1 dx. where is a large constant, i.e., 1, (7.6) and Px is of the order of unity. Indeed, if px is given by (7.5), the inequality (7.

Transcription

1 Lectures Nine an Ten The WKB Approximation The WKB metho is a powerful tool to obtain solutions for many physical problems It is generally applicable to problems of wave propagation in which the frequency of the wave is very high or, equivalently, the wavelength of the wave is very short The WKB solutions are approximate solutions, but sometimes they are surprisingly accurate In this chapter we ll iscuss this metho, which is applicable to linear equations only A The Zeroth an the First-Orer WKB We consier the secon-orer ifferential equation y p 2 y 0 (7) If p is a constant, the two inepenent solutions of (7) are expipx, waves with wave number p travelling in opposite irections of the x-axis If p is a function of x, it appears reasonable that the solutions are two waves with the phase pxx Thus we may surmise that the inepenent solutions of (7) are e ipxx, which are the zeroth-orer WKB solutions Let us see if these solutions satisfy (7) We have (72) x 2 p2 e ipxx ip e ipxx 073 Therefore, the WKB solutions (72) o not satisfy (7) unless p 0, or p is inepenent of x While we get a negative answer, (73) suggests that expipxx are goo approximate solutions of (7) provie that ip is negligible, or, more precisely, if p p 2, the left-sie of this inequality being a term insie the parenthesis of (73) The inequality above can be written as x p (74) This conition is satisfie if px is of the form px Px, (75) where is a large constant, ie,, (76) an Px is of the orer of unity Inee, if px is given by (75), the inequality (74) is x (77) Px Clearly, (77) is satisfie if, provie that x is not near a zero of Px We note that the phases of the solutions expipxx are functions of x, but the magnitues of these approximate solutions are inepenent of x Let us remember that, in Chapter, we have shown that if y an y 2 are two inepenent solutions of (7), then the Wronskian Wx y y 2 y y 2 is inepenent of x Now the Wronskian of expipxx an expipxx is easily shown to be equal to 2ipx, not a constant unless px is a constant This suggests that these approximate solutions still leave something to be esire Because the Wronskian of the approximate solutions miss by a factor px, let us try to fix it by aing an aitional factor / px to each of the approximate solutions The resulting approximate solutions are y WKB x px eipxx (78) which inclues both the zeroth orer an the first orer terms of the WKB approximations

2 The magnitue of these solutions varies with x like / px The Wronskian of y WKB x is now exactly a constant (See homework problem ) It is therefore tempting to surmise that, uner the conition (74) or equivalently, (77), y WKB x are even better approximations than e ipxx To see if this is true, we put y e ipxx v, an substitute this expression of y into (7) We get DipDipv p 2 v 0 or 2 x 2ip x ip v 0 We shall write the equation above as v p 2p v 2p i v (70) 79 where By (75), (70) can be written as v P 2P v i 2P v, / is a small number In the first-orer approximation, we ignore the right sie of (7) an we get which gives v P 2P v 0, vx Px (7) (72) (73) Thus (79) an (73) give, asie from an immaterial overall constant, the WKB solutions (78) We have mentione that if Px has a zero at x 0, the inequality (70) oes not hol at x 0 To see how far away from x 0 it must be for the WKB approximation to hol, let Px near x 0 be approximately given by Then (77) requires Px ax x 0 n,x x 0 /n (74) x x 0 a n (75) Eq (75) tells us how far away from x 0 it must be for the WKB approximate solutions to be vali If Px vanishes in the way given by (74), we say that Px has an n th -orer zero at x 0 Not all zeroes of Px are of finite orer, an an example of Px having a zero of infinite orer is given by homework problem 6 The range of valiity of the WKB solutions of homework problem 6 is given in homework problem 5b As we hve mentione, the WKB approximation is useful for escribing the propagation of waves with very high frequencies or, equivalently, very small wavelengths As an example, consier the problem of etermining the shaow cast on a wall by a point light source in front of a screen To obtain the exact solution of this problem, one solves the wave equation an make the solution satisfy the bounary conitions impose by the presence of the screen, which is a ifficult bounary-value problem On the other han, the shaow on the wall is very accurately etermine simply by rawing straight lines from the light source to the eges of the screen This is because when the wavelength of light is very small compare to the imensions of the screen, the WKB approximation can be use to justify the results obtaine with the use of geometric optics footnote As another example, we know that Newtonian mechanics is an approximation of quantum mechanics However, the behavior of atoms obeying the rules of wave mechanics is rastically ifferent from that of particles obeying the rules of Newtonian mechanics How oes one reconcile these two sets of rules? The answer again lies in the WKB approximation, in which the Schroinger equation is reuce to the Hamilton-Jacobi equation satisfie by the classical action of Newtonian mechanics The WKB approximation can also be use to solve problems in which the functional behavior is 2

3 rapily growing or rapily ecaying other than rapily oscillatory, an example being the problems of bounary layer which we will iscuss in Chapter 9 Consier the equation The WKB solutions are given by y 2 y 0 y WKB x x exx These solutions are goo approximations of the solutions of (76) if The counterpart of (75) is x x (76) (77) (78) x Nx, (79) where an Nx is of orer unity As before, ifis in the form (79), the inequality (78) is always satisfie unless x is near a zero of Nx The WKB approximation can be justifie uner other conitions For example, it is easy to verify that (74) is satisfie if px is of the form px Px,, (720) where we have a small parameter rather than a large parameter If we make the change of variable X x, then (7) becomes y X 2 2 P 2 Xy 0, where the large parameter appears Sometimes the large parameter is not explicitly exhibite As an example, consier the problem of solving the Airy equation y xy 0 when x is very large In this problem, x inherently contains a large parameter Inee, let x be of the orer of, with We may put x X, where X is of the orer of unity Then the Airy equation is X 2 3 X y 0 Comparing with (76), we have, if X is positive, X X /2, where 3/2 is the large paramater We also note that the integral px is imensionless an hence oes not change with a change of the scale of the inepenent variable In the example of the Airy equation, we have X /2 X x /2 x Inee, since the imension of the wavelength is that of length, same as that of x, the leftsie of (74) is imensionless Thus (74) can be expresse either with the variable X or with the variable x, as X X /2 x x /2 B Solutions Near an Irregular Singular Point We will now apply the WKB metho to obtain the asymptotic solutions near an irregular singular point of a secon-orer linear homogeneous equation While we have alreay given a metho in the preceing section to obtain these solutions, it applies only when the rank of the singular point is an integer In aition, the use of the WKB metho makes it very easy to obtain the leaing terms of the asymptotic series To start, we consier the leaing asymptotic terms for the solutions of the equation y xy 0 3 (72)

4 For x 0, we have, comparing with (76), Thus the WKB solutions are x /2 x /4 e 2 x 3/2 /3 (722) We conclue immeiately that, when x is large an negative, one of these solutions is an exponentially increasing function of x an the other is an exponentially ecreasing function of x When x is positive, we have, comparing with (7), Thus the WKB solutions are both being oscillatory functions of x C p x /2 x /4 e i2x3/2 /3, Higher-orer WKB Approximation We shall in this section fin the higher-orer terms of the WKB approximation For this purpose let us return to eq (7) Since this equation has a small parameter an is (723) linear, it is straightforwar to use it to erive successive corrections to the WKB approximations We put v v 0 v 2 v 2, (738) where v n, n 0,, are inepenent of The series in (738) is calle a perturbation series which is expecte to be useful whenis small We substitute (738) into (7) an get v 0 v 2 v 2 2P P v 0 v 2 v 2 2P i v 0 v 2 v 2 In the lowest-orer approximation we set in (739) to zero an get This equation gives v 0 P 2P v 0 0 v 0 x Px, which is, asie from an immaterial constant multiple, (73) Setting to zero the sum of terms in (739) which are proportional to, we get v P x 2Px v i 2Px Solving this first-orer linear equation, we fin that v x i 2 Px Pt Px Pt (739) (740) t # Now we are reay to give a justification of the WKB metho, which is approximating the solution of (7) by truncating the series of (738) Strictly speaking, truncating a series is justifie if we succee in proving that the sum of terms neglecte is much less than the sum of terms kept But proving this is sometimes ifficult to o We shall be content with proving that the n th term in the series is much less than the n th term ifis sufficiently small Thus we will accept the WKB solutions (75) are goo first-orer approximations if v v 0 (742) Since is small, (742) is satisfie provie that v x/v 0 x oes not blow up, which is true unless Px happens to vanish If Px vanishes at x 0, the ifferential equation (7) is sai to have a turning point at x 0 At a turning point of the ifferential equation, both v 0 x an v x blow up, an the WKB approximation fails How far away from the turning point it must be in orer for the WKB approximation to work? If, 4

5 when x is near x 0, Px goes to zero like x x 0 n, then v x blows up like xx 0 3n/2, (743) while v 0 x blows up like x x 0 n/2 Thus (742) requires x x 0 (744) /n Asie from a multiplicative constant, (744) is the same conition as (75) We may fin all higher-orer terms of the solution from (739) This is one by gathering all the terms in (739) proportional to m an setting the sum to zero We get v m 2P P v m 2P i v m (745) Thus v m x i x 2 Px Px x 2 v mx (746) From (746), we obtain the m th -orer term of the perturbation series once the m th -orer term of the perturbation series been foun Thus we obtain all v m by succesive iteration If Px has no zero, all v m are finite Whenis sufficiently small, we have v m v m (747) Thus the WKB approximation is justifie to higher orers Here we like to give the reaer a reminer: the WKB approximation has been justifie to higher orers only if px is of the form (75), or x is of the form of (79), an neither of them vanish I feel obligate to say it as I have seen the WKB approximations being too liberally applie We may show that, if Px vanishes at x 0, an is given by (74) near x 0, the conition (747) is satisfie provie that x is sufficiently far away from x 0 so that (75) is satisfie (See homework problem 3) Similarly, to obtain successive approximations to the WKB solutions of (76), we put y e xx v Then we have v 2 v 2 v We may use (749) to obtain successive approximations of the WKB solutions (748) (749) Homework ue next Monay (Oct 8, 04) : Chapter 7, 4a, 5b, 7b 5