Table of Contents. 2016, Eric D. Carlson i

Transcription

1 Table of Cotets I The Schrödger Equato A The Werdess of Quatum Mechacs B The Schrödger Equato 6 C The Meag of the Wave Fucto 8 D The Fourer Trasform of the Wave Fucto E Expectato Values ad Ucertates Problems for Chapter 4 II Solvg the Schrödger Equato 6 A Solvg the Schrödger Equato Free Space 6 B The Tme-Idepedet Schrödger Equato 8 C Probablty Curret 9 D Reflecto from a Step Boudary E Quatum Tuelg 4 F The Ifte Square Well 5 G Boud States from a Double Delta Potetal 7 Problems for Chapter 3 III Hlbert Space ad Vector Spaces 3 A Hlbert Space 3 B Drac Notato ad Covectors 33 C Coordate Bases 37 D Operators 39 E Commutators 4 F Adjot Operators 4 G Matrx Notato 44 H Egevectors ad Egevalues 46 I Fdg Egevectors ad Egevalues 49 Problems for Chapter 3 5 IV The Postulates of Quatum Mechacs 54 A Revstg Bass Choce 54 B The Postulates of Quatum Mechacs 55 C Cosstecy of the Postulates 58 D Measuremets ad Collapse of the State Vector 60 E Expectato Values ad Ucertates 6 F Evoluto of Expectato Values 64 G The Tme-Idepedet Schrödger Equato 66 Problems for Chapter 4 68 V The Harmoc Oscllator 7 A The D Harmoc Oscllator 7 B Workg wth the Harmoc Oscllator ad Coheret States 74 C Multple Partcles ad Harmoc Oscllators 77 D The Complex Harmoc Oscllator 80 06, Erc D Carlso

2 Problems for Chapter 5 8 VI Symmetry 83 A The Traslato Operator 83 B Rotato Operators 85 C Party ad the Fte Square Well 88 D Dscrete Rotatos 9 E Dscrete Traslatos ad the Krog-Peey Model 93 F Cotuous Symmetres ad Geerators 96 G Solvg Problems wth Cotuous Symmetres 98 Problems for Chapter 6 0 VII Agular Mometum 04 A Agular Mometum Commutato Relatos 04 B Geeralzed Agular Mometum 06 C Sphercally Symmetrc Problems 08 D Sphercal Harmocs E The Hydroge Atom 4 F Hydroge-lke Atoms 9 G Tables of Sphercal Harmocs ad Radal Wave Fuctos Problems for Chapter 7 VIII Sp ad Addg Agular Mometum 4 A Rotatos Revsted 4 B Total Agular Mometum ad Addto of Agular Mometum 7 C Clebsch-Gorda Coeffcets 30 D Scalars, Vectors, Tesors 33 E The Wger-Eckart Theorem 36 F Products of Sphercal Harmocs 38 Problems for Chapter 8 40 IX Electromagetc Forces 43 A Electrc Forces 43 B Electromagetc Felds ad Forces 44 C Workg wth Kematc Mometum 47 D Magetc Dpole Momets 49 E Smple Problems wth Magetc Felds 50 F The Aharoov-Bohm Effect ad Magetc Moopoles 53 Problems for Chapter 9 56 X Multple Partcles 58 A Tesor Products of Vector Spaces 58 B Multple Partcles 59 C Symmetry ad Iterchageable Partcles 59 D Idetcal Partcles 6 E No-teractg Partcles ad Ferm Degeeracy Pressure 65 F Atoms 68 06, Erc D Carlso

3 Problems for Chapter 0 7 XI Iterpretato of Quatum Mechacs 74 A The Tme Evoluto Operator 74 B The Propagator 76 C The Feyma Path Itegral Formalsm 78 D The Heseberg Pcture 8 E The Trace 83 F The State Operator, or Desty Matrx 84 G Workg wth the State Operator 88 H Separablty 90 I Hdde Varables ad Bell s Iequalty 93 J Measuremet 96 K The May Worlds Iterpretato of Quatum Mechacs 0 Problems for Chapter 04 XII Approxmate Methods 07 A The Varatoal Method 07 B The WKB Method C Perturbato Theory 8 D Degeerate Perturbato Theory Problems for Chapter 5 XIII Applcatos of Approxmate Methods 7 A Fte Nuclear Sze 7 B Sp-Orbt Couplg 9 C Hyperfe Splttg 3 D The Zeema Effect 34 E The Va Der Waals Iteracto 36 Problems for Chapter 3 39 XIV Scatterg 4 A Cross-Secto ad Dfferetal Cross-Secto 4 B The Bor Approxmato 45 C Scatterg from a Sphercally Symmetrc Potetal 49 Problems for Chapter 4 55 XV Tme-Depedet Methods 57 A The Sudde Approxmatos 57 B The Adabatc Approxmato 58 C Tme-depedet Perturbato Theory 6 D Harmoc Perturbatos 64 E Electromagetc Waves ad the Dpole Approxmato 67 F Beyod the Dpole Approxmato 7 G Tme-depedet Perturbato Theory wth a Costat Perturbato 7 Problems for Chapter , Erc D Carlso

4 XVI The Drac Equato 79 A The Drac Equato 79 B Solvg the Free Drac Equato 8 C Electromagetc Iteractos ad the Hydroge Atom 83 Problems for Chapter 6 88 XVII Quatzg Electromagetc Felds 90 A Gauge Choce ad Eergy 90 B Fourer Modes ad Polarzato Vectors 9 C Quatzg the Electromagetc Felds 93 D Egestates of the Electromagetc Felds 94 E Mometum of Photos 95 F Takg the Ifte Volume Lmt 96 G The Nature of the Vacuum 97 H The Casmr Effect 98 Problems for Chapter 7 30 XVIII Photos ad Atoms 303 A The Hamltoa 303 B Absorpto ad Emsso of Photos by Atoms 304 C The Self-eergy of the Electro 307 D Photo Scatterg 308 E A Dagrammatc Approach 309 F Thomso Scatterg 30 G Scatterg Away From a Resoace 3 H Scatterg Near a Resoace 35 Problems for Chapter 8 30 Appedx A Some Mathematcal Tools 3 A Vectors Three Dmesos 3 B Calculus wth Oe Varable 33 C Tables of Itegrals 36 D Calculus wth Several Varables 39 E Coordates Two ad Three Dmesos 33 F Fourer Trasforms 335 Idex 337 Uts ad Costats 345 I The Schrödger Equato v 06, Erc D Carlso

5 A The Werdess of Quatum Mechacs I The Schrödger Equato The world s a strage place, but t was t utl 900, whe Max Plack was studyg the thermal spectrum of lght (black body radato) that we bega to uderstad just how strage t was Although ths represeted the formal start of moder quatum mechacs, t s easer to start 877, wth Hertz s dscovery of the photoelectrc effect, whch was explaed 905 by Este Ths effect demostrated, effect, that whe you she a lght source of frequecy f oto a pece of metal, the lght acted ot as f t were made of waves, but rather as f t were composed of ty packets of eergy hf, where h was a ew costat called the Plack costat, gve by h J s ev s Ths s the bass of the moder photomultpler, a devce that s capable of detectg dvdual partcles of lght, also kow as photos We are t really terested the er workgs of a realstc photomultpler, but stead wll smply treat t as a dealzed detector that couts photos Although the formula E = hf was the formula orgally formulated by Plack, we wll fd t more useful to rewrte ths by wrtg the frequecy f terms of the agular frequecy = f, ad the exchage Plack s costat for the reduced Plack costat, h J s ev s I terms of whch the relatoshp betwee eergy ad agular frequecy s gve by E () The basc problem wth ths dscovery s that there was already ample evdece that lght was electromagetc waves (deed, f they are t waves, the what frequecy are we dscussg?) For example, the electrc feld of a electromagetc wave travelg the +x drecto would take the form t kx t t kx t E r, E cos or E r, E s () 0 0 For techcal mathematcal reasos, whch represet a coveece whe studyg electrcty ad magetsm but wll prove crucal here, t s commo to combe these two real fuctos by multplyg the latter by ad the addg them to get the complex feld t kx t E r, E 0 exp (3) Ths form of the electrc feld (together wth a correspodg magetc feld) ca be show to satsfy Maxwell s equatos provded ck Wth a lttle help from relatvty, aother mportat relatoshp ca be worked out For partcles movg at the speed of lght (such as photos), the eergy ad mometum are related by E cp, where c s the speed of lght ad p s the mometum Combg these two relatos wth Eq (), t s a momet s work to demostrate 06, Erc D Carlso I The Schrödger Equato

6 p k (4) Waves have propertes very dfferet from partcles; for example, partcles have a very defte posto, whereas waves ted to be spread out, or essece, have some ucertaty ther posto The waves gve by Eq () or (3) have a very defte value of the wave umber k, but they are fte extet, so that the posto x has o defte value; deed, for Eq (3) the waves are dstrbuted uformly throughout all of space Wthout gog to too much detal here, t ca be show that you ca combe waves wth dfferet wave umbers k to make a wave packet that s less spread out space but has a ucertaty wave umber k Ideed, t s a classcal result that for ay wave the ucertates the posto x ad wave umber k must satsfy the equalty xk, where x ad k represet the ucertaty the posto ad wave umber respectvely These ucertates wll be defed much more precsely later the course If we combexk wth Eq (4), we obta a correspodg relatoshp betwee the mometum ad posto of a photo, amely x p (5) I cotrast, partcles are ormally descrbed by gvg ther posto ad mometum, ad hece t s mplctly assumed classcal mechacs that the ucertaty each of these quattes s zero To help us uderstad some of the dffcultes ecoutered as we move to a quatum pcture of the world, a umber of gedake expermets may help to clarfy the stuato For example, cosder a half-slvered mrror; that s, a mrror that lets some lght through, but ot all Now, suppose we place a dm source frot of ths half-slvered mrror ad she a lght o t, detectg the resultg waves usg a par of photomultplers, as llustrated Fg - What happes ths expermet? Perhaps ot surprsgly, the rate of photos appearg each of the two detectors s exactly half the rate at whch they strke the half-slvered mrror If you ject the photos from the source very slowly, say oe at a tme, you fd that the photos appear ether detector A or detector B ot both, but always oe of them It s as f the half-slvered mrror sorts the photos oe way or the other, eve though there s apparetly o way to predct whch way ay sgle photo wll go It s perhaps worth potg out a couple of reasos for our cofuso If you work out the effects o Eq () or (3) of the half-slvered mrror, you would fd that the wave gets dvded equally to two waves, oe of whch goes to each detector But expermetally, each photo goes oe way or the other If you Source Half-slvered mrror Detector B Detector A Fgure - The half-slvered mrror somehow seems to selectvely deflect each photo to oly oe of the two detectors, but the process seems to be completely radom The Germa word gedake meas thought A gedake expermet s a thought expermet oe that s ot ecessarly gog to be performed, but stead s used to llustrate a partcular pot Actually, realstc detectors ever have 00% effcecy, so real expermets, some photos wll ot appear ether detector Though ths gedake expermet caot be perfectly performed, t s beleved that othg prcple makes such a expermet mpossble I The Schrödger Equato 06, Erc D Carlso

7 chaged the half-slvered mrror for oe that reflects some dfferet fracto, you wll fd that the photos aga each flow ether oe way or the other, though the probabltes wll shft Ideed, the probablty of the photos gog to oe detector or the other s proportoal to the testy, whch the complex otato of Eq (3) yelds * t P E r, E E (6) 0 0 Ideed, sce the power of a wave s proportoal to ths same quatty, ad each of the photos has exactly the same eergy (for a plae wave), ths result s hardly surprsg Aother expermet wll clarfy thgs eve more It s possble, wth the addto of a couple of covetoal mrrors ad a addtoal half-slvered mrror, to create a cofgurato alog the les of Fg - The lght comes from the source, s dvded two by the half-slvered mrror The lght the cotues to the two covetoal mrrors, whch the Source Half-slvered mrror Detector B Detector A Fgure - I ths expermet, the lght takes two paths to go from the source to the secod halfslvered mrror If the dstaces are set up carefully eough, all of the photos wll appear detector A, ad oe detector B sed t to a half-slvered mrror, whch recombes the two waves It s possble, f the dstaces are carefully cotrolled, to arrage thgs so that accordg to classcal wave theory the waves wll combe perfectly at the secod half-slvered mrror such that all of the lght goes to detector A ad oe of t goes to detector B It s mportat to ote that ths happes, accordg to the wave theory, oly f the lght wave follows both paths; t s the terferece of the two halves of the waves at the recombg half-slvered mrror that allows ths effect: the two waves cospre to cacel out as they head towards detector B, ad they cospre to add together (terfere costructvely) as they head towards detector A If we do ths expermet wth lght that s low testy, t s dffcult to recocle the wave ad partcle pctures I ths pcture, half the photos wll be deflected by the frst half-slvered mrror, ad half wll pass through Thus each photo wll follow ether the upper path or the lower path, ot both, ad therefore wll reach the secod half-slvered mrror from the upper path or the lower path, but ot both Sce we already kow that half-slvered mrrors reflect half the photos ad pass the other half, we would expect the photos to aga be dvded half, so the ed there are four equally lkely paths from the source to the detector, two of them edg up detector A ad two detector B But ths s ot what we fd expermetally We fd, stead, that the photos all ed up detector A, ad oe of them detector B Recall, the wave theory, ths ca oly happe f the wave passes alog both paths, but how ca we expla ths f we have oly oe partcle at a tme? It s as f the partcle behaves lke a wave betwee the frst mrror ad the secod, ad the behaves lke a partcle after passg through the secod To clarfy the stuato, mage replacg oe of the two covetoal mrrors Fg - by somethg that absorbs photos, say, aother photomultpler, as show Fg -3 I ths cofgurato, the system acts exactly as predcted f photos are partcles: half of them are deflected by the frst half-slvered mrror to detector C, ad the other half are subdvded by the secod mrror to half, so that ultmately oe-fourth of them ed up detectors A ad B But 06, Erc D Carlso 3 I The Schrödger Equato

8 whe the secod mrror s replaced, the photos dsappear aga I ths case, addg a path by whch the partcles could reach detector B causes a decrease the photos arrvg there It s mportat to remember that the terferece descrbed Fg - ca oly occur f the two path legths are carefully cotrolled I partcular, f oe of the mrrors s repostoed by, say, a modest fracto of a wavelegth of lght, the terferece wll be mperfect or eve completely destroyed, ad may photos wll make t to detector B Fortuately, moder optcal equpmet ca easly be adjusted much more accurately tha a fracto of a wavelegth, so ths expermet (or comparable equvalet expermets) ca easly be performed What we would lke to do s fd a expermet where we ca actually catch quatum mechacs the act, so to speak, where we ca both measure the path that the lght took (upper or lower?) ad also we ca create terferece effects, such as are show Fg - Ths proves to be surprsgly dffcult For example, cosder the followg mor modfcato of Fg -: replace oe (or both) of the covetoal mrrors wth a very lghtweght mrror; deed, let s make t so lght that whe a sgle photo bouces off of t, t wll cause the mrror to recol wth some measurable velocty Now we ject a sgle photo to the apparatus, let t pass through ad check whether t regstered detector A or B We also carefully measure the mometum of oe of the mrrors after the expermet s doe, to see whether the photo rebouded from t, trasferrg mometum to t If we set up the dstaces accurately, the the photo should always appear detector A, ever B, ad f ecessary, we ca repeat the expermet several tmes to check that t actually ever does appear B I the mea tme, for each repetto of the expermet, we ca check whch path the photo took Sce the terferece requres that the photo take both paths, ad measurg the recol of the mrror requres that the photo take oe path or the other, t seems we should be able to catch ature a bd What wll happe? The aswer turs out to be surprsgly subtle Recall, as I have sad before, that the mrror must be placed very carefully to make sure the terferece works; deed, t must be postoed much better tha a wavelegth I other words, f there s some ucertaty the mrror s posto, because we were sloppy settg up the expermet, we had to be very careful that ths m ucertaty satsfes x, or usg the relatoshp k, we must have where k s the wave umber of the lght, ad Source, k x (7) m Detector C Detector B Detector A Fgure -3: Whe you replace oe of the two mrrors wth a detector, the system behaves as oe would expect for partcles, wth half the partcles gog to detector C, ad oe-fourth each to A ad B x s the ucertaty the posto of the mrror Not surprsgly, ths expermet s oly a gedake expermet; such a small recol from a macroscopc mrror caot actually be measured What s surprsg s that ths expermet wll prove to be theoretcally mpossble as well Fudametally, ay such expermet s doomed to falure I The Schrödger Equato 4 06, Erc D Carlso

9 Of course, we are also tryg to measure a credbly ty mometum chage the mrror Whe t reflects a sgle photo the photo trasfers a mometum of order k to the mrror We must be certa there are o other forces o the mrror that are larger; for example, errat breezes Ths ca geerally be cotrolled by placg the expermet vacuum, etc, but we must addto be very careful that the ty mrror s ot movg wth ay sgfcat mometum tally, otherwse we mght mstake ths tal moto for the effects of the recol of the photo I partcular, we had better make sure that the mometum s kow wth accuracy smaller tha the mometum of the photo, so we eed p k (8) m We see, therefore, that the tal mometum ad posto of the mrror must both be kow to hgh accuracy, or the expermet s doomed to falure Now, f we multply Eqs (7) ad (8), we fd that the expermet s success wll requre x p (9) m The exact meag of the much less tha symbol Eq (9) s ambguous Roughly t came about because we ssted that we ot have ay photos passg to detector B If, for example, we let the combato k x m Eq (7) be equal to 4, we ca show that the terferece effects are completely destroyed, ad half the photos wll go to each detector, rather tha all to A So we wat Eq (9) to be qute a bt smaller, perhaps as small as 05 or eve less Now, f we could do the expermet descrbed, we really would be able to detect whch way the photo wet ad prove that t wet both ways Perhaps ot surprsgly, the expermet fals, ad the reaso s that Eq (9) smply caot be satsfed It s mpossble to specfy the posto ad mometum of a mrror, eve theoretcally, wth arbtrary precso The reaso s that quatum mechacs apples ot oly to lght, but also to mrrors as well Mrrors must satsfy the ucertaty relato Eq (5), as well as photos Ideed, as far as we kow, Eq (5) apples to electros, protos, eutros, atoms, molecules, mrrors, ad eve plaets Quatum mechacs s uversal All of physcs must be rewrtte to corporate quatum mechacs Everythg, ad ot just photos, acts lke waves (at least sometmes), ad our goal wll be to come up wth a wave theory that apples ot just to lght, but to everythg Although t s possble (at least prcple) to satsfy ether of the two relatoshps Eq (7) or (8), t s mpossble to satsfy both If we satsfy Eq (7), the terferece pheomea wll occur, ad all (or the vast majorty) of the photos wll fall to detector A, but we wll be capable of measurg whch of the two paths the photo took If we satsfy Eq (8), the we wll successfully measure whch way the photo wet, but the ucertaty the posto of the mrror wll destroy the terferece I essece, the process of measurg where the photo wet affects the photo Sometmes t s stated that quatum mechacs smply states that measurg systems dsturbs them, chages them, but quatum mechacs s far more profoud tha ths smple statemet That other partcles besdes lght have wave-lke propertes was frst proposed by de Brogle 94, ad cofrmed the case of electros by Davsso ad Germer 98 I 930, t was demostrated for atoms ad molecules, ad sce the for protos ad eutros It s ow beleved that everythg has both partcle ad wave propertes, though some cases (gravty, for example), ths has ot bee expermetally demostrated m 06, Erc D Carlso 5 I The Schrödger Equato

10 The de Brogle hypothess was that Eq (4) apples to electros, ad shortly thereafter t was smlarly speculated that Eq () apples as well We wll smply assume that these two relatos apply to all partcles B The Schrödger Equato I late 95, Erw Schrödger was gvg a talk about the de Brogle hypothess to a group of colleagues, whe Debye suggested that f you are gog to use wave relatos to descrbe electros, you probably should develop a wave equato, somethg ak to Maxwell s equatos, to descrbe that electro Ths goal was acheved by Schrödger, ad we attempt here to derve the Schrödger equato To obta ths equato, we wll have to make a umber of assumptos ad guesses I the ed, what assumptos we make are ot really mportat; what s mportat s the result, ad we ca oly ask ourselves whether the result s cosstet wth what s observed the world aroud us, or f t s cotradcted by t Hece we wo t worry too much about the fe mathematcal pots of our dervato We start by assumg that a electro (eve a sgle electro) oe dmeso s descrbed xt, Furthermore, we wll assume, as Schrödger dd, that t s a by a wave fucto complex wave fucto I electrcty ad magetsm, the troducto of complex electrc ad magetc felds s a mathematcal coveece, but quatum mechacs, t wll tur out to be a xt, has a sgle compoet, ulke the electrc or magetc ecessty We wll assume that feld, whch as vector felds have three compoets, ad also that free space (o forces or teractos) waves smlar to Eq (3) wll be solutos of our equatos I other words, we are lookg for equatos whch are satsfed by waves of the form x t N kx t, exp (0) There s o partcular reaso to thk ths wll work for arbtrary k or I fact, we kow that for o-relatvstc partcles of mass m, we expect there to be a relatoshp betwee the mometum p ad the eergy E gve by p E () m Now, we would lke to somehow relate Eqs (0) ad () wth the quatum mechacal Eqs () ad (4) We wat these the form of a wave relatoshp; that s, we wat expressos volvg thgs lke dervatves of the wave Eq (0) Notg that whe you take the dervatve of Eq (0) wth respect to x, you get a factor of k, t s easy to see that px, t kx, t x, t x It s temptg to avely cacel the wave fucto from both sdes of ths equato, ad wrte somethg lke () p (3) x I The Schrödger Equato 6 06, Erc D Carlso

11 I have wrtte ths wth a arrow, rather tha a equal sg, because I wat to make clear that ths should ot be vewed as a equalty, but rather as a trasformato Whe we move from the classcal to the quatum mechacal, we replace the classcal mometum wth the mometum operator, the expresso o the rght If we perform ths replacemet twce, t s easy to see that p x, t k x, t x, t x, t x x I a smlar maer, we ca covert eergy to a tme dervatve by usg Eq () We fd Ex, t x, t x, t, t whch suggests that whe we quatze a theory, we make the smlar substtuto (4) (5) E (6) t We ow use Eq (), a classcal equato, to relate Eqs (4) ad (6) to produce a wave equato: p Ex, t x, t x, t x, t m t m x Ths, the, s Schrödger s oe-dmesoal equato for a free o-relatvstc partcle To clarfy the steps we wet through dervg t, t s helpful to put them the proper order, rather tha the hodgepodge dervato we just completed These steps are: () Start wth a equato for the eergy terms of the mometum ad posto, () Multply t o the rght sde by the wave fucto xt,, (3) Replace the mometum by the dfferetal operator Eq (3), ad (4) Replace the eergy by the dfferetal operator Eq (6) For example, let us apply these steps for a o-free partcle, oe that s acted o by a force We eed to start wth a expresso for the eergy Most forces ca be wrtte as the dervatve of some potetal fucto; that s, oe dmeso, F x, t V x, t x The the potetal just cotrbutes to the eergy, ad the total eergy s p E V x, t m Followg our prescrpto, we multply ths o the rght by the wave fucto, I the example doe already, oly the mometum appeared, but more geeral cases the posto wll also appear The mportat thg to ote s that tme dervatves do ot appear; the velocty must be frst rewrtte terms of the mometum As we wll see later, tme may also occasoally appear 06, Erc D Carlso 7 I The Schrödger Equato

12 p Ex, t x, t V x, t x, t m We the replace E ad p by Eqs (3) ad (6) to gve us x, t,,,, t m x x t V x t x t ad we have the Schrödger equato, ow wth a potetal What do we do f we are more tha oe dmeso? Well, we smply start wth the threedmesoal formula for eergy p px py pz E V r, t V r, t m m We ow multply by the wave fucto o the rght, whch, ot surprsgly, ow wll be a fucto of all three space varables What we do wth the three mometum expressos s ot specfed by (), but t s t hard to guess that the correct procedure s to replace p x p y p z, or more succctly, x,, ad y z (7) p (8) It s the oly a mute s work to fd that the Schrödger equato for a partcle 3D takes the form,,,, r t r t m t V r t r t (9) Other possbltes come to md; for example, what f we have more tha oe partcle? Ca we deal wth the possblty of havg a ukow umber of partcles? What f the partcles are relatvstc? What f there are o-coservatve forces, lke magetsm? Is t possble that the wave fucto has multple compoets, much as the electrc ad magetc felds do? All of these are ssues that we wll deal wth tme, but for ow we wll treat Eqs (7) ad (9) as the Schrödger equato D ad 3D They wll provde ample examples of terestg ad sometmes dffcult problems to solve, but for ow, let us set these asde ad ask a bt about the terpretato of the wave fucto Before we move o, oe mor commet s order Eqs (7) ad (9) are complex equatos, because they explctly cota the magary umber Eve f we made the wave fucto real at the tal tme t = 0, t s easy to show that the Schrödger equato demads that t wll acqure a magary part at other tmes Hece complex umbers are a ecessty, ot merely a coveece C The Meag of the Wave Fucto We have derved a wave equato for the wave fucto xt, xt,, but we ever sad what s I cotrast, for example, electromagetsm, we ca defe the electrc feld I The Schrödger Equato 8 06, Erc D Carlso

13 Er,t as the force o a small test charge dvded by that charge Is there a smlar terpretato of xt,? The aswer s o There s o kow method to measure the wave fucto Or, at least, there s o kow way to measure the wave fucto wthout chagg t the very act of measurg t It ca be show that f we could somehow actually measure the wave fucto drectly, we would be able to produce all kds of mracles, such as commucatg faster tha lght But as far as we ca tell, quatum mechacs does ot allow oe to measure the wave fucto tself What you ca measure s the locato of a partcle However, as I attempted to llustrate wth all my gedake expermets secto A, the wave fucto must ofte be more tha oe place at a tme, eve though the posto of the partcle must oly be oe place Ths leads to cofuso about what the wave fucto represets If you look at Eq (6) t seems reasoable to assocate the square of the ampltude of the wave fucto wth the probablty of fdg the partcle there For example, oe dmeso, we assume the probablty desty to take the form x t x t * x t x t,,,, (0) Ths has uts of verse legth, sce probablty s dmesoless ad we are oe dmeso I partcular, f we have a rego ab, whch we are gog to look for the partcle, the probablty of fdg the partcle s posto x ths rego wll be gve by b * P a x b x, t dx x, t x, t dx a I partcular, f we have oe partcle, ad we look everywhere for t, the partcle must be somewhere, so the probablty of fdg t somewhere must be, so we have * x, t dx x, t x, t dx b a () Because the probablty desty s foud by squarg the ampltude of the wave fucto, the wave fucto s sometmes called the probablty ampltude I three dmesos, smlar statemets apply The probablty desty (ow wth uts of verse legth cubed) s foud by squarg the ampltude t t * t t r, r, r, r, () The probablty of fdg a partcle a rego V of space s gve by 3 * 3 P r V r, t d r r, t r, t d r If we tegrate over all of space, the probablty s oe V V 3 * 3 r, t d r r, t r, t d r (3) 06, Erc D Carlso 9 I The Schrödger Equato

14 The terpretato of the wave fucto (or, more geerally, the state vector) as a probablty desty s so dstasteful to may physcsts that may eschew ths formulato of quatum mechacs etrely For example, de Brogle-Bohm theory, the wave fucto s called a plot wave, a o-detectable fucto whch oetheless gudes the motos of actual partcles I the may worlds terpretato, o terpretato at all s gve to the wave fucto (t s what t s), ad probabltes do ot appear the theory at all, or at best are derved as a lmtg case I other terpretatos, quatum mechacs should ot be wrtte terms of the wave fucto tself, but rather terms of the state operator However, throughout most of these otes I wll treat the wave fucto (or state vector) as f t s real ad use the tradtoal Copehage terpretato of quatum mechacs A few commets are order I classcal mechacs, a partcle s descrbed at ay tme completely by gvg ts posto x ad ts velocty v, the tme dervatve of the posto It s ot suffcet to gve oly the posto, sce ts subsequet moto wll deped o the tal velocty I quatum mechacs, the partcle s completely descrbed ONLY by the tal wave xt, 0 Ths s because the Schrödger equato, Eq (7) or (9), are frst fucto order tme Gve the wave fucto at t 0, we ca determe from these equatos the frst tme dervatve of the wave fucto ad therefore ca determe the wave fucto a momet later Repeatg the process, we could actually umercally tegrate the Schrödger equato ad determe the wave fucto at arbtrary tme ( some cases, ths ca actually be practcal) Hece the wave fucto of the partcle s completely descrbed at ay tme by the wave fucto aloe; we do t eed ts tme dervatve Aother mor cocer has to do wth the ormalzato codto Eq () or (3) If we start wth a arbtrary wave fucto at t 0 ad let t evolve usg the Schrödger equato, how do we make sure that these equatos are satsfed at subsequet tmes? The aswer s that both versos of Schrödger s equato automatcally assure coservato of probablty, ad therefore o addtoal work s requred to mata the ormalzato codtos A more dffcult problem, ad oe we wll ot etrely resolve, has to do wth measuremet If a partcle s descrbed by a spread-out wave fucto, ad we measure ts posto, what happes? If we measure the same partcle repeatedly, wll the partcle jump from place to place erratcally betwee measuremets? The aswer to the latter turs out to be o If you measure a partcle, o matter how spread out, ad dscover that t s at a pot x, ad the mmedately measure the posto aga, t wll stll be at x Of course, the partcle may move betwee measuremets, gve eough tme, so you may have to repeat the expermet pretty quckly (photos, for example, move at the speed of lght) Ths mples that the process of measurg the wave fucto results a chage of the wave fucto Ths process s descrbed chapter four Oe last commet has to do wth the phase of the wave fucto Suppose we take a arbtrary complex wave fucto ad multply t, for example, by a factor of two The wave fucto wll stll satsfy the Schrödger equato, but wll o loger satsfy the ormalzato codtos, so t caot descrbe aythg realstc I cotrast, suppose stead that we multply the wave fucto by a arbtrary phase; that s, by a complex umber of magtude oe, so we have r t r t e r t,,,, where s a arbtrary real umber The the probablty desty wll rema uchaged I The Schrödger Equato 0 06, Erc D Carlso

15 * r t r t e r t e r te r t * r tr t r t,,,,,,,, I other words, f we detect the posto of the partcle, t wll have the exact same probablty r,t satsfes the Schrödger equato, Eq (9), dstrbuto It s also easy to see that f so also wll r,t Hece the two wave fuctos r,t ad,t r are physcally dstgushable, ad we ca treat them as detcal I some formulatos of quatum mechacs, they are detcal; we wll treat them as f they are dfferet prcple but detcal expermetally D The Fourer Trasform of the Wave Fucto Suppose at a gve tme the wave fucto oe dmeso takes the form x, t x Cosder ow the Fourer trasform, as gve by Eqs (A36): kx k x e, dx (4a) dk kx k e (4b) x x For example, suppose that our wave fucto takes the form 0 e k x The the Fourer trasform ca be foud wth the help of Eq (A34) to be k k k 0, so k cocetrated at k k0 Furthermore, the mometum, accordg to Eq (4), wll be k 0 Put smply, whe the Fourer trasform of the wave fucto s cocetrated at a partcular value of k, the mometum s smply tmes that value of k Now, whe the partcle s spread out x, we terpreted the wave fucto as a probablty dstrbuto for the posto x How do we terpret the Fourer trasform wave fucto k, whe t s spread out k? Probably ot surprsgly, the aswer s that t represets a probablty dstrbuto for the mometum p From Eq (4b), we see that f k s spread x out k, the the wave fucto s smply a lear combato of may dfferet waves of kx the form e The square of the ampltude of the Fourer trasform tells you the probablty desty that the mometum s aroud that value If you tegrate over dfferet k values, you ca fd out the probablty that the mometum les a gve rage Oe way to wrte ths s kb A B ka P k p k dk k Before acceptg ths equato as vald, the frst thg we should check s that the terpretato of ths as a probablty makes sese We should demad that the probablty of the partcle havg some mometum s equal to, so we fd P p dk k dx x, 0 s 06, Erc D Carlso I The Schrödger Equato

16 where we have used Eq (A37) together wth the ormalzato costrat Eq () to complete the proof Ideed, as we look at the wave fucto ad ts Fourer trasform, there s a kd of complemetary relatoshp betwee the two Gve ether, we ca fd the other usg Eqs (4) Oe descrbes the partcle terms of posto, the other terms of mometum Both ( geeral) wll represet oly a probablty dstrbuto for the posto or mometum Ether s a complete descrpto of the partcle There s o partcular reaso to thk of x as k the actual wave fucto, ad as smply somethg mathematcally related to t Ideed, the ext chapter we wll modfy our otato so that we ca refer to the wave fucto wthout specfyg whch of these two fuctos we are referrg to E Expectato Values ad Ucertates As we have already sad, whe you measure the posto or mometum of a partcle you caot geerally predct exactly what posto or mometum t wll have Istead, we get a dstrbuto, a sort of lst of probabltes Although t s most descrptve to gve a complete lst of the possble probabltes, whch for posto meas kowg x, sometmes t s suffcet to ask what the average outcome wll be f we perform a measuremet The expectato value a of some quatty a s the average of what you would get f you performed may measuremets If there are oly dscrete values of a that ca occur, the expectato value would take the form a apa, where a Pa s the probablty of some outcome a occurrg, ad the sum s take over all values of a that mght occur Ths sum s smply coverted to a tegral (ad the probablty P replaced by the probablty desty ) whe the varable a s cotuous, so we have a a a da Let us apply ths to the posto ad the mometum of a sgle partcle oe dmeso The probablty desty for the partcle to have posto x s x, whle for mometum the probablty desty for the partcle to have mometum k s k, so we have, x x x dx (5a) p k k dk (5b) Of course, eve though ths s the average value, the value you get o ay gve measuremet wll ted to be dfferet It s possble to defe ucertates x ad p as the Note that ths sum, there s o harm cludg values whch actually caot occur, sce they wll have probablty zero I The Schrödger Equato 06, Erc D Carlso

17 root-mea-square dfferece betwee the expectato value ad the value you get o partcular measuremets Mathematcally, these are gve by x x x x dx, (6a) p k p k dk (6b) It s these two quattes that satsfy the famous Heseberg ucertaty prcple Eq (5), whch we wll prove a later chapter The ucertaty prcple goes a log way towards uderstadg how quatum mechacs ca solve classcal coudrums Cosder, for example, the hydroge atom, whch cossts of a heavy proto (whch we treat as statoary) ad a lght electro orbt aroud t Classcally, the eergy of the hydroge atom s gve by p ke e E m r, (7) where r s the separato of the electro from the proto, p s the mometum of the electro, ad m s ts mass Ths s a problem, because the electro ca be as close to the atom as t wats to, whle keepg ts mometum small, ad hece the eergy ca be ftely egatve Before the advet of quatum mechacs, t was ot uderstood why the electro would ot smply spral to the proto, emttg electromagetc eergy Quatum mechacally, however, t s mpossble to both specfy the posto of the partcle precsely ad smultaeously lmt the mometum Suppose, for example, we wat the electro to be ear the proto We wat r = 0, but we caot demad ths precsely, because the electro s a wave fucto Istead we ask the electro to be wth about a dstace a of the proto We mght therefore approxmate r x a (8) We d also lke to get the mometum as low as possble, but because of Eq (5), we ca t actually force t to be qute zero Istead, we approxmate p p x a Substtutg Eqs (9) ad (8) to Eq (7), we fd that the eergy s gve by (9) ke e E 8ma a (30) We ow see how quatum mechacs solves the problem If we make the electro get too close to the proto, the ketc eergy term (the frst term) wll become very large, ad we wll ed up wth a postve eergy We ca fd the best possble sze for the separato by fdg where the dervatve of Eq (30) vashes, whch turs out to be a 4 e mk e The exact form of the Coulomb teracto, the last term Eq (7), wll deped o our choce of uts I ths class we wll always use SI uts, so e s the charge o the proto ad k e s Coulomb s costat 06, Erc D Carlso 3 I The Schrödger Equato

18 Substtutg ths back to Eq (30) yelds a approxmate eergy mk e E Ths aswer turs out to be too large by precsely a factor of four Part of ths ca be attrbuted to the fact that we used a ucertaty relato oe dmeso, whe we are really workg three; the remag (smaller) error s smply due to the fact that we worked wth a approxmato, so we should t expect to get the aswer exactly rght 4 e Problems for Chapter A partcle of mass m les oe-dmeso a potetal of the form V(x) = Fx, where F s costat The wave fucto at tme t s gve by x, t N t exp A t x B t x, where N, A, ad B are all complex fuctos of tme Use the Schrödger equato to derve equatos for the tme dervatve of the three fuctos A, B, ad N You do ot eed to solve these equatos For each of the wave fuctos oe dmeso gve below, N ad a are postve real umbers Determe the ormalzato costat N terms of a, ad determe the probablty that a measuremet of the posto of the partcle wll yeld x > a ( x) N x a (a) (b) ( x) N exp x a Nx x a for 0 x a, (c) ( x) 0 otherwse 3 A electro the groud state of hydroge has, sphercal coordates, the wave fucto ra ( r,, ) Ne, where N ad a are postve costats Determe the ormalzato costat N ad the probablty that a measuremet of the posto wll yeld r > a Do t forget you are workg three dmesos! 4 For each of the ormalzed wave fuctos gve below, fd the Fourer trasform k, ad check that t satsfes the ormalzato codto k dk /4 (a) x A exp Kx Ax (b) x exp x 5 For each of the wave fuctos questo 4, fd x, x, p, ucertaty relato x p s satsfed p, ad check that the I The Schrödger Equato 4 06, Erc D Carlso

19 V x m x 6 A partcle of mass m les the harmoc oscllator potetal, gve by Later we wll solve ths problem exactly, but for ow, we oly wat a approxmate soluto (a) Let the ucertaty the posto be x a What s the correspodg mmum ucertaty the mometum p? Wrte a expresso for the total eergy (ketc plus potetal) as a fucto of a (b) Fd the mmum of the eergy fucto you foud (a), ad thereby estmate the mmum eergy (called zero pot eergy) for a partcle a harmoc oscllator Your aswer should be very smple 06, Erc D Carlso 5 I The Schrödger Equato

20 II Solvg the Schrödger Equato We ow attempt to actually start solvg the Schrödger equato oe or three dmesos We start by wrtg t aga ( oe or three dmesos): x, t,,, t m x V x t x t (a), t,, t m V t r r r t (b) The frst thg I wat to ote about these equatos s that they are lear; that s, the wave fucto always appears to the frst power Suppose we have maaged, somehow, to fd two solutos, r,t ad r,t, to Eq (b), the t s easy to show that ay lear combato of these fuctos s also a soluto, e, so also s the combato r t c r t c r t,,,, where c ad c are arbtrary complex costats Obvously, ths ca be geeralzed to ay umber of fuctos A Solvg the Schrödger Equato Free Space We ow attempt to actually start solvg the Schrödger equato three dmesos for the case where there s o potetal, so r, t r, t m t () We already kow some solutos of ths equato; specfcally,, t kr t r e s kow to satsfy the Schrödger equato provded k m We ca add up a arbtrary umber of such solutos wth dfferet values of k Ideed, the most geeral soluto would be to add up all values of k, whch case the sum becomes a tegral, ad we therefore cosder the soluto 3 d k k r, t c 3/ kexp k r t (3) m Where c(k) s a arbtrary complex fucto of k Ths s a eormous umber of solutos; deed, as we wll demostrate a momet, ths actually represets all solutos To check that t really s a soluto, merely substtute Eq (3) back to Eq () ad check that the result s true We fd The factor of 3/ the deomator s a coveece, whch makes our work a ty bt smpler wthout chagg the outcome I oe dmeso t would be II Solvg the Schrödger Equato 6 06, Erc D Carlso

21 3 3 d k k d k k k c 3/ exp t c 3/ exp t m k k r m k k r m m 3 d k k ckexp k r t 3/ t m The most geeral problem we ca be gve for a free partcle s to specfy the tal wave r, t 0 r ad the be asked to fd the wave fucto at subsequet tmes fucto Settg t = 0 Eq (3), we see that d k r k 3 kr c 3/ e Comparso wth the Fourer trasform Eq (A38b) shows us that 3 d r kr ck k 3/ r e (4) We ca perform the Fourer trasform Eq (4) for arbtrary wave fucto r, ad the substtute the result back to Eq (3) Of course, actually performg the tegrato aalytcally may be dffcult, but ofte umercal methods wll work eve whe t s dffcult or mpossble to do these tegrals closed form Eqs (3) ad (4) ca be easly geeralzed to a arbtrary umber of dmesos As a example, suppose a partcle of mass m free oe-dmesoal space has tal wave fucto x Nx Ax exp (5) where A ad N are postve real costats To fd the soluto of Schrödger s Eq (a), we frst fd the costats ck usg a formula aalogous to Eq (4): dx dx Nk k c k e x N x Ax kx A A kx exp exp 3/ (6) where the tegral was performed wth the help of Eq (A3h) wth A ad k Substtutg ths to the oe-dmesoal verso of Eq (3), we the have m 3 A m dk k N k k x, t ck exp kx t k exp exp kx t dk A N t k k kx dk 3 A A m exp Ths tegral s aga of the type appearg Eq (A3h), except that the roles of x ad k have A t m ad x, so we bee reversed, ad ths tme we make the substtutos have 06, Erc D Carlso 7 II Solvg the Schrödger Equato

22 N x x Nx Ax xt, exp exp 3/ 3/ 3/ A t t 4 At m At m a m A m It s obvous that ths satsfes the boudary codto Eq (5); t s less obvous that t satsfes the Schrödger Eq (a) B The Tme-Idepedet Schrödger Equato Superposto s a very useful method for solvg a wde varety of problems It s partcularly useful cases whe the Schrödger Eq (a) or (b) has o explct tme depedece; that s, whe V s depedet of tme Suppose we are asked to fd solutos of Eq (b) Let us seek solutos of the form t t Substtute ths to Eq (b), ad the dvde by both r t : r, r (7) t V t t r r m Now, Eq (8), ote that left sde of the equato s depedet of r ad the rght sde s depedet of t Sce they are equal, they must both be depedet of both r ad t We gve ths costat a ame E, ad wrte d t dt t E V r The left equalty s relatvely easy to solve If you multply both sdes by dt ad tegrate, you wll fd t m r r r (8) (9) Et e (0) Ths s a equato for a wave wth agular frequecy E, whch allows us to detfy E as the eergy of the partcle The rght equalty Eq (9) ca be rewrtte what s kow as the tme-depedet Schrödger equato ( three dmesos) whch s V m r r r () E Techcally, there s a costat of tegrato we have gored here Ths costat would become a multplcatve costat Eq (0) Ultmately, ths multplcatve costat wll be cluded whe we use superposto to create the most geeral soluto to Eq (0), so there s o harm mometarly leavg t out II Solvg the Schrödger Equato 8 06, Erc D Carlso

23 Of course, we stll have to solve Eq (), whch wll deped o the partcular form of V r I geeral, there wll be a lst, geerally a fte lst (sometmes dscrete ad sometmes cotuous) of such solutos, whch we deote wth the dex, each wth ts ow eergy E, so that Eq () should be wrtte as r r r () V m E I geeral, we start wth the (hard) problem of solvg Eq (), ad call all the solutos egestates, the we substtute the resultg wave fuctos ad Eq (0) to Eq (7), ad fally take a arbtrary lear combato of the resultg solutos So the most geeral soluto of Eq (a) wll be r, t c r e E t (3) We wll leave the problem of fdg the costats c for a geeral tal wave fucto r, t 0 to a later chapter I geeral, however, we wll treat the Schrödger equato as solved as soo as we fd the wave fuctos r ad correspodg eerges E What s the advatage of solvg the tme-depedet Schrödger equato Eq () over smply solvg the orgal Schrödger equato? The prmary advatage s that there s oe less varable tha the orgal equato The reducto of the umber of varables s a vast advatage, ad ca ofte tur a seemgly tractable problem to a relatvely smple oe For example, suppose we are workg oe dmeso, whch the tme-depedet Schrödger equato takes the form d m dx V x x E x Ths s a ordary dfferetal equato, wth o partal dervatves Numercal methods for solvg such ordary dfferetal equatos are fast ad hghly accurate, so eve f we do t maage to solve Eq (4) aalytcally, we ca effectvely solve t umercally As a addtoal advatage, Eqs () ad (4) are both real equatos, so whe solvg them we ca set asde complex umbers ad work wth exclusvely real fuctos (4) C Probablty Curret I chapter I, secto C, we clamed wthout proof that the ormalzato codtos Eq () or (3) are preserved by the Schrödger Eqs () We wll ow demostrate ths ad dscover the probablty curret We start by takg the Schrödger Eq (b) ad multplyg * r,t o the left, ad the take the complex cojugate of the result: by 06, Erc D Carlso 9 II Solvg the Schrödger Equato

24 * * r, t r, t r, t V, t, t, t r m r * * r, t r, t r, t V, t, t t r m r We ow subtract these two equatos Note that there wll be a cacellato of the potetal V r, t s real term; ths cacellato occurs because * * * *, t, t, t, t, t, t, t, t r t r r t r m r r r r (5) We ote that the left sde of ths equato s smply a tme dervatve of the probablty desty, Eq () Also, the rght sde ca be wrtte terms of a dvergece, usg Eq (A8b), to show that r t r t r t r t r t r t,,,,,,, * * * * * * r, t r, t r, t r, t r, t r, t, t t t t t t t t * r, r, r, * r, * r, r, r, * r, (6) We defe the probablty curret jr,t by the equato or the more coveet form, t *, t, t, t *, t j r m r r r r (7) * t t t j r, Im,, m r r (8) Substtutg Eq (7) to Eq (6) ad usg Eqs (5) ad (), we see that m r, t jr, t, t m, t, t 0 t r j r (9) Eq (0) s a local verso of coservato of probablty There s a detcal equato,t jr,t s electrc curret To see that electromagetsm, where r s the charge desty, ad Eq (0) s coservato of probablty, tegrate t over a volume V, whch mght represet all of space d dt V r td 3 r d 3 r jr t,, O the left sde, the tegral s the probablty of a partcle lyg the rego V O the rght sde, use Gauss s Law Eq (A9) to rewrte t as a surface tegral: V II Solvg the Schrödger Equato 0 06, Erc D Carlso

25 d P r V ˆ, t ds dt j r (0) Eq (0) says s that the total probablty of the partcle lyg a gve volume chages oly due to the flow of the probablty to or out of the surface of the rego I partcular, f the volume V s all of space, ad f the wave fucto s small at fty, the the rght had sde of Eq (0) wll vash, ad the total probablty wll rema a costat (equal to ) So f the wave fucto s ormalzed so that Eq (3) s satsfed at oe tme, t wll always be satsfed Note that Eqs (9) ad (0) ca be trvally geeralzed to a arbtrary umber of dmesos kxt To get a better hadle o probablty curret, cosder for example the plae wave Ne The probablty desty at all places ad tmes ca be calculated trvally to be S r,t N, whch mght lead oe to mstakely beleve that there s o curret flow at all But the curret desty s ot zero; deed ts value s t t t t t N e e N e e t j r, Im Im, m m k r m k r k r k r k r k Sce k s the mometum of such a partcle, ths s just p m, the classcal velocty, whch meas oly that the resultg wave has ts probablty flowg at a steady rate, as we mght expect V(x) D Reflecto from a Step Boudary Cosder a partcle mpactg from the left wth eergy E o a step boudary, as llustrated Fg -, gve by V x 0 f x 0, V0 f x 0 Classcally the partcle should cotue owards past the step f E > V0, ad t should be reflected back to the left f E < V0 What happes quatum mechacally? Sce the potetal s tme depedet, we wll focus o the tme-depedet Schrödger s Eq (4) I the rego x 0 ths s just Fgure -: A partcle approaches a step potetal of magtude V0 from the left (rego I) Classcally, t wll be reflected f t has suffcet eergy to go over the step (gree arrows), whle oe that s above the step wll be trasmtted (red arrows) d I x E I x m dx () where the Roma umeral I just deotes that we are the rego x < 0 Ths s just a free partcle, ad we already kow solutos look somethg lke e kx If we substtute ths to Eq (), we fd k m E Ths equato has two solutos, whch I wll call k, where I V 0 II x 06, Erc D Carlso II Solvg the Schrödger Equato

26 The most geeral soluto for ths rego to Eq () wll be me k () kx kx I x Ae Be (3) Let s ow solve the same equato for x > 0 Schrödger s Eq (a) ths rego s Let us assume for the momet that E > V0 If we defe the there are two solutos of the form d II x E V0 II x m dx (4) k m E V 0 (5) e k x,, so the geeral soluto takes the form kx kx II x Ce De (6) We have solved the Schrödger equato regos I ad II What about the boudary? The potetal s everywhere fte, ad therefore every term the Schrödger equato wll be fte at x = 0 But the Schrödger equato has a secod dervatve wth respect to x The secod dervatve ca be fte oly f the fucto ad ts frst dervatve are cotuous, sce dervatves are oly defed for cotuous fuctos I other words, we must satsfy the boudary codtos 0 0 ad 0 0 (7) I II I II Substtutg Eqs (3) ad (6), ths mples A B C D ad k A B k C D (8) It s worth stoppg a momet to thk about the sgfcace of the four terms we have come up wth, Eqs (3) ad (6) To uderstad ther meag, look at the probablty curret Eq (8) for each of the four waves: k k k k ja A, jb B, jc C, ad jd D m m m m Wave A has a postve curret, whch meas t s movg to the rght, ad therefore t represets a comg wave o the left Wave B s a reflected wave, movg away to the left Wave C represets the trasmtted wave, movg off to the rght Ad what about D? It would represet a wave comg from the rght Such a wave mght exst, but we were askg specfcally about a wave comg from the left, so t s rrelevat to ths problem, ad we assume D 0 We are ow prepared to solve our smultaeous Eqs (8) If we solve them for B ad C terms of A, we fd k k k C A ad B A k k k k II Solvg the Schrödger Equato 06, Erc D Carlso

27 These two equatos tell us the magtude of the two waves, from whch we ca easly derve the relatve sze of the probablty desty However, we are tryg to fgure out f the wave s reflected or trasmtted, whch s askg somethg about the flow of probablty, or probablty curret I short, we wat to kow what fracto of the probablty flowg s reflected back ad what fracto s trasmtted forward These two quattes are deoted R (for reflected) ad T (for trasmtted), ad are gve by k k j C 4kk T A A k jb B C k R, j A k k j k k A (9) It s easy to demostrate that RT, whch smply meas that the wave s certaly ether reflected or trasmtted, as t must be These equatos ca be rewrtte terms of E ad V0 wth the help of Eqs () ad (5) f desred Oe cocer s that the wave fuctos we have foud are ot ormalzable The wave fuctos Eqs (3) ad (6) do ot dmsh as they go to fty Of course, the same could be sad of our plae wave solutos we foud secto A However, secto A, we were able to combe plae waves wth dfferet eerges to form wave packets whch we could ormalze The same s true here Bascally, ths s possble because the wave solutos we foud, Eqs (3) ad (6), are ot blowg up at fty, ad ths made t relatvely easy to combe them ad make wave packets whch do ot grow wthout bouds at fty We have solved the wave fucto oly uder the assumpto that Eq (4), we have a eergy E greater tha the potetal V0 We eed to recosder the case whe E V0, whe classcally the wave has suffcet eergy to peetrate the barrer To do so, we reexame Eq (4), whch I trvally rewrte the form We ow defe d m dx x V E x (30) II 0 II m V E The the solutos of Eq (30) wll be expoetals of the form x x 0 II x Ce De (3) It s tme to cosder the two solutos more detal before movg owards The term C represets a wave that des out the rego x > 0 I cotrast, the term D results a wave fucto that grows quckly; faster tha ay polyomal As a cosequece, t wll be dffcult to buld wave packets out of the D term, because the wave grows so fast Our cocluso s that the D term s o-physcal, ad we oce aga assume D = 0 We ow substtute Eqs (3) ad (3) (wth D = 0) to the boudary codtos Eqs (7), whch yelds the smultaeous equatos A B C ad k A B C These equatos ca be solved for B ad C terms of the comg wave A, whch yelds 06, Erc D Carlso 3 II Solvg the Schrödger Equato

28 k k B A ad C A (3) k k Now t s tme to fgure out the probablty of reflecto ad trasmsso If you substtute a x Ce to Eq (7) you just get j 0, whch makes the wave fucto of the form x trasmsso probablty zero Hece we quckly fd the reflecto ad trasmsso coeffcets C jb B k k jc R, T 0 j A k k j A A (33) Thus whe the eergy s suffcet to peetrate the barrer classcally, t eds up beg totally reflected, eve though the wave fucto peetrates a short dstace to the forbdde rego Whe E s greater tha V0, whe classcal theory predcts peetrato of the barrer, trasmsso s possble, eve quatum mechacally, but there s also a possblty of reflecto, as gve Eq (9) There s eve reflecto whe V0 s egatve; that s, whe the potetal barrer s egatve It would be as f you tred to ru off a clff, but stead reflected back! Fg - T s a sketch of the reflecto probablty as a fucto of R V0 E Eve though the partcle caot be trasmtted the V 0 E case E V0, we do ote that there s some peetrato of the wave fucto to the forbdde rego, though the wave fucto rapdly dmshes to zero Ths does Fgure -: The probablty of trasmsso (blue curve) ad reflecto (red curve) as a fucto of the sze of the barrer Note that there s always some reflecto except for the trval case V0 = 0 However, f the potetal s larger tha the eergy the wave s completely reflected brg up the possblty that f we had a barrer of fte thckess, would t be possble for t to peetrate the barrer, ad escape the other sde? Ths pheomeo, called quatum tuelg, wll be explored the ext secto E Quatum Tuelg Cosder ow a barrer of heght V0 ad thckess d, ragg from x d to x d, as llustrated Fg -3, wth potetal V x V 0 f d x d, 0 otherwse We wsh to cosder the case of a partcle of mass m ad eergy E V0 mpactg from the left I a maer smlar I II III -d/ V(x) V0 d/ Fgure -3: The fte barrer of secto E II Solvg the Schrödger Equato 4 06, Erc D Carlso

29 to before, we defe k me e ad mv E We expect oscllatg wave kx fuctos the allowed regos I ad III, ad expoetal fuctos the classcally forbdde rego II We do ot expect there to be a comg wave o the rght, so rego III kx we exclude the e soluto from ths rego Hece the most geeral soluto wll take the form 0 e x kx kx x x kx x Ae Be, x Ce De, x Fe I II III As before, we match the fuctos ad ther frst dervatves at each of the boudares x d, leadg to four boudary codtos: Ae Be Ce De kd kd d d kd kd d d k Ae Be Ce De,, ad Ce De Fe d d kd Ce De kfe d d kd These are four equatos fve ukows, ad we ca solve for ay of the varables terms of ay of the others We wll be partcularly terested the probablty of trasmsso, whch requres that we fd F Wth cosderable work oe ca show that The trasmsso probablty wll the be kd F ke A k cosh d k sh d 4 0 F E V E jf 4k T ja A 4k k sh d 4E V0 E V0 sh d where mv E 0,, (34) For a thck barrer, d, the hyperbolc se wll expoetally suppress ths rate, so the process wll be rare, but t ever vashes Note that each of the cases we have studed so far, a free partcle (secto A), a partcle approachg a step barrer (secto D), ad a fte thckess barrer, we foud solutos for all possble eerges The reaso ths occurred s because all these cases the eergy of the partcle was suffcet to get the partcle out to fty I secto F we wll explore a alteratve case, whe the eergy s assumed to be less tha the eergy at fty I such cases (called boud states), the wave fucto wll vash away at fty, ad the eergy wll tur out to oly come dscrete possble values F The Ifte Square Well Let s tur our atteto ow to a dfferet kd of problem, oe where the partcle s eergy s less tha the eergy at fty Cosder a partcle oe dmeso trapped a fte square well, wth potetal gve by V x 0 f 0 xa, otherwse 06, Erc D Carlso 5 II Solvg the Schrödger Equato

30 We start by tryg to solve Eq (a) The fte potetal has the effect that the eergy wll be fte for ay wave fucto lyg outsde the allowed rego, so certaly we wat a wave fucto that vashes outsde ths rego Furthermore, the dervatve terms wll cause problems f the wave fucto s ot cotuous We therefore demad that the solutos of Eq (a) 0 a 0 vash at the two boudares, It remas to solve Eq (a) the allowed rego, where the wave fucto wll satsfy d E x x (35) m dx Because Eq (35) s a secod-order lear dfferetal equato, there wll be two learly depedet solutos The fuctos whose secod dervatve are proportoal to ther egatves cos kx, so we wrte are s kx ad x A kx B kx Substtutg Eq (36) to Eq (35), t s easy to see that 0 0 wll demad that s cos (36) E k m The boudary codto wll be satsfed oly f we pck B = 0 The the boudary codto a 0 multple of, so k a Hece we ca label our solutos by a sgle postve teger, terms of whch x As x a We ow demad that the wave fucto be ormalzed; that s a x x dx A s dx a A a where the tegral was completed wth the help of Eq (A) Solvg for A, ad choosg t to be real ad postve, we fd A a Puttg everythg together, we ow have formulas for the egestates ad eerges x 0 a s x a for 0 x a, 0 otherwse, (37a) E ma (37b) s ka 0, whch mples that ak wll be a teger Fgure -4: The frst sx wave fuctos of the fte square well, for = through 6, red, gree, blue, black, brow, ad orage respectvely The frst few wave fuctos are plotted Fg -4 We ca geeralze ths soluto to ay umber of dmesos For example, suppose a partcle of mass m s cotaed a three-dmesoal potetal gve by There s a mplct assumpto here that E > 0 If E < 0, the the solutos to Eq (35) wll take the form e x or e -x ; however, o lear combato of these two fuctos ca satsfy both boudary codtos II Solvg the Schrödger Equato 6 06, Erc D Carlso

31 V x, y, z 0 f 0 x a, 0 y b ad 0 z c, otherwse Our wave fuctos must vash at the sx boudares Isde these boudares, t s easy to see that the Schrödger equato s smply m E It s t hard to guess the solutos to ths equato Products of se fuctos each of the three dmesos wll work cely We wll have three dces, x, y, z labelg our states, ad the eerges wll be fuctos of all of these The wave fucto ( the allowed rego) ad the eerges wll be gve by x, y, z 8 x y z x, y, z s s s, abc a b c E m a b c x y z x, y, z G Boud States from a Double Delta Potetal Cosder a partcle of mass m oe dmeso wth potetal so that the Schrödger equato takes the form V x x a x a, d E x x a x a x m dx where a ad are postve costats I other words, the partcle s free except at two pots where the potetal s egatvely fte, as sketched Fg -5 We wll be seekg boud states wth eergy E < 0 Accordg to classcal physcs, the partcle wll be stuck oe of the two ftely arrow wells What does quatum physcs predct? Away from the two delta fuctos, the partcle s free, satsfyg d E x x (39) m dx, Keepg md that E s egatve, we defe me We the fd that the solutos to Eq (39) take the form e x However, (38) x Fgure -5: The potetal s zero except at the two pots, where t s egatvely fte 06, Erc D Carlso 7 II Solvg the Schrödger Equato

32 there s o reaso to beleve that the form of the equato wll match the three regos marked I, II, ad III Fg -5, so the coeffcets of the three waves wll ted to be dfferet Furthermore, we do t wat solutos whch dverge at fty, so rego III we reject waves x that look lke e, ad rego I we reject e x We therefore guess the solutos wll take the form x x x x x Ae, x Be Ce, x De (40) I II III What about boudary codtos? It s temptg to assume that the fucto ad ts dervatve wll be cotuous, but ths s ot correct To fgure out what to do at the boudary, tegrate Eq (38) across oe of the boudares, say from a to a We fd x a a a d m a dx a a E x dx dx x x a x a dx The frst term o the rght sde s easy to evaluate, because the tegral of the secod dervatve of a fucto s just the frst dervatve, by Eq, (A) The tegral of a delta fucto s The other boudary pot smlarly easy to evaluate usg Eq (A4a): t smply yelds x a s ot the rage of tegrato We have a a E x dx III a a a m a II Note that the dervatve terms are evaluated usg the approprate fuctos each rage Now, cosder the lmt as 0 The tegrad o the left s fte, ad the tegral wll become very small So ths smplfes to m (4) III a a a II Sped a momet aalyzg ths equato It says specfcally that the dervatve s ot cotuous at the boudary: t has a fte dscotuty there However, a fte dscotuty mples that the fucto must at least be cotuous, so we have II a a (4) III Ths s fortuate, sce otherwse equato Eq (4), we would t kow whch fucto to use o the rght sde of the equato, but we see from Eq (4) that t does t matter We ca smlarly tegrate Eq (38) across the boudary at x a Ths leads to two more boudary codtos m, ad (43) a a a a a II I I II Let s wrte out Eqs (4), (4), ad (43) terms of our explct waves Eqs (40) We fd a a a Ae Be Ce, (44a) a a a De Be Ce, (44b) II Solvg the Schrödger Equato 8 06, Erc D Carlso

33 m a a a a, Ae Be Ce Ae (44c) m a a a a De De Be Ce (44d) To fd the o-trval solutos to Eqs (44), we substtute Eqs (44a) ad (44b) to Eqs (44c) ad (44d), whch yelds m a a a m a a a Be Ce Be ad Be Ce Ce Note that these equatos are detcal save for the terchage of the role of B ad C If ether B or C vash, the the other wll as well, ad the resultg soluto s a trval oe If they do t vash, we ca dvde by ether of them, ad we ca rewrte the two equatos as B C e a C B m Now, the oly umbers that are ther ow recprocal are, so substtutg ths ad rearragg a bt, we must have a e (45) m The left sde of ths equato s a smple le The rght sde s two fuctos, oe of whch starts at 0, the other at, ad the both asymptotcally approach It s easy to see, as llustrated Fg -6, that the lear fucto must cross the upper curve evetually, so there s always oe soluto to Eq (45) The other curve crosses the le at 0, but Fgure -6: Graphcal soluto of Eq (45) We are tryg to fd where oe of the straght les crosses the red (for +) or gree (for -) curves The straght les correspod to, 050, 075, 00, 5, ad 50 (bottom to top) The solutos for the ad (avy le) are (dotted red ad gree les respectvely) Note that wll exst oly f ths turs out to lead to oly a trval soluto to our equatos However, f s ot too small, t s easy to show that there wll be a addtoal crossg, whch we deote I other words, we have oe or two solutos to Eq (45), whch satsfy the two equatos 06, Erc D Carlso 9 II Solvg the Schrödger Equato