( ) Unbound States 2. x = E < V 0. x = -a/2. x = a/2

Transcription

1 Uboud Stts As promisd lst tim i lctur, w ow tur to scttrig from fiit squr wll. Howvr, w will strt by discussig scttrig from fiit squr brrir: rfl i r κ A brr B κ trs t < -/ / m m k, κ o simplify mttrs, I v tk t mplitud of t icomig wv s, d t mplitud of t rflctd d trsmittd wvs s r d t rspctivly., R r, t d r t. Isid t rgio of t pottil brrir, otic tt I v trms i bot gtiv d positiv pots. Wy? Now, w follow our prscriptio for dlig wit uboud stts, w look t t boudry coditios t ll t cgs i pottil: t /, cotiuity of t wvfuctio givs: r A κ κ B d cotiuity of t first drivtiv givs: r κ A κ B κ t /, cotiuity of t wvfuctio givs: κ κ A B t d cotiuity of t first drivtiv givs: κ κ κ A B t

2 Aftr muc lgbr tt I v o ittios to go troug (or do I pct you to), w v [it s trivil to sow tt ]: t ( kκ ) ( k κ ) si ( κ) ( kκ ) cos ( κ) wit our dfiitios of k d κ, si κ d R c b foud by R. It s vry importt r to otic t distictio btw wt w s r for qutum mcics d wt w would pct from clssicl mcics. A prticl, impigig from t lft coms to pottil brrir tt is grtr t its rgy. I clssicl mcics, t prticl would stop d compltly rflct. Hr, bcus of t boudry coditios d t coctio btw t wvfuctio, t probbility dsity d t wy i wic t rgy is drivd, t wvfuctio must ot go to zro discotiuously, so tt tr is still probbility for t prticl to b foud wll witi t brrir. If t brrir t ds, t prticl c cotiu o (wit lowr probbility mplitud of cours). is is wt w cll qutum tulig! W v lrdy s tis i t cs wr I tlkd bout prity ffcts t doubl smi-ifiit squr-wll pottil: b d lt s cosidr t cs wr t prticl is iitilly isoltd to o sid of t ctr brrir. I tt cs t wvfuctio is just:

3 Ψ (, ) wr v v φ φ Lft Lft φ φ Rigt Rigt Now, sic t v d odd igstts v diffrt rgis (for fiit widt ctrl brrir), t tis is ot sttiory stt: Ψ(, t) wr v i t o i t v t < Ψ(, t) wr i t v i t v, t wvfuctio c b writt i trms of t lft d rigt d portios: Ψ(, t) Ψ i t i t φ i t Lft i t i t φ φ Lft Lft i t φ Rigt φ i t i t i t i t i i t i t Lft Rigt (, t) [ φ cos( t ) iφ si( t ) ] Rigt φ Lft iφ Rigt φ Rigt i t W w look t t probbility dsity, w tk t compl squr of t wvfuctio d t ps i frot gos wy, s dos t i i t scod trm: Ψ [ ] Lft Rigt (, t) φ ( t ) ( ) cos φ si t

4 d w c s tt t probbility dsity oscillts bck d fort btw t two sids wit frqucy of /ћ! is is clld tulig oscilltios. Now, lt s rtur to our pottil brrir d ivstigt t cs wr >, t κ wr k m ( ) m( ), κ z z Now, rmmbrig our dfiitios of yprbolic si: ( z) ( ) si i i i i si si so tt [( cos i si ) ( cos( ) i si( ) )] [ cos i si ( cos i si )] t w do t v to go troug ll tt lgbr tt w did t go troug i t first plc, d w c just gt for > : si ( k ) Puttig ts two trsmissio cofficits togtr to plot tm s fuctio of t rgy:

5 d w s tis pculir bvior tt t prticulr rgy, tr is % trsmissio, but t it drops gi d sows oscilltory bvior. I prticulr, w s tt wvr k π (wr t si trm bcoms zro), t t trsmissio is %. W c udrstd tis i trms of t wvlgt: w t brrir widt is itgrl umbr of lf wvlgts, t brrir bcoms trsprt. It is prps illumitig to lso cosidr t rgis wr tr is prfct trsmissio: k ( k ) m ( ) m ( ) π ( ) m π π wr r our llowd rgis of t o dimsiol ifiit squr-wll pottil! Now lt s gt to wt I promisd, ow bout fiit pottil wll, istd of brrir: rfl i r wll -/ A B / trs t < k ( ) m m, k is is ctly t sm problm tt w just d (wit brrir d wit > ), wit t sig of ow cgd, so tt our solutios r ow: si ( k )

6 is s v mor drmtic bvior wit rgy: is bvior of trsmissio of prticls scttrd from pottil wll is kow s t Rmsur ffct d c b s i t low-rgy scttrig of lctros from toms. ttrctiv wll rprsts t Coulomb fild of t uclus, wic is scrd by t tomic lctros, suc tt t pottil turs o bruptly s is t cs for squrwll. is typ of pom sould b fmilir to you from clssicl wv mcics. k t itrfc btw ir d glss: ir glss ir wr if tr is itgrl umbr of lf wvlgts isid t glss, t tr will b prfct costructiv itrfrc btw t (twic) rflctd wv d t trsmittd wv cusig prfct trsmissio. is is kow s Fbry-Prot ffct.