onedim - (om; . Specifically. forlight. the enetgy E and...

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Unformatted text preview: (om; . Specifically. forlight. the enetgy E and the frequency u are Hill E =hv : (g'klj'+[kyjt+fkriyflefiit.t+E.v+tyc-Mt and for matter war-es, the momentum p and the wavelength Jr. are related by = “"4" Since we want p = Mt to multiply tit on the right-hand side. we multiply both a = Ml sides by not; We will assume that berth of these properties apply to matter waves. The equation _yyl1yqy = HILL. _ 4,. for energy. E = uh LI1 Et'tt'l'ctfipt'tndfi. [n — p In summary. we now have two operators for which I]! is an eigenfunctiun; an: E = (hizfllhul = “'5” gives the WW E 33 I“: Eigenvalue. and the outer gives the momentum F as the eigenvalue; and the momentum equation. p 2 ML becomes _ Elli" a =tfitlrrtt1=rtlt= at: [33} tH-fir— =. go We can generalize this equation from one oirrtention to three dimensions. In three —t'fiVHl" = pill dimensions. It points in the direction of— mrrtimt ol' the pflflicifi, which is also the direction of the momentum vector. p. Hence. Equation (3.7) becomes We now derive an equation which con-esponds to Equation (3 El. Applying the momentum operator —:'.w twice to o produces two factors of p: {—ih?) " {fiih‘fi'tllt = P 1 pt? ——. 3lI-I' For a non—relativistic particle with mass till end with no potential energy. the rela— '0 . . . . . Hence, ttonshtp between p and E; is given by at??? = ? 2m mica" “it” ‘5' 1-“ themed for ‘i' - ‘i' = filtaxi + stray? +32I3zZ—J Then _It 1'5 now possible to generate expressions for E and p from Equation til-.5} hr 1 p: by taking the appropriate derivatives. Taki ng the derivative of Equation {3.5‘] Wiflt _ E v' ll’ = m 4' [3.10; - 'l to t' . ct. ' - - - reaper- 1rne We at wfli I} = Heitmpmj le'lfi right-hand srtles ofIEquatLOns {3.9J and [3.10) are manifest” equal. Since E — —ituBr”"""°“" mic-titan Euiuriililitmlirtmmm Sides “"1”: eEll-mime aim us a dif- fir _ n so s y : 3.5 s Multiplying both sides by in git-es energy on the right-hand side'., I i _ Fl' v2!” _ In Hill g -‘ 3 [3.11; ii'lI-I' . . . . . “l.— = hmflf'tt'rnm So far. all we have shown is that the wave funenon given In Equation {3.5) d: _' satisfies- Equation {3.1 l). and further. that il is an eigenfunction oflhe ope-rams on = e o {1'50 “1.: Momma mi .1- gh't-hsmd sides of Equation {3. I t l with eigenvalllflfi Fitz»: and E, respectively. It is at this point that we malu: an. unjustified leap: what happens if we now add a potential energy V to the system? In a classical system, the total energy is just the sum ofthc kinetic and potential energies: Note that ihtiiifiiltl is a linear operator. and '4' is an. eigenfunction of This operator with cigsnt'm'ue E. New weneedto find asimilar operator which gives the momenmm p. Expanding the clot product in Equation {3.5} gives * d-wl- ~ m 5- M~ 2 II'L “it t . “‘5‘ p— + V = E (fr-*BJE 551345 Eur EM- “fifijpcq will; win” : Bantt.t'+t.y+t_-:—ofl Th' . st that mod'f t' 3.” tot-earl 50th“! I if? E? {Pry-‘- tssugge s we IyEquatoni ]I '- it?- 53L flu! is +t +s t ..--'—"L —' I 1‘- VJ' ri'4|“ _..rk_,b'e (inf? 1— .” :52. shin fl =ikyflEIH-ll-l-Jty_'t'+it;1-II|T:| #- E's aw film; f rig? _ = ik‘Beiu.xlk.J-I~k_.: mr| ii: " Hy [Notethatunlikethe ltjnetit: energy ahdtotal energy. which etuTesp-ondto operators containing yarims derivatives. the operator mrresptntding to the potential energy is simply multiplication ol' '4' by VJ Equation (3.1 1} is theS-chrfidingcr equatitut, Tutsst'hly the most important equa- rP lye. 3 1572;;- 2.1 STA'I'IDI‘HHRY STATES [def/'72s) In Chapter I we tallted a lot about the wave function. and how you use ll to calculate various quantities of interest. The time has come to stop procrastinating. and confront what is. logically. the prior question: How do you get tilts. t'] in the first place? We need to solve the Schrodinger equation. ,hao n1 taintI W a} 1 t —-— = ————.- lit 2m 312 + ' [" J for a specified potentialt 'v'fx. .r}. In this chapter {and most of this book} I shall assume that V is independent of r. to that case the Schrodinger equation can be solved by the method of separation of variables tthe physicist’s first line of attack on any partial differential equation}: We look for solutions that are simple products. tl'f.r.r} = arts-toot. t2 1t where tit {hover-case} is a function of it alone. and to is a function of l' alone. [In its face. this is an absurd restriction. and we came: hope to get more than a tiny subset of all solutions in this way. But hang on. because the solutions we do obtain turn out to be of great interest. Moreover {as is typically the case with separation of variablesl we will be able at the end to patch together the separable solutions tn such a way as to construct the most general solution. For separable solquns we have so_ as shit die at at' as2 as! [ordinary derivatives, now}. and the Schrodinger equation reads as slain 'it —=—_=— v . ’ “but anax2w+ W Dr. dividing through by itrw: _ [dip s- ldzitr 1 __=__.__._. _ -s mth thllretxl +5; [ 1 Now. the left side is a function of t alone. and the right side is a function of x alone.2 The only way this can possibly be true is if both sides are in fact cemtont—dd'terwise. by varying t. [ could change the left side win-tout touching the right side. and the two would no longer be equal. {That’s a subtle but crucial argument. so it‘ it's new to you. be sure to pause and think it through.) For reasons that will appear in a moment. we shall call the separation constant E. Then 1 do: 'h——=E. I melt or . it” = _Eg~ [2.41 dt it and CIT [1-5] Separation of variables has turned a partial differential equation into two units nary differential equations {Equations 2.4 and 2.5}. The first of these [Equation 2.4: it? HS)" to solve [jue'l multiply through by tit and integrate}: the general solittittti is C expt-t' Erin}. but we might as well absorb the constant E into it [since the quantity of interest is the product drip}. Then —r.ElI."h left] = e [lo] The HEt-‘Ond [Equation 2.5.1 ts called the timevindEpendent Schrodinger equation; We can go no further With it until the potential Hit} is specified. The real or lhifi Chapter will he devoted to solving the tithe—independent Schrodinger actuation. for a variety of simple potentials. But before i get to that you have everyr right to ask: What's so great about segutt'ttbic striations ." After all. most solutions to the {time dependent} Schrodinger equation {to our take the form tttid'lyttt‘l. I offer three answers—two of them physical. and one mathematical: 1. "ll-icy are stationary states. Although the wave function itself. 4r{_tlr}=1fi['x}E—ffififfll "'1 does {obviously} depend on t. the pmbttbt'it'ry density. salami = ufo = ats+i5‘i”as"'5'i“ = Ian-n3. i2.:.~t| does t‘ltl!--1I'IE lime—dependence cancels out.‘ The same thing happens in calculat- iog ll'lt: expcewlifln value of any dynatttiea] ‘t'flIiliblc: Equation tJo reduces to ill of too. to} = f we (r. 2'") it or. 12.91 refit Every caper-rattan: value it murmur in tin-re; we might as wei] drop the factor tutti altogether. and simply use tit in plane of Iit". tindecd. it is common to relet' to tit an :‘the wave function." but this is sloppy language that can be dangerous. and it is Il'l'tpDI‘L'IJll to remember that the one wave function always carries that exponential tttne~dependent I‘actont In particular. tr} is constant. and hence iiiquatton 1.33:: {p} 2 ti. Nothing ever happens in a stationary state. 2. They are states of definite nitnt energy. In classical mechanics. the total enters")r {kinetic plus potential] is called the Hamiltonian: 1 Ht.t.n‘t= g—m+vta‘]. _ [lint THE INFINITE SQUARE WELL _.______'_'—.________u_'__ Suppose if t] 5 .r 5 a. . _ LL H” '- I oo. otherwise [1W] {Figure 2.1}. A particle in this potential is completely free. except at the two ends is = I] and .t = a}. where an infinite I'orce prevents it from escaping A classical model would be a can on a frictionless horizontal air track, with pertEt-tly elastic butnpers— it just keeps bouncing back and forth forever. [This potential in urtiliv cial. of course. but I urge you to treat it with respect. Despite its simplicity—or rather. precisely because of its simplicity—it serves as a wonderfully aCCd‘s‘ni- ble test case for all the Fancy machinery that comes later. We'll refer back to it frequently.) utxi EilGURE 2.1: The infinite square well poten- ” x nal {Equation 2.13}. (Janine the well. arts] = t! (the probability of finding the particle mere is zero}. inside the well. where l’ = D. the timesindiependent Schrodinger equation {Equation 1.5i reads a? iii-a _EE = so, [2.2a] 131' 1 d— E‘i—j = 421a. where t a flaw. [2.21] the general solution is tltfri = A sin it: + B cos let. [2.31 where A and B are arbitrary constants. Typically. these constants are fixed by the boundary conditlons of the problem. What are the appropriate boundary oorn- ditiona for shirt)? Ordinarily. he!!! hit and dihde are continuous. hill Where the potential goes to infinity only the first of these applies. tl'll prove these boundary conditions. and amount [or the exception when V = no. in Seclion 2.5; for new I hope you will trust me.) Continuity of iglrtix} requires that infill = with} = t]. 11.23] so as to join onto the solution outside the well. Whal does this tell us about A and B'.‘ Well. with = Asinfl+ Email: 3. w B = t}. and hence 1M1”) = A stnlta'. [2.24] Then tarts} = A smite. so either at = I] [in which ease we‘re Jet‘t with the triv— iaJ—non-normalizabIe—solutiort ttrtr} = [I], or else sin in = t}. which means that he =t'l. in". i'En'. iii-Ir. [2.15] But I: = [II is no good (again, that would imply ttrtx} = lit}, and the negative solutions give nothing new. since sini—h‘i = —airttft5"] and we can aheor'h the minus sign into A. So the distinct solutions are r323, wima=t.2.3. 12.26] rt i E z " = 1 _ 2.1? L 1 2m Euro- j I I In radical contrast to the classical case. a quantum particle in the infinite snuaie well cannot have just any old energy—it has to be one of these special allowed values.“ To find A. we norm-ooze tir: ||t_I “ 1 . 1: sir 1t 1 IAI' stn“{k.t‘irt‘.t‘ = |A|‘— = 1. so |A|‘ = . u 2 “.1 E_,__ _. 1L|"tiJ|'J' Val-Tl titan} = find {I 1.1 THE STATISTICEL INTERPRETATION list slat crane it the "one henna " and can that it in or rate one 1»: get it'.’ After at]. a particle. by its nature. is localized at a point. whereas the wave function {as its name suggests} is spread out in space tit's a function of .t. for any given time ti. How can such an object represent the state of a pru'ra'ie T The answer is provided by Bern's stat'fifical interpretation of the wave function. which says that tutti. r}|j gives the probability offintling the particle at point .t. at time .r—or. more precisely" F‘h—_ | t ” probability ol‘ finding the particle ] I'll-I'[.'t.F]II:t.l.t = {in i (I between it and h. at time t. Probability is the iii-en under the graph of |.l'Ll |l. For the wave function in Figure Ll. you would be quite likely to find the particle in the vicinity of point at. where |ttt|- is large and relatively unlikely to find it near point 3. 1M“ FIGURE 1.2: A typical wave function. The shaded area I'CPIEII“! the probability elf finding the particle between a and it. Tltc particle would be relatively lilter to be found near A. and unlikely to be found near B. The statistical interpretation introduces a kind of indeterminacy into quan- tum mechanics. for even if you know everything the theory has to tell you about the particle {to wit: its wave functionl. still you. cannot predict with certainty the outcome of a staple experiment to meat-tun: its position—alt quantum mechan- ics has to offer is .rl'ctt'itt'it'af information about the prifl'ib‘fr’ results. This inde- terminacy has been profoundly d'tsmrhirtg to physicists and philosophers alike. and it is natural to wonder whether it is a fact of nature. or a defect in the theory. Suppose i do measure the positimi of the particle. and I rind it to be at point C4 Question: Where was the paniele just define I made the measurement? There are three plausible answers to this question. and they serve to characterize the ntain schools of thought regarding quantum indeterminacy: I. The realist position: Titrput'rt'r-le was E" C‘. This certainly seems like a sen- sible response. and it is the one Einstein advocated. Note. however. that it‘ this is true then quantum mechanics is an int-implied theory. since the particle rerriij.‘ tent at f‘. and yet quantum mechanics was unable to tell us to. To the realist. indeter- minacy is note feet of nature. but a reflection of our ignomnce. its tl‘Eepegnet put it. "the position of the particle was never indeterminate. but was trierer unknown to the eitpet'irntmter."5 Evidently '41 is not the whole story—some additional infor- mation [known as a hidden variable} is needed to provide a complete description of the particle. 2. The orthodo't position: The pnnicfe wasn't really nnytt'hrte. [I was the act of measurement that forced the particle to "take a stand" {though how and why it decided on the point C we dare not ask]. Jordan said it most starkly: ‘flhservations not only efiti'ttl'b what is in be measured. they pmduce it We {'tNTJp-P-l {the 401 Wane. no measuring instrument on pert'eelly precise: what i mom l:- thtlt the particle was found in the intuit}. oi C. to within. the tolerance of the equipment. We return now to the statisticat interpretation of the wave function [Equation 1.3}. which says that I'-I-"l.'l.'. l'll2 is the probability density for finding the particle at point x. at time L It follows {Equation 1.16} dial the integral of i'tl'i‘E must he I (the particle‘s got to he .t‘emtewherej: | W . If tet.r.:i|-tt.t=t. [too] it: ...
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