Lecture 38 (Trapped Particles) Physics Spring 2018 Douglas Fields

Transcription

1 Lecture 38 (Trpped Prticles) Physics 6-01 Sprig 018 Dougls Fields

2 Free Prticle Solutio Schrödiger s Wve Equtio i 1D If motio is restricted to oe-dimesio, the del opertor just becomes the prtil derivtive i oe dimesio: i V t Ad the the wve fuctio, of course, is lso just fuctio of oe dimesio (plus time):, t Ae Ae e i p Et i p iet Now, this solutio works for whe V() = 0 everywhere, but fils whe ot. However, whe the solutio hs defiite eergy, the geerl form is:, t e iet

3 Time Idepedet Schrödiger s Wve Equtio Pluggig this ito the 1D Schrödiger s equtio gives: iet iet iet i e e V e t iet iet iet E e e V e Ad we c divide both sides of the equtio by the time depedet prt to get: E V This is clled the time-idepedet (1D) Schrödiger s equtio, which we c use to solve for the positio depedece of the wve fuctio. Oe must remember though, tht the full wve fuctio eeds the time depedet prt put bck i: iet, t e

4 Emple E V

5 Emple 0 E V ik ik ik ik ik ik A 1e Ae ika1 e ikae k A1 e k Ae k k E k p This is wht we epect, sice E KE the potetil eergy is zero.

6 Prticle i 1D Bo The simplest potetil to uderstd, besides the free prticle (V=0 everywhere) is V=0 over some regio, d V= otherwise. For those of you who hve see wves i your previous clsses, thik of strig stretched betwee two wlls The wve (prticle) c be ywhere betwee the wlls, but owhere else. I will develop the solutios s if you hve t see the wves o strig solutio before. E V

7 Prticle i bo So, we strt with the 1D time-idepedet wve equtio: E V With the potetil defied s: 0 0 0, V V The, betwee 0 d, we c use the solutios we kow work: ik ik Ae Be Where we hve icluded both positive d egtive epoets to iclude wve motio i both the positive d egtive directios. This turs out to be ecessry to obey the boudry coditios s well.

8 Prticle i bo Now, before we try our solutios, let s cosider the boudry coditios of our problem. We kow tht the prticle cot be outside of the bo defied by the regio where V=0, so: 0 0 ik 0 ik 0 0 Ae Be A B 0 B A ik ik Ae Ae Acos k i si k cosk i si k A cos k i si k cos k i si k Aisi k C si k Where we hve coglomerted the costts together ito oe for coveiece for ow. Now, let s cosider the other boudry, t =. C si k C si k 0 k k

9 Prticle i bo So, we hve for our wve fuctio: Ad we hve oly oe step remiig for complete solutio, we eed to determie C, the ormliztio costt. C si k 0 0 si si k C C C 1 4k 0 4 C C si k C si k C si d d si So, we hve our completed sptil wve fuctios:

10 Eergy Eigevlues Now, these wve fuctios represet series of solutios with defiite eergies, E. si 0 E V E si si si E h 8m The sttiory sttes re clled eigesttes, d their eergies re clled eergy eigevlues. Oe thig to ote is tht the higher the eigevlues (eergy), the more wiggly the eigefuctios. This is becuse the kietic eergy (ll there is here, sice the potetil is zero i the llowed regio) is give by the mometum opertor squred, which is proportiol to the secod derivtive of the wve fuctio with respect to positio its curvture!

11 Prticle i bo We c lso plot ψ(), d the probbility distributio for ech of theses solutios: Note tht there ARE loctios iside the bo where you would ot epect to fid the prticle. But, if we put prticle (sy, electro) i 1D bo (sy, o-wire) i prticulr iitil loctio, the it my ot be i oe of the eigesttes it could be i superpositio of severl (or my)

12 Prticle i bo But, we cot forget the time depedece:, si iet t e Why? Becuse while sttes of defiite eergy re sttiory sttes (their positio probbility distributio remis costt with time), if we set the iitil coditio s certi positio withi the bo, the prticle will ot hve defiite eergy it will be superpositio of severl sttiory sttes with time depedeces tht will crete cross terms i the probbility distributio, tht will chge with time. Some trjectories of prticle i bo ccordig to Newto's lws of clssicl mechics (A), d ccordig to the Schrödiger equtio of qutum mechics (B-F). I (B-F), the horizotl is is positio, d the verticl is is the rel prt (blue) d imgiry prt (red) of the wve fuctio. The sttes (B,C,D) re eergy eigesttes, but (E,F) re ot. - UA_17%3A_Advced_Geerl_Chemistry_I/06%3A_The_Schr%C3%B6diger_equtio%3A_Predictig_eergy_levels_ d_the_prticle-i--bo_model

13 Orthoorml Sets The set of wve fuctios form complete set for this geometry: si iet, e t for y fuctio f ( t, ) tht obeys the boudry coditios:, w, f t t 1 Let s look t the probbility desity for oe of the eigesttes : t, si e 1 1 ie t 1 * 1 1,, si ie t ie t P t t e si e s i e si i E E t Time idepedet!

14 Orthoorml Sets Let s just look t the probbility desity for superpositio of two eergy eigesttes : 1 1 ie1t iet, t 1, t, t si e si e * 1 ie 1t iet ie1t iet P, t, t si e si e si e si e 1 si si si si 1 e i E 1 E t si si si si cos Et e i E 1 E t Time depedet!

15 Eigesttes d mied sttes So, eigesttes re sttes where the eergy is