Wave Equations. Michael Fowler, University of Virginia

Transcription

1 Wave Equatons Mcael Fowle, Unvesty of Vgna Potons and Electons We ave seen tat electons and potons beave n a vey smla fason bot exbt dffacton effects, as n te double slt expement, bot ave patcle lke o quantum beavo. We can n fact gve a complete analyss of poton beavo we can fgue out ow te electomagnetc wave popagates, usng Maxwell s equatons, ten fnd te pobablty tat a poton s n a gven small volume of space dxdydz, s popotonal to E dxdydz, te enegy densty. On te ote and, ou analyss of te electon s beavo s ncomplete we know tat t must also be descbed by a wave functon ψ ( x, yzt,, ) analogous to E, suc tat ψ ( x, y, z, t) dxdydz gves te pobablty of fndng te electon n a small volume dxdydz aound te pont (x, y, z) at te tme t. Howeve, we do not yet ave te analog of Maxwell s equatons to tell us ow ψ vaes n tme and space. Te pupose of ts secton s to gve a plausble devaton of suc an equaton by examnng ow te Maxwell wave equaton woks fo a sngle-patcle (poton) wave, and constuctng paallel equatons fo patcles wc, unlke potons, ave nonzeo est mass. Maxwell s Wave Equaton Let us examne wat Maxwell s equatons tell us about te moton of te smplest type of electomagnetc wave a monocomatc wave n empty space, wt no cuents o cages pesent. Fst, we befly evew te devaton of te wave equaton fom Maxwell s equatons n empty space: dvb = 0 dve = 0 B cule = t 1 E culb = c t To deve te wave equaton, we take te cul of te td equaton: v 1 E cul cule = culb= t c t togete wt te vecto opeato dentty

2 to gve cul cul = gad(dv) - 1 E E c t = 0. Fo a plane wave movng n te x-decton ts educes to E 1 E = 0 x c t Te monocomatc soluton to ts wave equaton as te fom Ext (, ) = Ee kx ( ωt) 0 (Anote possble soluton s popotonal to cos(kx - ωt). We sall fnd tat te exponental fom, altoug a complex numbe, poves moe convenent. Te pyscal electc feld can be taken to be te eal pat of te exponental fo te classcal case.) Applyng te wave equaton dffeental opeato to ou plane wave soluton. 1 ω x c t c ( kx ωt) ( kx ωt) Ee 0 = k Ee 0 = 0. If te plane wave s a soluton to te wave equaton, ts must be tue fo all x and t, so we must ave ω = ck. Ts s just te famla statement tat te wave must tavel at c. Wat does te Wave Equaton tell us about te Poton? We know fom te potoelectc effect and Compton scatteng tat te poton enegy and momentum ae elated to te fequency and wavelengt of te lgt by E = ν = ω p = = k λ Notce, ten, tat te wave equaton tells us tat ω = ck and ence E = cp. ( To put t anote way, f we tnk of e moe natual to wte te plane wave as kx ωt) as descbng a patcle (poton) t would be

3 3 Ee 0 ( px Et) tat s, n tems of te enegy and momentum of te patcle. In tese tems, applyng te (Maxwell) wave equaton opeato to te plane wave yelds o 1 E x c t c ( px Et) ( px Et) Ee 0 = p Ee 0 E = c p. = 0 Te wave equaton opeato appled to te plane wave descbng te patcle popagaton yelds te enegy-momentum elatonsp fo te patcle. Constuctng a Wave Equaton fo a Patcle wt Mass Te dscusson above suggests ow we mgt extend te wave equaton opeato fom te poton case (zeo est mass) to a patcle avng est mass m 0. We need a wave equaton opeato tat, wen t opeates on a plane wave, yelds E = c p + m c 0 4 Wtng te plane wave functon ϕ ( xt, ) ( px Et ) = Ae 4 wee A s a constant, we fnd we can get E = c p + m0 c by addng a constant (mass) tem to te dffeentaton tems n te wave opeato: 1 mc 1 E x c t c ( px Et) ( px Et ) 0 Ae = p + m 0c Ae = 0. Ts wave equaton s called te Klen-Godon equaton and coectly descbes te popagaton of elatvstc patcles of mass m 0. Howeve, t s a bt nconvenent fo nonelatvstc patcles, lke te electon n te ydogen atom, just as E = m 0 c 4 + c p s less useful tan E= p /m fo ts case. A Nonelatvstc Wave Equaton Contnung along te same lnes, let us assume tat a nonelatvstc electon n fee space (no potentals, so no foces) s descbed by a plane wave: ( px Et) ψ ( xt, ) = Ae.

4 4 We need to constuct a wave equaton opeato wc, appled to ts wave functon, just gves us te odnay nonelatvstc enegy-momentum elatonsp, E = p /m. Te p obvously comes as usual fom dffeentatng twce wt espect to x, but te only way we can get E s by avng a sngle dffeentaton wt espect to tme, so ts looks dffeent fom pevous wave equatons: ψ t m x ( x, t) ψ( x, t) = Ts s Scödnge s equaton fo a fee patcle. It s easy to ceck tat f ψ ( x, t ) as te plane wave fom gven above, te condton fo t to be a soluton of ts wave equaton s just E = p /m. Notce one emakable featue of te above equaton te on te left means tat ψ cannot be a eal functon. How Does a Vayng Potental Affect a de Bogle Wave? Te effect of a potental on a de Bogle wave was consdeed by Sommefeld n an attempt to genealze te ate estctve condtons n Bo s model of te atom. Snce te electon was obtng n an nvese squae foce, just lke te planets aound te sun, Sommefeld couldn t undestand wy Bo s atom ad only ccula obts, no Keplelke ellpses. (Recall tat all te obseved spectal lnes of ydogen wee accounted fo by enegy dffeences between tese ccula obts.) De Bogle s analyss of te allowed ccula obts can be fomulated by assumng tat at some nstant n tme te spatal vaaton of te wave functon on gong aound te obt pq ncludes a pase tem of te fom e, wee ee te paamete q measues dstance aound te obt. Now fo an acceptable wave functon, te total pase cange on gong aound te obt must be nπ, wee n s an ntege. Fo te usual Bo ccula obt, p s constant on gong aound, q canges by π, wee s te adus of te obt, gvng. so 1 pπ = nπ p = n, te usual angula momentum quantzaton. Wat Sommefeld dd was to consde a geneal Keple ellpse obt, and vsualze te wave gong aound suc an obt. Assumng te usual elatonsp p = / λ, te wavelengt wll vay as te patcle moves aound te obt, beng sotest wee te patcle moves fastest, at ts closest appoac to te nucleus. Neveteless, te pase

5 5 cange on movng a sot dstance Δq sould stll be 1 pδq, and equng te wave functon to lnk up smootly on gong once aound te obt gves pdq = n Tus only cetan ellptcal obts ae allowed. Te matematcs s nontval, but t tuns out tat evey allowed ellptcal obt as te same enegy as one of te allowed ccula obts. Ts s wy Bo s teoy gave all te enegy levels. Actually, ts wole analyss s old fasoned (t s called te old quantum teoy ) but we ve gone ove t to ntoduce te dea of a wave wt vaable wavelengt, cangng wt te momentum as te patcle moves toug a vayng potental. Scödnge s Equaton fo a Patcle n a Potental Let us consde fst te one-dmensonal stuaton of a patcle gong n te x-decton subject to a olle coaste potental. Wat do we expect te wave functon to look lke? We would expect te wavelengt to be sotest wee te potental s lowest, n te valleys, because tat s wee te patcle s gong fastest maxmum momentum. Peaps slgtly less obvous s tat te ampltude of te wave would be lagest at te tops of te lls (povded te patcle as enoug enegy to get tee) because tat s wee te patcle s movng slowest, and teefoe s most lkely to be found. Wt a nonzeo potental pesent, te enegy-momentum elatonsp fo te patcle becomes te enegy equaton p E = + V( x ). m We need to constuct a wave equaton wc leads natually to ts elatonsp. In contast to te fee patcle cases dscussed above, te elevant wave functon ee wll no longe be a plane wave, snce te wavelengt vaes wt te potental. Howeve, at a gven x, te momentum s detemned by te local wavelengt, tat s, ψ p =. x It follows tat te appopate wave equaton s: ψ( xt, ) ψ( xt, ) = + V( x) ψ ( x, t ). t m x Ts s te standad one-dmensonal Scödnge equaton.

6 6 In tee dmensons, te agument s pecsely analogous. Te only dffeence s tat te squae of te momentum s now a sum of tee squaed components, fo te x, y and z dectons, so becomes + + =, and te equaton s: x x y z ψ ( xyzt,,, ) (,,, ) (,, ) (,,, ). t m = ψ xyzt+ Vxyzψ xyzt Ts s te complete Scödnge equaton.