but a prtty good on a or accurat calculation rplacs th by.5. Forcs btwn a s Hydrogn to and a p Hydrogn to With on ato in th and th othr in, say, a typical lading ordr tr would b ( )( xx )( ) ( x )( x ) =, and this is crtainly ro, as ar all th othr lading trs. ay (Lcturs on Quantu Mchanics) concludd fro this that thr is no lading ordr nrgy corrction btwn two hydrogn atos if on of th is in th ground stat. This is incorrct: th first xcitd stat of th two ato syst (without intraction) is dgnrat, so, xactly as for th D sipl haronic oscillator tratd in th prvious lctur, w ust diagonali th prturbation in th subspac of ths dgnrat first xcitd stats. (For this sction, w follow fairly closly th xcllnt tratnt in Quantu Mchanics, by C. Cohn Tannoudji t al.) Th spac of th dgnrat first xcitd stats of th two nonintracting atos is spannd by th product spac kts: ( ), ( ), ( ), ( ), ( ) ( ) ( ) ( ),,,. = + in this ight dinsional Th task, thn, is to diagonali V ( x x y y ) subspac. W bgin by rprsnting V as an 8 8 atrix using ths stats as th basis. First, not that all th diagonal lnts of th atrix ar ro. Scond, writing V = ( r r ), it is vidnt that V is unchangd if th syst is rotatd around th axis (th lin joining th two protons). This ans that th coutator [ V, L ] =, whr L is th total angular ontu coponnt in th dirction, so V will only hav nonro atrix lnts btwn stats having th sa total L. Third, fro parity (or Wignr ckart) all atrix lnts in th subspac spannd by ( ), ( ) ust b ro.

5 5 This rducs th nonro part of th 8 8 atrix to a dirct product of thr atrics, corrsponding to th thr valus of L =. For xapl, th = subspac is spannd by ( ), ( ). Th diagonal lnts of th atrix ar ro, th off diagonal lnts ar qual to ( )( ), whr w hav kpt th unncssary labls, to ak clar whr this tr cos fro. (Th x and y trs will not contribut for =.) This is now a straightforward intgral ovr hydrogn wav functions. Th thr hav th for atrics k k / / (following th notation of Cohn Tannoudji) whr, and th nrgy ignvalus ar k a ± k /, with corrsponding ignkts ( / ) ( ) ± ( ). So for two hydrogn atos, on in th ground stat and on in th first xcitd stat, th van dr Waal intraction nrgy gos as /, uch or iportant than th / nrgy for two hydrogn atos in th ground stat. Notic also that th / can b positiv or ngativ, dpnding on whthr th atos ar in an vn or an odd stat so th atos sotis rpl ach othr. Finally, if two atos ar initially in a stat ( ), not that this is not an ignstat of th Hailtonian whn th intraction is includd. Writing th stat as a su of th vn and odd stats, which hav slightly diffrnt phas frquncis fro th nrgy diffrnc, w find th xcitation ovs back and forth btwn th two atos with a priod h / k =.