/16/17 Qutiztio rises whe the loctio of prticle (here electro) is cofied to dimesiolly smll regio of spce qutum cofiemet. Usig Qutum Mechics i Simple Systems Chpter 15 The simplest system tht c be cosidered to study the effect of cofiemet is to cosider sigle electro system restricted to spce i sigle dimesio. To uderstd the chrcteristics of prticle i oe dimesiol bo system it is ecessry to solve the SE. Solvig Schrodiger equtio (SE) yields s which c be used to obti vlues for mesurble qutities like eergy. The prticle i 1D bo Uderstdig of the solutio to the prticle i 1-D bo cse, would llow us to gi some isights ito the ture of qutum systems. Tht is, the ide of qutum behvior d the pperce of discrete sttes with specific eergies. 1D bo system: A coceptully simple system where electro cofied to spce i directio of legth d completely free (of zero potetil eergy) to move i tht spce of legth but ot llowed to move outside the spce of legth. origi = The prticle cot leve the bo, i eergy terms, the prticle is icpble of overcomig eormously lrge eergy brrier for > >. Tht is, for > > the potetil eergy V =, d for < <, V =. 1
/16/17 Solvig for s of strtig with SE Usig SE; The potetil eergy outside the lie is mkig the prticle totlly cofied withi the lie, cot peetrte the potetil brrier. (V iside = for coveiece, c be y costt.) Iside the bo, V() = H d ( ) k ( ) d Solvig the SE would yield s s solutios for llowed sttes. Solutios to this differetil equtio re of the bove form: Solvig the SE would yield s, wvefuctios s solutios for llowed sttes. SE is lier, the of is solutio the b is solutio, b = costt. The lierity of the SE is importt d the cosequece is ot trivil. This lierity is i qutum mechicl mplitude. Determiig k, A d B For to be well behved fuctio, for > >. At = d = ; the wve fuctio is ot llowed to cquire vlue d lso cot become ifiite. Thus () = d () = (boudry coditios). Apply BC; A geerl solutio of SE is ot ormlized fuctio. Lierity llows ormliztio of the wve fuctio. which mkes, A si k =
/16/17 Also; A, if ot there is o fuctio t ll; which mkes, si( k) k k Where = 1,, d. ( ) = Asi = qutum umber Ech defies stte. A wvefuctio is ssocited with stte boud stte, meig the prticle s eistece is cofied/boud ito the bo. Qutum umbers re the outcome of the pplictio of boudry coditios i the solutio of the differetil equtios. For the prticle i 1D bo, stte is ssocited with sigle qutum umber,. A sigle qutum umber is geerted becuse there is oly oe set of boudry coditios ivolved here. Geerlly ech set of boudry coditios led to qutum umber. A geerl solutio of SE is ot ormlized fuctio. Normliztio to fid the Amplitude A. ( ) A =? =Asi ( ) d = 1 = 1,, d. ( ) d A si d = 1 A si d 1 A 1 A si Complete set of wvefuctios: defies stte with the ssocited stte fuctio. For prticle i ifiitely deep potetil well there re ifiite umber of boud sttes. Prticle i ever free i.e. lwys cofied ito the bo. Full wve fuctio is of the form: spce * time 3
/16/17 Well behved s. The solutios of the prticle i bo (SE) forms complete set of fuctios. Note; the wve fuctios (solutios of SE equtio) bove re eige fuctios of the totl eergy opertor. Wve fuctios tht re eige fuctios of certi opertor re ot ecessrily be eige fuctios of other opertor; they my be eige fuctios of the other opertor or my be ot. Stdig wves Note withi ; #odes = -1 The time evolvig wve fuctio, - stdig wves. The time evolvig wve fuctio, - stdig wves. Full wve fuctio - form: spce * 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 wvefuctio. The sttes (B,C,D) re eergy eigesttes, but (E,F) re ot. The time vritio of the wve fuctios will be ectly s previously see. = 8 Oe prt show. http://e.wikipedi.org/wiki/prticle_i bo 4
/16/17 Clcultio of eergy of sttes 1D Bo: Eergy opertor = H H ( ) E( ) H ( ) E ( ) Time idepedet Schrodiger Equtio Key to clcultio of observbles H ( ) E ( ) d d si md md d si md m H E Note eigeeergies re of precise vlues. Mesurig the eergy of eigestte lwys produce the sme vlue. Eergy qutized d icreses with. Lowest eergy o-zero! Zero poit eergy. BC dicttes the umber of wve legths betwee two cosecutive sttes is /. Wve vector: me k Eergy Stbility h 16 8 m h 9 8 m h 4 8 m h 1 8 m 5
/16/17 Prity of fuctios The eigefuctios re symmetricl. = 1, 3 mirror imge reltioship, eve prity, eve fuctio =, 4 iverted reltioship, odd prity, odd fuctio Fuctios with differet prities re lwys orthogol. So itegrls ivolvig products of fuctios with differet prities re zero. Mthemticlly useful 1 me ( V( )) h 1 me h 1 me me h h me k For to be well behved fuctio, t > >, d t = d = the wve fuctio is ot llowed to become ifiite. Thus () = d () = (boudry coditios). The boudry coditio tured the eigefuctios of SE from trvellig wves to stdig wves ssocite with discrete vlues for eergy; h E 8m 6
/16/17 Probbility desity = Probbility i spce d * d d = d * *( ) ( ) d The qutiztio of eergy levels rises s turl outcome of the solutio of SE, s opposed to rbitrry ssumptio (Bohr model). The wvefuctios ssocited with the eergy levels re turl cosequece of the mthemticl solutio of the SE. SE of y system, more complicted th the 1D prticle i bo would hve the sme geerl properties d chrcteristics. It is the cofiemet of the prticle by wy of trppig the prticle i potetil well tht leds to qutiztio! As icreses qutum system morphs ito clssicl system. =5 Further liftig the cofiemet ( >> ) mkes qutum system rech it s clssicl limit - correspodece priciple. Prticle foud with ~sme probbility every where. = Correspodece Priciple. =1 7
/16/17 Rmifictio of completeess of the set of wve fuctios The solutios of the prticle i bo (eigefuctios ) forms complete set (of fuctios). for ll. Ay wve fuctio f() tht stisfies the requiremets of cceptble well behved wve fuctio c be costructed from the members of the complete set of fuctios. The costructed wve fuctio would be sum with ech term i the sum is ssocited with coefficiet (i.e. weight/mplitude) tht is mesure of the cotributio by tht member. Ay f() is epded i terms of the ormlized eigefuctios. si f( ) b b si 1 1 ormlized set of eigefuctios (Bsis set - bsis) The set of epsio coefficiet b is the represettio i f() from the bse fuctio. Determitio of Epsio coefficiets; b m f( ) b 1 * pre-multiply by d itegrte; * * mf ( d ) m b d b d b b * m m m m m f( ) d * m m overlp itegrl Eergy opertor for prticle i 3D bo: E KEPE m y z H m y z 8
/16/17 D d 3D boes I SE: Etedig the results from 1-D to -D/3-D is deceptively esy. For bo of size b c with BCs, Divisio by (,y,z), the time idepedet SE; Solutios re of form; d E = E +E y +E z Seprtio of vribles H h X( ) E The wve equtio where c be writte s sum of terms tht do ot shre coordites c be resolved ito the set (DE with three vribles trsforms to three DEs, ech with oe vrible); Solutios ~ 1D solutios!! Y( y) Z( z) si y y si b b z z si c c E E y z 8m h 8mb h z 8mc y E, y, z, y, z 8 y y z z ( yz,, ) si si si bc b c h y z 8m b c 9
/16/17 If the totl eergy c be writte s sum of idepedet vribles correspodig to differet degrees of freedom, the wve fuctio is product of idividul terms, ech correspodig to oe of the degrees of freedom. For cube of side ; E, y, z, y, z 8 y z ( yz,, ) si si si h y z 8m h E 8m, y, z y z y z 3 Severl i combitios c yield the sme E j,k,l - degeerte sttes Leds to the cocept of degeercy. Degeercy is the umber of wys system c chieve certi specified eergy. Cosolidtio of QM Postultes with solutios of Prticle i Bo For the prticle i 3D bo, stte is ssocited with three qutum umbers. Three qutum umbers re geerted becuse there re three sets of boudry coditios ivolved i 3D bo. More th oe eigefuctio (stte) would be ssocited with the sme eigevlue (eergy) i degeerte sttes. I geerl c be comple fuctio with terms crryig i = (-1); If is rel the use; (, td ) * (, t) (, t) d 1
/16/17 Note: eigevlues from opertor re of precise vlues. Mesurig the vlue of property ssocited with the opertor of eige stte lwys produce the sme vlue. i.e. Wht if the stte is described by fuctio tht is ot eige fuctio of the opertor (yet well behved, d cceptble wve fuctio)? The the eergy ssocited with tht wve fuctio would ot yield precise vlue (like eigevlue) every time the property is mesured. Wht we would be ble to clculte i such cse is the verge vlue of the multiple sigle vlues tht eperimet would produce; d is lso clled the epecttio vlue. The sigle determitios re the eige vlues tht the opertor would produce. Cosider ormlized wve fuctio costructed by ddig two wve fuctios of prticle i bo, =1 d = sttes s follows; c 1 si d si c'si d 'si This is cceptble wve fuctio, well behved first d secod derivtives, BCs stisfied. 11
/16/17 d md H d md c d H 'si 'si c'si d 'si c 4d E E 'si m 'si ' #!?! But is ot eigefuctio of the eergy opertor. d d d d c si d si c cos d cos c si 4d si Normlized fuctio however; c si d d 1 si si si si si c cd d d c d 1 Normliztio =1 = =1 u-ormlized. Note the coditio: c d 1 c 1 d d 1 c d d re iterpreted s the probbility tht the stte emultes eigesttes =1 d = respectively Becuse wve fuctio is rel d ormlized * ( ) ( d ) ( ) ( d ) 1 I geerl E * ( ) H ( ) d 1
/16/17 Clcultig mometum: Solutios of 1D bo: is eigefuctio of the Hmiltoi ot eigefuctio of mometum opertor. The QM opertor for mometum p is; d p i d So we clculte the verge, <p> usig postulte 4. p ( ) p( ) d d p si i si d d p ( ) p ( ) d 13
/16/17 Clcultig positio: Solutios of 1D bo: Agi it is eigefuctio of the Hmiltoi ot eigefuctio of positio opertor. Note: Clculted <p> = ; despite KE = p /m!! Epli. The QM opertor for positio is; So we clculte <>; me usig postulte 4. ( ) ( d ) ( ) ( d ) ( ) ( ) d si si d Clculted <> = /; the verge vlue of the positio is hlf wy betwee the eds if 1D bo. Motio of prticles i cofied boudries leds to qutiztio (discrete vlues for properties) d the qutized sttes re described by stdig wve like wvefuctios. Motio of prticles i ifiite spce, i.e. ucofied spce (free prticles) leds to rge of vlues for the properties. The wvefuctio i such situtios resemble propgtig wve. 14