Lctur 37 (Schrödingr Equation) Physics 6-01 Spring 018 Douglas Filds
Rducd Mass OK, so th Bohr modl of th atom givs nrgy lvls: E n 1 k m n 4 But, this has on problm it was dvlopd assuming th acclration of th lctron was givn as an objct rvolving around a fixd point. In fact, th proton is also fr to mov. Th acclration of th lctron must thn tak this into account. Sinc w know from Nwton s third law that: F p m a a p F m a p p m m If w want to rlat th ral acclration of th lctron to th forc on th lctron, w hav to tak into account th motion of th proton too. p a
Rducd Mass So, th rlativ acclration of th lctron to th proton is just: m a a a a a, rl p mp m m p m 1 a a m p m p m m F ma m m p p mrd Thn, th forc rlation bcoms: 1 4 And th nrgy lvls bcom: 0 v mm n p m rd, mrd rn r n m m p E n 1 k m n 4 rd
Rducd Mass Th rducd mass is clos to th lctron mass, but th 0.0054% diffrnc is masurabl in hydrogn and important in th nrgy lvls of muonium (a hydrogn atom with a muon instad of an lctron) sinc th muon mass is 00 tims havir than th lctron. Or, in gnral: m rd m mm p m 1836. m 0.99946m m m p m 1836. m rd mm p m 8.880 m 0.898m m m p m 8.880 m m1m m 1 m1,rd m1 m 1 m1 m m1 m m 1 1 m
Hydrogn-lik atoms For singl lctron atoms with mor than on proton in th nuclus, w can us th Bohr nrgy lvls with a minor chang: 4 Z 4. For instanc, for H +, E n 1 k m n 4Z 4 rd
Uncrtainty Rvisitd Lt s go back to th wav function for a travlling plan wav: x, t Acoskx t Notic that w drivd an uncrtainty rlationship btwn k and x that ndd bing an uncrtainty rlation btwn p and x (sinc p=ћk): xp x
Uncrtainty Rvisitd Wll it turns out that th sam rlation holds for ω and t, and thrfor for E and t: Et W s this playing an important rol in th liftim of xcitd stats. Each stat has a charactristic width in nrgy, invrsly proportional to how long it taks to d-xcit.
Problms for Bohr Modl Thr wr many problms with th smi-classical modl of Bohr: H quantizd orbital angular momntum, but an lctron with orbital motion would produc a magntic dipol momnt, and hydrogn in its ground stat dosn t hav a magntic dipol momnt. It couldn t b xtndd to multi-lctron atoms. Sinc th lctrons movd in circular orbits (say in th x-y plan at z = 0), thn thy also had no momntum in z. This didn t oby th uncrtainty principl in th z-dimnsion. W nd a mor comprhnsiv modl of th atom, and for that w nd to undrstand th consquncs of mattr wavs mor thoroughly. This was th goal of Erwin Schrodingr in 196.
Schrödingr s Wav Equation If particls bhavd as wavs, thy must thn hav an associatd wav quation (lik light or a guitar string). In a papr publishd in 193, Erwin Schrödingr dvlopd such an quation using th following rasoning: H startd by xamining plan wavs, whos wav function would b: k r pr x, t A A i t i Et
Som Mathmatics If you havn t workd with imaginary numbrs bfor (or mayb vn if you hav), som of what w ar going to covr will sm strang. First, w dfin i, as th squar root of -1. Thn, w hav: ipr Et x, t A cospr Et sin pr Et i Im R R Im 3 4 5 6 7 8 ix ix ix ix ix ix ix ix 1 ix! 3! 4! 5! 6! 7! 8! 3 4 5 6 7 8 x ix x ix x ix x 1 ix! 3! 4! 5! 6! 7! 8! 4 6 8 3 5 7 x x x x x x x 1 i x! 4! 6! 8! 3! 5! 7! cos x i sin x. So that our fr particl wav function is just a combination of cos and sin functions with both a ral part and imaginary part.
WHY??? But why introduc complx numbrs??? Hr is a hand waving answr: W want th wav natur of th particl whn w ar daling with its wav proprtis (lik intrfrnc, tc.). But w don t want th wiggls in th wav function whn w want to dal with it s particl natur. Lt s look again at th fr particl wav function, and dfin th probability distribution of finding it (dtction is a particl aspct) within a rgion dx at tim t as th squar of its wav function: r, t A i pr Et * r, r, t r, t r, t i Et i Et p r p r P t A A A S? No wiggls, and uniformly distributd in spac (sinc it has a dfinit momntum). What would w hav gottn without th complx wav function? r, t Acosp r Et r, t * r, r, t cos p r P t A Et
Wav Packts Th sam is tru for a wav packt. Th particl s wav natur is ncodd in th wav function s ral and imaginary parts, but th complx conjugat squard is ral, and has th typ of probability distribution that w ar looking for!
Intrprtation of th Wav Function Hr, w nd to spnd a minut talking about what th wav function is. As I said on th prvious slid, th probability distribution is givn by: Pr,t r, t This mans if you want to know th probability of finding th particl at a crtain point in tim and ovr a crtain rang in spac, you hav to intgrat th probability distribution ovr that rang: 3 r, r, P a b t t d r Thn, an additional condition on th wav function is that th total probability of finding th particl ovr all spac must b = 1: P Total All Spac b a 3 r, t d r 1 Not that th fr particl wav function is non-normalizabl! Nd to us wav packts.
Schrödingr s Wav Equation OK, w start with th fr particl (plan wav) wav function k r pr x, t A A i i t i Et Now, w notic that for th E&M and mattr wav quations, w hav drivativs with rspct to position and tim involvd. Lt s tak th first spatial drivativ of th wav function and s what w hav: i pr Et pa p OK, now lt s dfin th momntum oprator such that whn it oprats on th wav function, it givs us back th momntum tims th wav function: pˆ i p pˆ i i
Schrödingr s Wav Equation i k t i Et x, t A r pr A And now w do th sam thing only taking th first tim drivativ of th wav function: ie i Et ie pr A t And w s that w can dfin an nrgy oprator in th sam way: Eˆ i E t Eˆ i t
Schrödingr s Wav Equation k r pr x, t A A i t i Et And now w just stat consrvation of nrgy using our nw oprators and th wav function: p p ˆ p p E V E V m m with pˆ i and Eˆ i, t i V t m Schrödingr s Wav Equation E KE PE Total
Schrödingr s Wav Equation in 1D If motion is rstrictd to on-dimnsion, th dl oprator can just b rplacd by th partial drivativ in on dimnsion: i V x t m x And thn th wav function, of cours, is also just a function of on dimnsion: x x x, t A A i p x Et i p x iet Now, this solution works for whn V(x) = 0 vrywhr (FREE PARTICLE SOLUTION), but fails whn not. Howvr, whn th solution has a dfinit nrgy, th gnral form is: x, t x iet
Tim Indpndnt Schrödingr s Wav Equation Plugging this into th 1D Schrödingr s quation givs: iet iet iet i x x V x x t m x iet x iet iet E x V x x m x And w can divid both sids of th quation by th tim dpndnt part to gt: x E x V x x m x This is calld th tim-indpndnt (1D) Schrödingr s quation, which w can us to solv for th position dpndnc of th wav function. On must rmmbr though, that th full wav function nds th tim dpndnt part put back in: iet x, t x
Exampl x E x V x x m x
Exampl x 0 x E x V x m x ikx ikx ikx ikx ikx ikx A 1 A ika1 ika k A1 k A k x x x k E x x m k p E KE m m