called the state descriptors of Newtonian physics. The evolution of the state represented by the trajectory is given by m r t V r t

Transcription

1 Revew of Phys 3 The Classcal Pot of Vew A system s a collecto of artcles that teracts themselves va teral forces ad that may teract wth the world outsde va exteral felds. Itrsc roertes of a classcal system (e.g. rest mass ad charge) are deedet of ts hyscal evromet ad therefore do t deed o the artcle s locato ad do t evolve wth tme. Extrsc roertes of a classcal system (e.g. osto ad mometum) evolve wth tme resose to the forces o the artcle. Accordg to the classcal hyscs, all trsc ad extrsc roertes of a artcle could be kow to fte recso ad we could measure the recse value of both osto ad mometum of a artcle at the same tme (classcal hyscs descrbes a determate uverse). Classcal hyscs redcts the trajectory (.e., the values of ts osto ad mometum for all tmes after some tal tme ) of a artcle, r t, t; t t trajectory, where the lear mometum s gve by d t m r t m t dt v wth m the mass of the artcle. Trajectores are called the state descrtors of Newtoa hyscs. The evoluto of the state rereseted by the trajectory s gve by m r t V r t where d dt t, V r, t s the otetal eergy of the artcle. To obta the trajectory for, oe oly eed to kow V r, t ad the tal codtos ( the values of r ad at the tal tme t ). t t As a result, classcal hyscs (Newto s laws) redcts the future of ay system that s kow ts tal codtos.

2 Near the ed of the 9 th cetury, theoretcal hyscs was based o three fudametals: Newto s theory of mechacs Maxwell s theory of electromagetc heomea Thermodyamcs ad ketc theory of gasses At the ed of the 9 th cetury ad the begg of the th cetury, a crss aeared hyscs. A seres of exermetal observatos could t be exlaed by cocets of classcal hyscs. Blackbody radato Photoelectrc effect Comto effect Partcle dffracto The reeated cotradcto of classcal laws made t ecessary to develo some ew cocets ad the develomet of these cocets emerged Quatum Theory that troduced ew cocets: Partcle Wave Blackbody radato behavour of radato Photoelectrc effect Comto effect behavour of artcle Partcle dffracto Quatzato of hyscal quattes Electros at the atomc states wth L mvr Quatum hyscs s a theory descrbg the roertes of matter at the level of mcro heomea (molecule, atom, ucleus, elemetary artcle, ). Ths theory rovdes the aswers to may questos whch remaed usolved classcal hyscs.

3 The Quatum Pot of Vew The cocet of a artcle does t exst the quatum world, artcles behave both as a artcle ad a wave (wave-artcle dualty). Ulke the classcal hyscs, ature tself wll ot allow osto ad mometum to be resolved to fte recso (Heseberg ucertaty relato), xt t where x x t s the mmum ucertaty the measureme of the osto the x-drecto at tme ad x t s the mmum ucertaty the measuremet of the mometum the x- drecto at tme. Posto ad mometum are fudametally comatble observables (the uverse s heretly ucerta). Therefore, f we caot kow the osto ad mometum of a artcle at, we caot secfy the tal codtos of the artcle ad hece caot calculate the trajectory. t t Physcal quattes (lke eergy) take some certa values (quatzato). t 3

4 Heseberg ucertaty relatos The stadard devato or ucertaty for two oerators A ad B are gve by A A A A ΔA = = ΔA ΔB AB, A A The quatty mea for A. ΔA= A A measures the sread of values about the Wave-artcle dualty Partcles behave both as a artcle ad a wave the quatum world Wave Photo Matter Matter Matter Wave Physcal quattes for artcles E Physcal quattes for waves h h / What s the matter wave? (kx wt) Plae wave D: ψ(x, t) A e Δx dx v lm Δt Δt dt kx wt costat dx w v dt k v hase π ν νλ π /λ de Brogle seed artcle seed E h h (/ )mv mv v lae wave caot rereset a artcle. 4 v

5 Wave acket: the suerosto of the waves that s aroxmately localzed sace at ay gve tme) dk x dw t costat v g de Brogle seed v artcle seed The wave acket s a suerosto of waves. Suerosto of waves Fourer tegrate xet / ( x, t) d () e dx dw de/ mvdv v dt dk d/ mdv wave acket reresets a artcle. v g that s a soluto of a artal dfferetal equato called Schrodger Equato a free artcle of mass m ψ(x, t) - t m ψ(x, t) x Schrodger Equato the resece of a otetal V(x) wll be ψ(x, t) - t m ψ(x, t) x V(x)ψ(x, t) where (x,t) s the soluto of Schrodger equato, mathematcal descrto of the wave acket. It s to be hyscally accetable soluto (called wave fucto) f t satsfes square tegrable (fte), sgle-valued ad cotuous roertes. Ay soluto of Schrodger equato that becomes fte must be dscarded. 5

6 The Schrödger equato has two mortat roertes The equato s lear ad homogeeous. A mortat cosequece of ths roerty s that the suerosto rcle holds. Ths meas that f ad the the lear combato of these fuctos s also a soluto ψ (x,t) ( x, t) C ( x, t ). ψ (x,t) are solutos of the Schrödger equato, The equato s frst order wth resect to tme dervatve (meag that the state of a system at some tal tme to determes ts behavor for all future tmes) frst order tme dervatve ( x, t) ( x, t) - t m x It has oe tal codto. Thus, f ( xt, ) s kow, ca be - x/ foud. Frstly, s calculated from ( ) dx ( x,) e. The, (x-et)/ usg, ψ(x, t) s obtaed from ψ(x,t) d () e. π () () ( xt, ) Every soluto of Schrodger equato s ot a wave fucto (hyscally accetable soluto). To be a accetable soluto, a egefucto ( xt, ) ad ts dervatve ( x, t) / x are requred to have the followg roertes: ( xt, ) ad ( x, t) / x must be cotuous. ( xt, ) ad ( x, t) / x must be sgle valued. ( xt, ) ad ( x, t) / x must be fte (square tegrable). 6

7 Probablty terretato of the wave fucto: Fte roerty requres the terretato of robablty desty. ψ(x, t) s geeral a comlex fucto ad s aaretly ot a measurable quatty. However, the wave fucto s a very useful tool for calculatg other qute meagful, mathematcally real quattes. s always real. It s large where the artcle s suosed to be, ad small elsewhere. We also kow that s sreadg wth tme (t meas that as tme asses, t s less robable to fd the artcle where t s ut at t=). P(x, t) ψ(x, t) s ψ(x, t) called the robably desty ad requres: ormalzato codto. - ψ(x, t) ψ(x, t) dx that s called the Coservato of Probablty The robablty desty s defed as x, t * x, t x, t. We kow that xt, satsfes the Schrödger equato. It s also true that * xt, satsfes the Schrödger equato. It satsfes the followg equato called cotuty equato t x, t j( x, t) x where d d * j * m dx dx s the flux (robablty curret desty that s the umber of artcles er secod assg ay ot x ) I 3D, they wll be t We the fd t - - r, t j( r, t) ad j( r, t) * * m 7 ds j(x, t) 3 3 d x P(x,t) - 3 d x P(x, t) costat d x j(x, t) A chage of the desty a rego d dt b S as s a x b s equal to a et chage the flux to that rego dx P( x, t) dx j( x, t) j( a, t) j( b, t) a b a x

8 Exectato value: How ca we calculate the measurable quattes (osto, mometum, eergy, etc) from the wave fucto? A average value of a measuremet (statstcal average of a large umber of measuremets) s called the exectato value. The exectato value of the osto of a artcle: I geeral, x-coordate of a artcle caot have a certa value a rego f that rego. It s ossble to talk about the average value of x-coordate. It s called the exectato value of x-coordate ad s gve by x ψ(x, t) - x P(x, t) dx - ψ * (x, t) x ψ(x, t) dx The exectato value of the mometum of a artcle: where - ˆ x * ψ (x, t) ψ(x, t) dx x Alteratvely, mometum sace where s the mometum oerator x-sace. * * ad - - x d () () d () () xˆ s the osto oerator mometum sace. For a arbtrary fucto, the exectato value - * * f (x) ψ (x, t) f(x) ψ(x, t) dx ad f () ψ (x, Examle: - * x ψ (x, t) x ψ(x, t) dx ad ψ The ucertates x ad Δx Δ x x ad - * - t) f (x, t) x ψ(x, x ψ(x, t) dx t) dx The quatty Δx x x measures the sread of values about the mea for x ad Δ for. 8

9 Hermta Oerator The oerator oerators oerators. ˆ x The eergy oerator : s comlex, but ts exectato value s real. Such (osto, mometum, eergy, etc) are called Hermto Ê t Hamltoa of the system: Ĥ V(x) m x Oerators lay a cetral role Quatum Mechacs. Products of oerators eed careful defto, because the order whch they act s mortat., Bˆ Bˆ - Bˆ s called a commutato relato betwee oerators Commutato relato betwee mometum ad osto:, x Ths relato s deedet of what wave fucto ths acts o. Ths s a fudametal commutato relato quatum hyscs. Heseberg ucertaty relato becomes Δx Δ ˆ, x Δx Δ 4 Imortat cocets: The otato of a wave acket as reresetg a artcle Schrodger equato as fudametal equato quatum hyscs The wave fucto ψ(x, t) that has the robablty terretato Heseberg ucertaty relato Statstcal average of a large umber of measuremets as exectato value 9

10 Egevalues ad Egefuctos Tme-deedet Schrodger Equato ψ(x, t) - t m ψ(x, t) V(x)ψ(x, t) x The method of searato of varables: f the satal behavor of the wave fucto does ot chage wth tme, we use some tme-varyg multlyg factor frot of the satal art of the wave fucto. ψ(x, t) dt(t) -Et/ ET(t) T(t) Ce dt T(t)u(x) d u(x) - V(x)u(x) Eu(x) m dx The egevalue equato Ô f(x) λ f(x) oerator egevalue egefucto egefucto ˆ of Ô wrt f ( x) of O of Oˆ The egevalue equato states that the oerator actg o certa fuctos f(x) (egefucto) wll gve back these fuctos multled by a costat λ (egevalue). Qˆ Eergy egevalue equato H u E(x) E u E (x) -Et/ Although the wave fucto ψ(x, t) ue(x) e deeds o tme, the robablty desty ψ(x, t) u (x) does ot deed o tme. E The tal satal wave fucto as a lear suerosto of the eergy egestates ψ(x,) Cu( x) The evoluto of the wave fucto tme s gve by a smle lear suerosto of these egestates ψ(x, t) C u (x)e -E t/

11 Alteratvely ad equvaletly, a tme-evoluto oerator to a tal state to obta the evaluated state t as ψ(x,t ) at tme t - H t t / ( x, t ) e ( x, t ) e - H t / s aled ψ(x,t ) at tme Examle : a artcle a fte box, x V(x), x a, x d u(x) me u(x) u(x) Askx Bcoskx dx Eergy egefucto ad egevalues u (x) E π x s a a π ma,,3,... Orthoormalty codto a dx u (x) u (x) * m δ m whem whem The exaso ostulate: Ay fucto orthogoal fuctos. ψ(x) ca be exaded terms of where ψ(x) a A u (x) * A dx u (x) ψ(x) s the rojecto of ψ(x) oto u (x). The exectato value of eergy s gve by ψ(x) H ψ(x) A E where A s the robablty that a measuremet of the eergy for the state E. ψ(x) yelds the egevalue

12 Mometum egefucto oe u (x) u (x) u (x) e π x/ Degeeracy: f more tha oe egefucto corresods to the same egevalue, ths egevalue s sad to be degeerate. Party Pˆ ψ(x) ψ(-x) λψ(x) λ Pˆ s a costat of moto Pˆ,H Eergy egefucto s also ege fucto of arty Imortat cocets: The egevalue equato Partcle a box roblem The exaso ostulate

13 Oe-Dmesoal Potetals The Potetal Ste a) Whe a eutro wth a exteral ketc eergy K eters a ucleus, t exereces a otetal b) Whle a charged artcle moves alog the axs of two cyldrcal electrodes held at dfferet voltages, ts otetal eergy chages very radly whe assg from oe to the other. Potetal eergy fucto ca be aroxmated by a ste otetal. The Potetal Well a) The moto of a eutro a ucleus ca be aroxmated by assumg that the artcle s a square well otetal wth a deth about 5 MeV. b) A square well otetal results from suermosg the otetal actg o a coductg electro a metal. 3

14 The Potetal Barrer a) scatterg roblems b) Emsso of α artcles from radoactve ucle c) Fuso rocess d) Tuel dode e) Cold emsso electros The Harmoc Oscllator 4

15 The Geeral Structure of Wave Mechacs Postulates Postulate : The dyamcal state of a artcle ca be descrbed by a wave fucto whch cotas all the formato that ca be kow about the artcle. Postulate : A arbtrary fucto ca be exressed as a lear suerosto of a set of orthoormal fuctos. Postulate 3: The Schrodger equato descrbes the behavor of the wave fucto sace ad tme. Postulate 4: Each observable quatty q ca be drectly assocated wth a lear, Hermta oerator. The value q s a egevalue of the oerator. Hermta roerty: ψ Qψ Qψ ψ Theorem : The egevalues of a Hermta oerator are real. Theorem : The egefuctos of a Hermta oerator are orthogoal f they corresod to dstct egevalues. Postulate 5: The exectato value of a measuremet of a varable q s gve mathematcally as q ψ Q ψ ψ ψ Each observable quatty q ca be drectly assocated wth a lear, Hermta oerator. The value q s a egevalue of the oerator. Commutg Observables: The cummutator of two oerators ad Bˆ s defed by 5, Bˆ Bˆ - Bˆ. If the commutator vashes whe actg o ay wave fucto, the oerators ad Bˆ are sad to commute, Bˆ Bˆ., Bˆ, ther observables A ad B are sad Whe the oerators commute, to be comatble. Observables are o-comatble f, Bˆ. If two observables are comatble, ther corresodg oerators have the smultaeous egefuctos ad A ad B are sad to be smultaeously measurable. Thus, comatble observables ca be measured smultaeously wth arbtrary accuracy, o-comatble observables caot.

16 Bˆ ψ, Bˆ Bˆ ψ Bˆ ψ a Bˆ ψ s a egefucto of belogg to the egevalue. Sce s o-degeerate, ca oly dffer from by a multlcatve costat whch we ca call Bˆ ψ b Bˆ ψ ψ b Thus we see that s smultaeously a egefucto of the oerators ad belogg to the egevalues ad, resectvely. If are a set of commutg oerators, there s a smultaeous egefucto of, Bˆ, Ĉ,... wth the egevalues a, b, c,... Bˆ, Bˆ, Ĉ,... ψ ψ a ψ b a. a Tme Deedece ad Classcal Lmt: The exectato value of a oerator t - ψ * (x, t) ψ(x, t) dx The exectato value vares wth tme as If d dt t t t H, t has o exlct tme deedece, t H, t a costat of moto f the oerator commutes wth H. d dt s. The observable A s 6

17 Oerator Methods Quatum Mechacs Oe dmesoal harmoc oscllator Hamltoa s H mω x where m, x The roblem s how to fd the eergy egevalues ad egestates of ths Hamltoa.. Polyomal Method / α α x / u (x) e H (α x) π! E ω,,,.... Oerator Method Dmesoless osto ad mometum oerators are defed as q mω x mω x - q Two o-hermta oerators are troduces terms of q ad as (q ) (q ) mω mω Sce x ad are Hermto, Commutator x x mω x mω, gves, x s deed the hermta cojugate of.. The Hamltoa terms of these oerators becomes where H ω Nˆ Nˆ s kow as the umber oerator whch s Hermta. H, They have smultaeous egefuctos. H Nˆ E 7

18 The other commutato relatos become H, ω H H ω Cosder H, ω H H ω H E. Multly both sdes wth as H Hω E H (E ω) s a egefucto of H wth the egevalue eergy E s lowered by oe ut of. Multly aga wth H Hω (E ω ω) H (E ω) s also a egefucto of H wth the egevalue the eergy E s lowered by. ω ( E ω) ( E ω) such that the such that The oerator s called a lowerg oerator. Sce the harmoc oscllator has oly ostve eergy states cludg zero, there must be a lower boud o the eergy. There s a state of lowest eergy, the groud state. So that eergy caot be lowered ay more. H Now multly ω ω H E ω wth E ω H H ω E H (E ω) s a egefucto of H wth the egevalue ( E ω) eergy E s rased by oe ut of ω Multly aga wth H H ω (E. ω) H 8 (E ω) such that the s also a egefucto of H wth the egevalue ( E ω) such that the eergy E s rased by ω.

19 9 The oerator s called a rasg oerator. We obtaed the eergy sectrum of the harmoc oscllator wthout solvg ay dfferetal equato as,,,... ω E What are the values of C ad D? D C Usg, Nˆ ad, Nˆ, we get C ad D. The egestates are! :. : : 3 The exlct for of the egestates are / q / q / q (q)e H N (q) u e C q q! u e C u qu dq du dq d, u ) (q

20 The tme deedece of oerators: The soluto of tme deedet Schrodger equato s The exectato value of a oerator Bˆ s ψ(t) e Ht / ψ() Bˆ t ψ(t) Bˆ ψ(t) ψ() e Ht / Bˆ e Ht / ψ() ψ() Bˆ (t) ψ() Bˆ (t) Pctures Schrodger cture: Oerators are tme-deedet. Tme evaluato of the system s determed by a tme deedet wave fucto. Heseberg cture: Oerators are tme-deedet. Wave fuctos are tme-deedet. The result s the same whatever cture we use. Tme varato of gve by d Bˆ (t) dt H, Bˆ (t) H H (t) e (t) e ω t As a examle: osto ad mometum oerators of a artcle () x(t) x()cos ω t sω t mω (t) ()cosω t - mω x()s ω t ω t S S Bˆ (t) s

21 Lev-Cvta Tesor Coordate trasformato 3D ca be wrtte as x ' 3 λ j j x j,,3 The magtude of a vector s varat uder coordate trasformato (rotato) r r' x y z x' y' z' r r' or r r' 3 3 ' ' ' x xx λjx j λkx k λjλ k x jx k = j k jk j r r x j z, z' λ λ "orthogoalty codto" j y' k θ y jk x' θ x x' cosθ y' -sθ z' sθ cosθ x cosθ y sθ z x x' ftes mal rotato y y' -θ z z' -sθ cosθ x' y' z' θ x y z A B A eˆ B eˆ A B eˆ eˆ A B j j j j j j C A B A eˆ B eˆ A B eˆ eˆ A B eˆ C j j j j k jk k j j j eˆ A B jk j jk mk jm m j jk k j jk k ε jk f, j, k form a eve ermutato of,, 3 f, j, k form a odd ermutato of,, 3 whe ay two dces are the same