, so the state may be taken to be l S ÅÅÅÅ

Transcription

1 hw4-7.b: 4/6/04:::9:34 Hoework: 4-7 ü Skuri :, G,, 7, 8 ü. bel the eigefuctio of the qure well by S, with =,,. The correpodig wve fuctio re x = px i Iitilly, the prticle i kow to be t x =, o the tte y be tke to be l S, where l i orliztio fctor, which Skuri llow u to leve upecified. The iitil tte y be projected oto the eigette, l S = l S S = l S c S with c = = pi i = p i ÅÅ The reltive probbility to be foud i the tte i the P Sc W i ÅÅ p = odd = 0 = eve At tie t > 0, the idividul copoet ocillte t their ow frequecie, o Yx, t = S c x e -iw t ü coet o orliztio Hvig etblihed the reltive probbilitie, retur to the quetio of orliztio. The bolute probbility to be i tte S i P = Sl c W. Sice there re ifiite uber of tte with -odd, d S P =, the orliztio ut vih l Ø 0. Wheever oe h itutio where oe i delig with ter of the ort 0 ÿ =, oe i dvied to tret the ifiitie liit. I the preet ce, igie the d-fuctio the liit of erie of fuctio dx - x' = i Dx - x', where defe the width of orlized fuctio, Ø0 x Dx, = d the locliztio property i relized i the liit i Ø0 x' Dx - x', f x' = f x. For exple, D could be tke to be qure pule with width d plitude Å. Next coider wve fuctio proportiol to D, i.e. y = l D. The the orliztio would be deteried by = x y = 0 x l Å Å = l

2 hw4-7.b: 4/6/04:::9:34 To preerve the orliztio of prticle i the tte, l = H, coitet with the ide tht l ut vih i the liit tht D Ø d. At the e tie, the coefficiet c lo deped o. For ll, the ode fuctio re vryig lowly roud the poit x =. For ll, the reltio c = l p i ÅÅ = ÅÅÅ p i ÅÅ i vlid. For lrge, however, the ode fuctio ocillte y tie withi the fiite width of the D-fuctio. I thi liit, oddly eough, c Ø 0, eve for the odd vlue of. The cro-over betwee ll- d lrge- behvior occur t ~ Å. The orliztio of probbilitie i expreed the u, = S P = S Sc W = S Sc W = S Å = where the fctor of ÅÅÅ = i glibly ierted to ccout for -odd. The rithetic here h bee doe very roughly, but the decriptio i correct. A the width of the D-fuctio becoe rrower, ore d ore ode cotribute to the orlized, loclized tte, but they do o with progreively ller plitude. ü. é For the phericl ce, I hd i id tht the prticle be locted t the ceter. Soe people plced the prticle o hell of rdiu. I either ce, it i ecery to decribe loclized tte i phericl coordite. Deote the orlized ditributio by dr, ǹ; r', ǹ' = d 3 x - x', where r, ǹ re the rdil d directio coordite. The ditributio hould hve the propertie tht V dr, ǹ; r', ǹ' = r ` iq r q f dr, ǹ; r', ' = ` V ' dr, ǹ; r', ' f x' = f r, ǹ i.e. the ditributio i orlized d loclized. By ipectio, the ditributio dr, ǹ; r', ǹ' = Å r iq dr - r' dq - q' df - f' will do the job. Oe c eprte thee ito "rdil-d" fuctio, d r r, r' = Å r dr - r', d "directiol-d" fuctio d W ǹ, ǹ' = iq dq - q' df - f'. The directiol d W will coe up gi i the dicuio of phericl hroic d orbitl gulr oetu. It rei to defie rdil poitio-ket, Sr, with the orliztio r r' = d r r, r' = Å r dr - r' d copletee = r r Sr rw. The ier product with rbitrry tte give the rdil wve fuctio, r = R r, orlized o tht = r r r r = r r SR rw = The correpodig coordite-d fuctio y be expreed dr - r' = r r r' = r S r r' = r S R * r' R r

3 hw4-7.b: 4/6/04:::9:34 3 With thi preble, the wve fuctio which tify the phericl ifiite well (yr = = 0) boudry coditio re Sl, with x l = b l j l k l r Y l ǹ, where k l i the th -root of j l, d b l i orliztio cott. Becue of the phericlly yetric boudry coditio, oe eed oly coider l = = 0, for which the wve fuctio of iteret re r = b j 0 k r. Sice j 0 x = ix I x, the llowed vlue of k re k = ÅÅ p, =,,, d b i give by b = r r S j 0 k rw - = r r ik r k r - = k 0 r i k r - = k = p ÅÅÅ 3 or b = b 0 with b 0 = p Å 3 The prticle i iitilly loclized t the origi, d i decribed by tte l Sr = 0. Expdig i eigette l S0 = l S S 0 = S c S where c = lb 0 j 0 0 = lb 0 If the prticle i loclized o hell r =, the the correpodig clcultio i l S = l S S = S c è S where c è = lb 0 j 0 k = lb 0 p i ÅÅ Å ÅÅ p = 3I l ä odd 0 eve I the ce where the prticle i o hell of rdiu, the reult i very iilr to tht for the lier qure well - ll odd tte cotribute with the e probbility. However, whe the prticle i t the origi, the reltive probbility goe P ~ d both eve d odd -tte cotribute. The reo for thi i the behvior of j 0 x~ x t lrge x. For fixed qure well boudry, the higher -tte re ore trogly weighted t the origi, d hve greter overlp with r = 0. If oe took the loclized tte to be of fiite extet i the uppletry dicuio for the lier ce, the I believe tht oe would fid P ~ for < Å, but for lrger, c would be uppreed by the ocilltig overlp itegrl. ü. Thi proble w dicued i cl. The lecture ote re vilble t

4 hw4-7.b: 4/6/04:::9:34 4 et the tregth of the potetil be give by v 0 = V. Surizig the reult fro the lecture - for the regio where V = 0, the geerl olutio i y < = A e k x + B e -k x y > = C e k x + D e -k x Sice the boud tte ut be orlizeble, B = C = 0. Cotiuity require A = D. Norliztio give, A = H k. The tregth of the dicotiuity give k = V. The eergy i give by E = k, or E = V. There re o excited tte, ice there i oly oe olutio to the tchig coditio. ü.7 The probbility flux i give by jx, t = Iy* y where i the reduced of the electro i the hydroge to. For the hydroge to (d phericl potetil i geerl), the boud tte re give by, y l = R l r Y l q, f = R l r c l P l co q e i f where c l i cott d P l i "ocited egedre polyoil". Both R l r d P l co q c be tke to be rel, d o j r = j q = 0. The oly o-vihig copoet to j i j f = Iy* Å r i q ÅÅ f y = R lr c l P l co q Å = r i q y l r i q Ie-i f The ee of circultio deped o the ig of. ÅÅ f ei f ü.8 The free prticle propgtor i

5 hw4-7.b: 4/6/04:::9:34 5 Kx, x, t = x Ut x = d p x p pw e - i p = d p x p p x e - i = p d p e i = p d p e - i = p d p e - i t = p d p e - i t = p d p e - i t = p d p è e - i t = ÅÅÅ p it e i t = ÅÅÅ t Sx p t px -x e - i p t p t-px -x ÅÅÅ p -p x -x Å t ÅÅÅ p - p q ÅÅÅ p- q e ÅÅÅ i t ÅÅÅ pè e ÅÅÅ i t q p it e i x -x Å t q q Where I hve ued q = x -x Å t, d p è = p - q I 3-dieio, the e ipultio c be pplied to vector qutitie. The itegrl c be eprted i crtei coordite. There i fctor of p 3 for the 3-d wve fuctio x p p x. Kx, x, t = x Ut x p it 3I e i x -x t = ÅÅÅ