SUPPLEMENTARY NOTES ON THE CONNECTION FORMULAE FOR THE SEMICLASSICAL APPROXIMATION


 Thomasine Casey
 6 months ago
 Views:
Transcription
1 Physics 8.06 Apr, 2008 SUPPLEMENTARY NOTES ON THE CONNECTION FORMULAE FOR THE SEMICLASSICAL APPROXIMATION c R. L. Jffe 2002 The WKB connection formuls llow one to continue semiclssicl solutions from n llowe to forbien region n vice vers. However these formuls re subtle n must be use with cre. The purpose of these notes is to lin how to use the them n why it is necessry to be creful. Note tht we re not eriving the connection formuls here. The erivtion is either long n ull (e.g., Griffiths or Merzbcher) or short, elegnt n obscure (e.g., Lnu n Lifschitz). In ny cse, the proper use of the connection formuls is quite inepenent of how they re erive. The semiclssicl pproimtion is vli whenever the rte of chnge of the e Broglie wvelength is smll, λ () where λ = /p() n p() = 2m(E V ()). Eq. () cn be stisfie in clssiclly llowe region, where E > V (), n λ is rel, or in clssiclly forbien region where E < V (), n λ is imginry. As erive in clss the wvefunctions in the semiclssicl limit re given by: ψ() = c + p() i p( ) + c ī p( ) p() h (2) in the clssiclly llowe region, n ψ() = + κ( ) + κ( ) κ() κ() (2b) in the forbien region, where κ() = 2m(V () E). Notice tht ll the integrls hve been written s inefinite integrls. This is becuse chnge in the lower limit mounts to chnge in the vlue of the constnts c ± n ±. In specific pplictions the lower limits n the constnts re
2 chosen to suit the problem. To mke use of the semiclssicl metho it is lmost lwys necessry to continue the wvefunction from the llowe region to forbien region, or vice vers. The trouble is tht these regions re seprte by clssicl turning point, 0, where E = V ( 0 ), so p( 0 ) = 0 n λ/. So the semiclssicl pproimtion breks own t clssicl turning point. The question is, then, how oes one continue solution from n llowe region through clssicl turning point into forbien region, or vice vers? It shoul be possible becuse we re tlking bout the solution to secon orer ifferentil eqution (the Schröinger eqution). Once one hs specifie two constnts of integrtion, the solution is completely etermine, so specifying the solution in one region shoul fi it in the forbien region. In fct, this is not quite true, n tht s the subtlety of the Connection Formuls. Wht re the Connection Formuls? First, let s summrize the formuls n their omin of pplicbility. The formuls epen on whether the clssiclly forbien region lies to the left or right of the clssiclly llowe region. To be complete we give the formuls for both cses. Figure shows the sitution: in () the forbien region is on the right; in (b) it is on the left. () V() V() (b) E E = = b Figure 2
3 I. Forbien region to the right. If the wvefunction is known to be onentilly flling in the forbien region, then it s phse n mplitue re known in the llowe region: κ( ) 2 cos p( ) π κ() p() (3) A wvefunction 90 out of phse in the llowe region continues into growing onentil in the forbien region s follows: cos p( ) + π κ( ) p() κ() () II. Forbien region to the left. If the wvefunction is known to be onentilly flling in the forbien region, then it s phse n mplitue re known in the llowe region: κ() b κ( ) 2 cos p() b p( ) π (5) A wvefunction 90 out of phse in the llowe region continues into growing onentil in the forbien region s follows: cos p( ) + π b κ( ) p() b κ() (6) 2 Are the Connection Formuls equlities? (This section is written by Hong Liu, 2007) Cn we use the connection formuls in the irections opposite to the rrows in (3) (6)? The nswer is yes, but etr cre must be pi in reversing the rrows in equtions () n (6). In contrst, the rrows in (3) n (5) cn be reverse reltively strightforwrly. 3
4 To illustrte the subtleties, let us consier specific emple: Suppose the wvefunction in forbien region to the right is given by ψ() + κ() + κ() κ( ) ( + O()) κ( ) ( + O()). (7) Then using () in the irection opposite to the rrow n (3), we obtin the wve function on the left + cos p( ) + π ( + O()) p() + 2 cos ( + O()). (8) p() p( ) π The bove proceure is correct provie in (7) we know the coefficient before the onentilly flling term κ( ) precisely. In most circumstnces, however, the ccurcy of the WKB pproimtion is not enough for us to know ectly. This is ue to tht the secon term in (7) is so smll compre with the first term, tht it is normlly roppe completely. Note tht for ech term in (7), we hve roppe terms of orer O(), s inicte in the eqution. The O() contribution in the first term, which ws lrey roppe, is much lrger thn the onentilly suppresse secon term in the smll limit. Thus it is completely legitimte to rop the secon term in (7). In such sitution we re simply left with ψ() + κ( ) (9) κ() n it is then incorrect to use eqution () bckwrs to conclue tht the wvefunction on the left is of the form + cos p( ) + π. (0) p() In compring (0) with (8), we see tht (0) misses the secon term in (8), which is of the sme orer s the term in (0). Nevertheless, one oes encounter situtions in which in (7) is known precisely, often ue to symmetry. For emple, consier sitution in which
5 = 0 n both ψ() n κ() re even functions of. Then the symmetry requires tht + =. (We will see such n emple in the problem of ouble well potentil in pset 8). Sometimes it is lso possible to use more sophisticte mthemticl methos to keep trck of the coefficient precisely even without symmetry. In these situtions one cn then use the connection formuls in both irections. 3 An Emple A worke emple will help show how the Connection Formuls re to be pplie. Consier prticle trppe between the origin t = 0 n high potentil V(). Figure 2 gives n illustrtion. For energy E, the clssicl turning point is t = (E). is the point where E = V (). The semiclssicl solution which vnishes t the origin is V() E Figure 2 ψ() = sin 0 p( ). () This form is vli for <. Wht o we o s pproches the turning point t =? We rewrite the sin... s the liner combintion of the two cosines for which we hve connection informtion: ψ() = sin 0 = sin cos p( ) p( ) π 5 + cos cos p( ) + π (2)
6 where = p() π 0. (3) This much is just ppliction of trig ientities. If the coefficient of cos p( ) + π is not zero, then the wve function continues into growing onentil for > ccoring to eq. (). The only thing we cn sy for certin is tht ψ() hs n onentilly growing term in the forbien region: ψ() cos κ() κ( ). () ψ() woul in generl lso contin onentilly flling terms in the forbien region, but we on t hve the ccurcy to compute them. Tiny corrections to the onentilly growing term, which we i not keep trck of in our WKB pproimtion will be much lrger thn the onentilly flling term. However, if cos = 0, then there is no onentilly growing term in the forbien region. Thus, if we know tht ψ() flls onentilly in the forbien region cos must be zero. Since boun stte wvefunction must fll onentilly in the forbien region we lern tht the WKB conition for boun stte is cos = 0, or 0 p() = (n + 3 )π (5) which is the BohrSommerfel Quntiztion conition when there is hr wll on one sie. The problem set contins other problems which require creful use of the Connection Formuls. Deriving the Connection Formule This section c Krishn Rjgopl, 200 Let us consier the cse epicte in the left pnel of Fig., with turning point t = with the llowe region t < n the clssiclly forbien region t >. We know tht in the forbien region ψ() = κ( ) for (6) κ() 6
7 n in the llowe region ψ() = c + i p() p( ) + c ī p() h p( ) for. (7) Our tsk is to relte c + n c to, n we ect to fin the reltion escribe by the connection formul (3). I ll tke you through this erivtion, n leve eriving the other three connection formule to you. In the vicinity of the turning point, the potentil is pproimtely liner n we cn write V E b( ) where b V > 0. (8) = This linerize potentil is goo escription for < L, where L is the length scle over which V curves. The semiclssicl forms for the wve function, (6) n (7), re not vli too close to. For emple, (6) is vli where κ() (9) n we must sk whether there re vlues of ( ) tht re simultneously lrge enough tht (9) is vli, but not so lrge tht (8) breks own. If there is rnge of ( ) in which both (9) n (8) re vli, then within this omin (9) becomes 2mb( ) = We cn therefore conclue tht s long s L 8mb ( ). 8mb, (20) there is region where (9) is vli, mening tht (6) escribes the wve function, n where (8) escribes the potentil. The sme conition (20) implies tht there is rnge of, in the llowe region in which is big enough tht (7) is vli escription of the wve function n smll enough tht (8) still escribes the potentil. The conition (20) must be stisfie by the potentil ner its turning points in orer to ensure the vliity of the nlysis we re pursuing (with semiclssicl wve functions 7
8 fr wy from turning points n connection formule prescribing how they re connecte cross the turning points.) One wy of reing (20) is tht it is lwys stisfie in the 0 limit, n inee this is why the metho is clle semiclssicl. Perhps better wy to re the conition, though, is to leve fie it is fter ll constnt of nture n view (20) s the sttement of how smooth the potentil must be ner its turning points. Henceforth, we ssume tht (20) is stisfie. Within the omin of where both (8) n (9) hol, the wve function is given by ψ() = = 2mb( ) 2mb( ) / / 2mb( ) 2 2mb ( ). (2) 3 Now, we wnt to nlyticlly continue this ression from > 0 to < 0. The trick is to consier ( ) comple vrible, which we shll write s ( ) = ρ iφ, n to strt with φ = 0 n ρ in the rnge such tht ll our pproimtions re vli n then to continuously chnge φ from 0 to π, ll the while keeping ρ fie. In this wy, we en t point in the llowe region (with < 0) where both (8) n (9) hol. Lets see wht hppens to the wve function upon performing this proceure. First, we rewrite the wve function s ψ() = 2 2mb ρ 3iφ (22) (2mbρ) / iφ 3 2 which, for φ = 0, is wht we h before. For φ = π, nmely < 0, the wve function becomes ψ() = +i 2 2mb ρ. (23) (2mbρ) / iπ 3 So, this is the wve function in the llowe region tht we obtin by strting from the wve function in the forbien region n nlyticlly continuing. Note tht the turning point is t ρ = 0, n we never went ner it. By turning into comple vrible, we were ble to strt with > 0, en with < 0, n never go ner = 0. We must now compre 8
9 (23) to the form of the wve function we were ecting to get in the llowe region, nmely (7). Using (8) n, note tht we re in the region where this is vli we cn rewrite the wve function in the llowe region (7) s follows: ψ() = = = c + 2mb( ) i / 2mb( ) c + ī / 2mb( ) h c + +i 2 / 2mb ( ) 2mb( ) 3 c + 2mbρ c + / 2mb( ) +i 2 2mb ρ 3 / c + 2mbρ / i 2 3 i 2 2mb ρ 3 2mb( ) 2mb ( ). (2) We now see tht if we choose c + = iπ (25) then the wve function (23) tht we obtine by nlyticlly continuing the forbienregion wve function to the llowe region is the sme s the c + term in (2)! This looks goo, but wht hs hppene to the c term??? Let s try to figure out why we foun the c + term, but not the c term. To o this, we strt with (2), incluing both the c + n c terms, n try to nlyticlly continue it in the opposite irection to wht we i before, bck to the forbien region. We o the nlyticl continution by chnging φ from π to 0. Note tht the imginry prt of ( ) is positive uring the continution. You cn esily check tht if you strt with (2) n perform this proceure, s you begin the continution into the comple plne (i.e. s you strt reucing φ from π) the mgnitue of the c term becomes onentilly smll compre to the mgnitue of the c + term. As we iscusse in lecture, the semiclssicl pproimtion entils ropping such onentilly smll terms. So, if we strt with (2) n continue bckwr, we 9
10 lose the c term n the c + term turns into the correct wve function in the forbien region once φ is bck to 0. Anlogously, when we strte with the forbienregion wve function n continue forwr, we only obtine the c + term. Now tht we unerstn why we lost the c term, how cn we fin it?? Simple. Strt with the forbienregion wve function (22) gin. This time, chnge φ from 0 to π. As before, we strt in the forbien region n en in the llowe region. This time, though, the imginry prt of is negtive uring the continution. This mens tht ner φ = π, the mgnitue of the c + term is onentilly smller thn tht of the c term, so we ect this time to lose the c + term. An, lo n behol, the wve function in the llowe region tht we obtin by strting from the forbien region n continuing φ from 0 to π is ψ() = (2mbρ) / ( ) i 2 2mb ρ, (26) iπ 3 which is not the sme s (23). Inste, it is precisely the c term in (2), s long s we choose c = ( ). (27) iπ By oing the nlytic continution from φ = 0 to φ = π, we hve lost the c + term n obtine the c term! By performing these two ifferent nlytic continutions, we re ble to strt from the wve function in the forbien region n etermine the complete wve function in the llowe region. Wht we fin is tht in the llowe region, the wve function is given by (2) or, equivlently, (7) with c + n c specifie by (25) n (27). Tht is, in the llowe region ψ() = 2 cos p( ) π (28) p() which is the connection formul (3) we set out to prove. Elegnt, n estce ps? 0