11. EQUATIONS OF MOTION

Transcription

1 QM P453 F95 (Zon) Equations of Motion page EQUATIONS OF MOTION In this section we develop a ationale fo the Schödinge equation, the nonelativistic equation of motion fo the pobability amplitude of electons and othe paticles with finite est mass. We begin by showing how some familia, classical equations of motion can be obtained by a method of inductive easoning (in contast to the moe familia deivations that employ deductive easoning). We then conside equations of motion that goven continuous media, choosing fo ou examples (1) the stetched sting, and () the electomagnetic field. Fo both of these we show how the equation of motion can be obtained eithe by deduction based on fist pinciples and fundamental laws, o by induction in easoning that the obseved function is natue's solutions to the equation of motion. Then, by asking how that solution changes in time and in place, tease out an equation that would poduce that sot of solution. Having shown how the inductive method gives esults in which one can have confidence, we then apply it to the poblem of finding an equation of motion fo the pobability amplitude Ψ(x,t) that pedicts the behavio of electons. To do this we use expeiments that show the empiical fom of Ψ(x,t), and we examine the elationship between the tempoal and spatial behavio of that empiical fom. The esult will be the Schödinge equation, and we appoach it inductively because thee is no diect and simple deductive appoach. GRAND GENERAL PRINCIPLES (e.g. spatial and tempoal symmeties and consevation laws DEDUCTIVE REASONING Geneal Pinciples e.g. enegy consevation Subsidiay Pinciples & Laws Hooke's Law, invese squae laws fo gavity & electostatics EQUATION OF MOTION SPECIFIC SOLUTIONS as pedictions that must be tested against new expeiments detailed examination of specific solution including study of tempoal, spatial behavio and study of the effect of changing paametes FUNDAMENTAL EXPERIMENT that clealy demonstates a SPECIFIC SOLUTION that canbe used fo the stating point of inductive easoning INDUCTIVE REASONING Fig File 811 Eqs Motion Septembe, 001

2 QM P453 F95 (Zon) Equations of Motion page Equations Of Motion Fom Inductive & Deductive Reasoning SATELLITE IN ORBIT An equation of motion enables us to pedict the behavio of a system at time t if we know its condition at t=0. Conside the motion of a satellite (m) that obits a heavy, spheically-symmetic mass (M).. The satellite motion is govened by the geneal F = ma law of Newtonian mechanics, and in this case the foce is fom an invese squae law gavitational inteaction, so we can quickly obtain an equation of motion: F = m d M dt F = G M m ˆ geneal law specific situation (gavity) m d { dt = G M (11.1) ˆ 13 tempoal behavio spatial behavio PLANET'S EQUATION of MOTION MASS ON A SPRING As anothe example, conside a simple hamonic oscillato compised of mass m confined to one dimensional motion and subject to a sping's linea estoing foce. As befoe, we elate time dependence to space dependence to get the equation of motion: Again the geneal law is F= ma, and in this case the foce of the sping is fom Hooke's law F = -kx k m F x = m d x dt F x = kx specificsituation (sping) geneal law d x { dt tempoal behavio = k m x 1 3 spatial behavio OSCILLATOR'S EQUATION OF MOTION (11.) In both of these cases we have used a deductive, Aistotelian easoning [~ 350 BC] in poceeding fom the geneal to the specific: The pinciple of Newtonian mechanics F = ma togethe with a geneal law of foce (invese squae law of gavitation, o Hooke's law of spings) leads to a deduction of the equation of motion fo the situation at hand... this is what we usually call a deivation fom fist pinciples. We then use solutions of this deived equation of motion fo ou desciptions of expeiments. File 811 Eqs Motion Septembe, 001

3 QM P453 F95 (Zon) Equations of Motion page 11.3 Altenatively, we could have used inductive easoning [F. Bacon ~ 160], a easoning which poceeds fom paticula examples to the geneal pinciple. Even if we do not know the fundamental laws that goven the motions of a paticula system, we can still egad that system as displaying the solutions to its own equations of motion (as yet unknown by us). By obseving the displayed behavio of the system unde vaious conditions, we can attempt to discen the equation of motion and the undelying genealities. We begin by obseving the system: We wite empiical functions F(x,t) to epesent ou obsevations of the displayed solutions. We then find how the time dependence of F(x,t) elates to its dependence on its spatial coodinates and so obtain, by induction, an equation of motion. By taking data with diffeent values of the system paametes (e.g. mass, chage, sping constant...) we can attempt to find how those paametes elate to the empiical constants. Fo example, obsevation of the mass undegoing simple, one-dimensional hamonic motion is descibed by the function ( ) = A cos( ωt) 1 x t (11.3) paametes A and ω ae chosen to match obsevations which we egad simply as an empiical desciption with paametes A and ω coming fom measuements of the amplitude and the fequency. We now take time deivatives of the obseved fom of x(t) : ( ) = A cos( ωt) 1 x t take two time values of A and ω fom expeiment deivatives inductivelydetemined equation of motion fo the hamonic oscillato d x { dt = 1 ω 3 x tempoal behavio spatial behavio (11.4) So we have an equation of motion because it elates the acceleation, a tempoal behavio, to the value of the coodinate x. Fom expeiments with changes in mass and sping constant we discove the elationship ω = k m between the constant k (usually measued unde static conditions) and the fequency of oscillation (measued on the moving system) d x dt = k m x (11.5) So we have found an equation of motion fo the simple hamonic oscillato without explicit efeence eithe to Newton's Foce Law o to Hook's Law of spings. File 811 Eqs Motion Septembe, 001

4 QM P453 F95 (Zon) Equations of Motion page 11.4 The discussion has been on the motion of a classical mass with a definite location. But ou concept of being govened by an equation of motion can be used fo moe than just the coodinates of a mass: We can also use it fo the desciption of continuous media such as the displacement of a sting, the density of gas in a box, o the intensity of an electomagnetic field. EQUATIONS OF MOTION FOR CONTINUOUS MEDIA Suppose that a system is descibed by a function F(x,t). An equation of motion coelates the tempoal and spatial behavios of that function: If we know the value of F(x,t) fo a ange of times aound t, then the equation of motion enables us to calculate values of F(x,t ) fo all x in the spatial neighbohood of x. Convesely, If we know F(x,t ) fo a ange of space aound x, then the equation of motion enables us to calculate values of the function F(x,t) fo all t in the tempoal neighbohood of t. In the immediately following paagaphs and figues we compae the deductive and inductive appoaches to classical continuous media. Using an inductive appoach quite simila to what we have just used fo the hamonic oscillato, we find equations of motion (1) fo a stetched sting without efeence to Newton's laws and () fo the electomagnetic field without efeence to the Maxwell equations. We give these examples of easoning since we need the inductive appoach to find the Schödinge equation, an equation of motion fo the pobability amplitude (a continuous function) that descibes the behavio of paticles. We need the inductive appoach to the Schödinge equation because thee is no diect and simple deductive ationale fo this fundamental expession of quantum mechanics. File 811 Eqs Motion Septembe, 001

5 QM P453 F95 (Zon) Equations of Motion page Continuous Medium: The Vibating Sting The deductive appoach to the equation of motion fo a vibating sting begins by applying F = ma to an element (length x, mass m) of a sting that is constained to move only to a limited extent in the y-diection. m= x T T x Tension povides the y-diected foce Y F = T dx x Fig The sting is assumed to be ideally flexible and the tension is of constant magnitude. 6 mass 78 F = ρ x d Y dt F = T d Y dx x geneal law F=ma specific situation fom 4 geomety 444 of 4 stetched 444 sting 3 d Y { dt tempoal behavio A solution of the equation of motion is = ρ d Y T 14 dx 43 spatial behavio STRING'S EQUATION of MOTION (11.6) Y(x,t) = A(x) cos T ρ t (11.7) whee A(x) is of the fom sin(nπx/l) fo a sting constained at x=0 and x=l. The equation of motion follows; it shows how the paametes of tension T and mass pe unit length influence the velocity of popagation and thus the fequency of the sting oscillation. File 811 Eqs Motion Septembe, 001

6 QM P453 F95 (Zon) Equations of Motion page 11.6 To use the inductive appoach, the obseve sets up the stetched sting, measues the displacement Y(x,t 1 ) at some instant of time t 1 fom a photogaph, measues the time vaiation of displacement Y(x 1,t) at a fixed location x 1, and then wites an empiical functions Y(x,t,) that descibe the esults. The obseve then asks "now what sot of equation of motion would have that function as a solution?" take snapshot at time t 1 measue wavelength k=π/λ Y(x,t 1 ) = A sin( kx) empiical solution measue fequency ω at fixed location x Y(x 1,t) = A cos( ωt) Y(x,t) = A sin( kx) cos( ω t ) compae the time & space deivatives of the empiical solution to get STRING'S EQUATION OF MOTION d Y { dt tempoal behavio = ω d Y k 14 4 dx 3 spatial behavio (11.8) It is then up to the expeimente in futhe eseach to find how the values of k and w depend (o do not depend) on paametes such as tension, mass pe unit length, elastic modulus, sting diamete, and the like. Eventually one can discove the identification of (k/w) with /T on expeimental gounds alone, and this would be quite satisfying when one finally compaes notes with the theoist who obtained the equation of motion in a puely deductive manne. File 811 Eqs Motion Septembe, 001

7 QM P453 F95 (Zon) Equations of Motion page 11.7 EQUATION OF MOTION FOR A VIBRATING STRING Stingeason1 #6 x1 x Inductive Method Fom expeimental obsevations on stings we find that the simplest standing waves ae descibed by a function of the fom: : y(x,t) = A sin(kx) sin(ωt), Deductive Method Fom the geomety of the situation we see that the net y-diected foce is elated to the tension (T) and the cuvatue of the sting: : F y =T dy T dy = T d y x dx dx dx x But by Newton' s law, F y = ma y,= m d y so the dt mass element m moves accoding to: F y = m d y dt = m a y so we can wite: d y dx = ( ) d y m x T o if we define linea density as dt we get the equation fo waves on a sting: x1 d y dx = ρ T d y dt ρ ( m x) and we want to find the equation of motion that govens the y-behavio. The spatial behavio of y is: so we find dy = ka cos(kx) sin(ωt); dx d y dx = k A sin(kx) sin(ωt) y= 1 d y k dx On the othe hand, the tempoal behavio of y is: so we find dy dt = ωa sin(kx) cos(ωt);, d y dt = ω A sin(kx) sin(ωt) y = 1 ω d y dt The wave : equation that we obtain is.. If this is to wok, then ρ T =k, and this can ω can be tested by seeing how the mass/length and the tension affect the feqency and wavelength of the sting oscillation. d y dx = k ω d y dt Whee k π and ω π ν λ ae constants fom the expeimentally obseved wavelength and fequency of the sting Fig Compaison of Inductive, Deductive methods fo stetched sting equation File 811 Eqs Motion Septembe, 001

8 QM P453 F95 (Zon) Equations of Motion page Continuous Medium: The Electomagnetic Field The deductive appoach to the equation of motion fo a popagating electic field in vacuum begins fom the Maxwell equations as they ae witten fo a chage-fee vacuum: I) E = III) Coulomb' s Law II) E = B t Faaday induction B 14 4 = 30 IV) B E = µ 0 ε t 3 no magnetic poles Ampee's Law (11.9) Fom these fundamental elationships we can find the elationship between the spatial and tempoal vaiations of E. Take cul of (II) and use (IV) fo cul B: ( E ) = t ( ) = t B Now use vecto identity E = E E E µ 0 ε 0 = µ 0 ε 0 t ( ) E E E =0 in vacuum t (11.10) The equation of motion fo E (the wave equation fo electomagnetic adiation in fee space) is then: E = µ 0 ε 0 E t (11.11) This shows how the paametes of electic and magnetic susceptibility contol the speed of light. It is woth ecalling that the Maxwell equations and the values of µ o and ε o come fom expeiments done unde nealy static (ν < 10 HZ) conditions, yet they accuately pedict the behavio of high fequency (ν» 10 1 Hz) phenomena. File 811 Eqs Motion Septembe, 001

9 QM P453 F95 (Zon) Equations of Motion page 11.9 To use the inductive appoach, one begins by doing expeiments that show the adiation can be descibed by an empiical function of the fom E(x,t) = E 0 sin(kx-ωt). (This function has a zeo (at kx=ωt) that moves with a (phase) velocity v = dx/dt = ω/k.) The paametes k and ω ae detemined by measuements of the wavelength and the speed of popagation of the adiation. One then exploes the elationship of the spatial and the tempoal behavios: spatial behavio of E E x,t 6 E 44 x = k E cos( kx ωt) ( ) = E x = k E 0 sin kx ωt ( ) ( ) E 0 sin kx ωt E x = k ω E t o, if we ae woking in thee dimensions E = k ω tempoal behavio of E 6 E t E t E t = ω E 0 cos( kx ωt) = ω E 0 sin kx ωt ( ) (11.1) It is then up to the expeimente in futhe eseach to find how the values of k and w depend (o do not depend) on paametes such as dielectic constant, adiation intensity, tempeatue, and the like. Eventually one can discove the identification of (k/ω) with µ o ε o on expeimental gounds alone, and this would be quite satisfying when one finally compaes notes with the theoist who obtained the equation of motion in a puely deductive manne. The electomagnetic field, as govened by the wave equation, is paticulaly impotant because it can be egaded as the pobability amplitude fo pedicting the behavio of photons that ae the obsevable manifestation of the adiation field. Indeed this intepetation is essential fo explaining adiation phenomena in the limit of weak intensities. File 811 Eqs Motion Septembe, 001

10 QM P453 F95 (Zon) Equations of Motion page THE SCHRÖDINGER EQUATION OF MOTION Nicht wie es wiklich wa, sonden wie es hätte sein sollen. The inductive easoning towad the Schödinge equation given hee does not follow the histoic development Inductive Appoach To A Quantum Equation Of Motion The inductive appoach to the Schödinge equation follows fom the need to descibe intefeence effects seen with individual paticles. We have discussed this at some length in Sec 3, but the desciption thee is in tems of elatively abstact pobability amplitudes Ψ. The abstact appoach has many advantages so we will etun to it late, but we now need moe specific functions fo Ψ Diffaction expeiments on electons and neutons suggest that the pobability amplitude fo a paticle moving in a egion of unifom potential enegy will have the fom of a popagating wave: Ψ( x,t) = A e i( kx ω t) (11.13) The paametes of fequency (ω) and wave numbe (k =π/λ) ae to be detemined by expeiment.. The constant A is chosen so that the we have unity pobability of finding the paticle within the appaatus. Planck Fequency The photoelectic effect shows the fequency of a photon to be associated with its enegy. We claim that this association should also hold fo a paticle: ω = E h (11.14) whee E is the total (kinetic plus potential) enegy of the paticle. deboglie Wavelength If a beam of paticles having momentum p ae incident on an odeed aay of atoms (e.g. a cystal), then the distibution of the scatteed paticles is pedicted by using a pobability amplitude having deboglie wavelength) λ: λ = h 1 3 p DeBoglie wavelength k π λ = p h wave numbe (scala) hk = p 1 3 (11.15) popagation vecto The diffaction of a paticle beam was fist seen in the electon scatteing expeiments done (196-7) by Davisson and Geme in the US and by G.P. Thompson in England. Beyond poviding an expeimental ancho fo quantum mechanics, paticle diffaction is an impotant contempoay tool fo the study of mateials. Low Enegy Electon Diffaction (LEED) and High Enegy Electon Diffaction (HEED) ae used fo the examination of sufaces. Complementay to this is neuton diffaction which gives infomation on the inteio stuctue since neutons, being electically neutal, can penetate solids easily. File 811 Eqs Motion Septembe, 001

11 QM P453 F95 (Zon) Equations of Motion page Obtaining the Equation of Motion We exploe the elationship of Ψ 's cuvatue (d /dx ) to its ate of change (d/dt) and find that this lead to Schödinge's equation of motion. We begin with the function that epesents the paticle in fee space Ψ( x,t) = A e i( kx ω t) (11.16) This function changes at a ate that depends on its total enegy (E= E kin + E pot = hω) Ψ t = i E Ψ { h ω Ψ t = i p h m + V Ψ E(total) (11.17) By taking spatial deivatives we find an expession fo the kinetic enegy p /m: Ψ x = k Ψ cuvatue of Ψ popotional to squae of momentum p=hk Ψ x = p h Ψ h m Ψ x = p m Ψ the opeato fom fo kinetic enegy needed fo use in the time equation (11.17) (11.18) Putting this diffeential opeato expession fo kinetic enegy (11.6), we find an equation that elates Ψ's tempoal and spatial behavios: Ψ t = i h h m Ψ 14 x 43 + V p m Ψ eaange ih Ψ t = h m x + V Ψ (11.19) Fom this we get Schödinge's wave equation that govens the pobability amplitude Ψ(x,t):.ih t Ψ ( x,t h )= m x + V( x,t) Ψ( x,t) Hamiltonian opeato H o (11.0a,b) ih t Ψ( x,t) = Η ( x,t )Ψ ( x,t ) The Schödinge equation File 811 Eqs Motion Septembe, 001

12 QM P453 F95 (Zon) Equations of Motion page Sepaation Of Time Dependence Fom Space Dependence The foces that bind electons to atoms, o that bind atoms into molecules, do not depend explicitly on time, so the potential descibing such foces can be witten as a function of position only: V = V(x). With the potential independent of time, the Hamiltonian opeato depends only on coodinates and we can make pogess by witing the state function with vaiables sepaated: Ψ( x,t) = ψ(x) F(t) (11.1) Using this in the Schödinge equation gives: ih h ψ(x) F(t)= t m x + V x ( ) ψ(x) F(t) (11.) We now gathe tems so that one side depends only on coodinates, the othe on time: ih F ( t ) t 14 F(t) 4 3 depends only on time = Hψ(x) 1 ψ(x) 3 depends only on position = C{ (11.3) sepaation constant Fom the time-dependent elationship we find that the sepaation constant can be identified as the total enegy: ih F( t) t = C F(t) F( t) = e i C h t C = E (11.4) Having this identification, we then wite out the position-dependent pat of Eq. (11.11) to obtain the Schödinge equation fo the time-independent pat of the poblem:: h m x + V( x) ψ n (x) = E n ψ n (x). (11.5) The Time Independent Schödinge equation The subscipt n indicates that thee is an entie set of functions ψ n (x), each associated with an enegy E n, that solve the time-independent Schödinge equation Hψ n =E n ψ n. We see that these solutions ψ n ae functions that suvive the mastication of the opeato H in almost unalteed fom... they ae just multiplied by a constant E n even though the opeato involves both diffeentiation and multiplication by V(x). The ψ n (x) ae called eigenfunctions of the opeato H; The E n ae the eigenvalues associated with those eigenfunctions File 811 Eqs Motion Septembe, 001

13 QM P453 F95 (Zon) Equations of Motion page Some Popeties Of State Functions We will devote much effot to developing solutions fo the time-independent Schödinge equation because its enegy eigenvalues E elate diectly to what we can measue most accuately, the fequencies of spectal lines. A NORMALIZED SUM OF EIGENFUNCTIONS IS ALSO A SOLUTION The Schödinge equation is linea, so a sum of solutions is also a solution. Ψ(x,t) = Thee mattes a n Ψ n ( x,t) n a n ψ n ( x) e i E n / h n ( 4 3 )t (11.6) spatial eigenfunction phase factos of unit amplitude 1) The coefficients a n may be dependent on time (as we shall see late), but fo now we take them as constants detemined by the initial conditions of the situation. If we have a constant oveall pobability of finding the paticle somewhee in the system, then: NORMALIZATION Ψ(x,t) dx = integal ove entie volume of system a n ψ ( n x )e iω t n dx = 1 (11.7) fo supeposition states n ) It is of the utmost impotance to note that each of the spatial eigenfunctions ψ n (x) has its own time-dependent phase facto, and this will be of paticula elevance when making physical intepetations fom expessions such as (11.14) Fo example, if the wavefunction fo a given situation is fomed by the coheent addition of two o moe eigenfunctions we will find that the expectation values have an explicit time dependence. Indeed this enables us to descibe how atoms adiate. 3) The function epesenting a system, whethe an eigenfunction Ψ(x,t) o a weighted sum of such eigenfunctions = Σ a n Ψ ν (x,t) should be chosen such that the spatial integal of the pobability density is unity when the integal is ove the entie ange of the system. When the limits of integation ae finite, nomalization is usually just a matte of choosing coefficients in a suitable way. In this text we show this explicitly fo the squae well, the hamonic oscillato, the igid oto, and the hydogen atom. Nomalization is somewhat tickie fo fee paticle wave functions as encounteed, fo example, in plane wave epesentations and in scatteing poblems because the associated functions do not vanish as one goes to infinity. One usually does something like nomalizing not to unity oveall but to one paticle pe unit volume. We will not woy too much about this. File 811 Eqs Motion Septembe, 001

14 QM P453 F95 (Zon) Equations of Motion page (x,t) MUST BE REPRESENTED BY COMPLEX FUNCTIONS The obvious question about ou choice of state function is "why an exponential? Why not just a simple sine o cosine?" The unsuitability of the simple tigonometic fom is seen by consideing the standing wave that is to epesent a paticle confined between two boundaies: The tigonometic fom G( x,t) =A sin( kx) sin( ωt) (11.8) will be zeo fo all x wheneve t = π, π,..., s o oveall pobability is not constant in time, in contadiction to ou hypothesis that the paticle is always thee. Howeve the standing wave using a complex exponential fo time dependence Ψ(x,t)= A sin( kx) e iωt (11.9) has a modulus that is constant in time and thus conseves oveall pobability. CONTINUITY OF THE STATE FUNCTION AND ITS DERIVATIVE The squae of the state function Ψ(x,t) epesents pobability density, so a discontinuity in the function Ψ(x,t) would imply an abupt step in pobability. Since this seems unphysical, we ague that Ψ(x,t) should be continuous as a function of x fo all x. The cuvatue of the state function is an opeato measue of the paticle's kinetic enegy, [see Eq (11.6)], so a discontinuites the fist deivative ( Ψ/ x) would lead to infinities in that enegy. On this basis we ague that Ψ/ x must be continuous eveywhee. Cuves should join so that pobability is continuous acoss bounday (x) Slopes should match The cuvatue of the wave function is negative in the allowed egion but positive in the fobidden egion. Classically allowed egion (E (total) >V) Classically fobidden egion (E (total) <V) V(x) x Taken togethe, these d ψ imply that the cuvatue is zeo at the bounday; dx i.e. that E(kinetic) vanishes at the bounday. Fig Bounday contdition between allowed and classically fobidden egions File 811 Eqs Motion Septembe, 001