Clssicl Limit B Lecture Notes WKB Method Hmilton Jcobi Eqution We strt from the Schrödinger eqution for single prticle in potentil i h t ψ x, t = [ ] h m + V x ψ x, t. We cn rewrite this eqution by using ψ x, t = e is x,t/ h : S [ t ψ = m S i h ] m S + V ψ. Assuming ψ 0, this leds to n eqution S t = m S i h m S + V. 3 Now tking the forml limit h 0, it becomes the sme s the clssicl Hmilgon Jcobi eqution S t = m S + V. It is often sid tht the h 0 limit is indeed the clssicl mechnics becuse it reduces to the Hmilton Jcobi eqution. 3 h Expnsion The Eq. 3 is exct s long s ψ 0. consider n expnsion One cn use this eqution, nd S x, t = S 0 + hs + h S +. 5
This is n expnsion in h, nd hence clled h-expnsion or semi-clssicl expnsion. Plugging in the expnsion into Eq. 3, we find S 0 t S t = m S 0 + V, 6 = [ i S 0 + S m 0 S ], 7 nd similrly for higher terms in h. The leding eqution hs only S 0, nd it is exctly the sme s Hmilton Jcobi eqution. Once you solve these equtions strting from S 0, S, etc, you hve solved the wve function ψ in systemtic expnsion in h. There is one importnce difference between the clssicl limit which reduces to the Hmilton Jcobi eqution nd the ide of expnsion in h. In the clssicl limit, S is of course rel becuse it does not mke sense otherwise. However, in the sense of n expnsion in h, we don t know if S 0 hs to be rel. In fct, in mny interesting cses, S 0 turns out to be complex. We will see these exmples below. WKB Approximtion WKB Approximtion, due to Wentzel, Krmers, nd Brillouin, keeps terms up to O h in the h expnsion. It is used mostly for the time-independent cse, or in other words, for n eigenstte of energy E. Then the wve function hs the ordinry time dependence e iet/ h. We lso restrict ourselves to the one-dimensionl problem. In terms of S, it corresponds to S x, t = S x Et. 8 Therefore only S 0 hs the time dependence S 0 x, t = S 0 x Et, while higher order terms S i = S i x for i 0 do not depend on time. The lowest order term S 0 stisfies the Hmilton Jcobi eqution see Eq. 6 E = m S 0 + V x. 9 The differentil eqution cn be solve immeditely s S 0 x = ± me V x dx = px dx 0
up to n integrtion constnt which cn be determined only fter imposing boundry condition on the wve function. We used the nottion px = ± me V x becuse it is nothing but the momentum of the prticle in the clssicl sense. Once we know S 0, we cn lso solve for S. Strting from Eq. 7, nd using S / t = 0, we find which hs solution S x = i S 0S = is 0, S 0 x S 0x dx = i log px + constnt. Therefore the generl solution to the Schrödinger eqution up to this order is ψx, t = e is 0x+ hs x/ h e iet/ h = c ± px exp ī me V x / h dx e iet/ h, 3 nd the overll constnt c is of course undetermined from this nlysis. This solution mkes it immeditely cler tht this pproximtion breks down when px goes to zero. Or in other words, the pproximtion is bd where the clssicl prticle stops nd turns becuse of the potentil. Such points re clled clssicl turning points.. Vlidity of the WKB Approximtion The pproximtion to stop with S in the h expnsion is vlid only when S is much smller thn S 0. Or in other words, if the term with h in Eq. is much smller thn the other terms. In prticulr, we require S h S. In the one-diimensionl time-independent cse discussed bove, this is px h p x. 5 Using the definition of px = ± me V x, we find hdv x/dx. 6 E V xpx 3
Agin we find the sme conclusion: the WKB pproximtion breks down close to the clssicl turning point V x = E e.g., px = 0. For exmple, tke hrmonic oscilltor V x = mω x. The vlidity condition Eq. 6 cn be rewritten s 8 E mω x 3 hω mω x. 7 This inequlity is lwys stisfied exctly t the origin x = 0, but once wy from the origin, it is impossible to stisfy unless E hω. In this sense, we re indeed in the clssicl regime. However, even for lrge E hω, the pproximtion is not vlid close to the clssicl turning points E = mω x. Here is the surprise. The vlidity condition Eq. 6 my be stisfied even in the region where the prticle cnnot enter clssicly E < V x. For exmple with the hrmonic oscilltor gin, the vlidity condition is lwys stisfied for lrge x E/mω for ny vlue of E. In other words, the WKB pproximtion is good wy from the clssicl turning points both where clssicl prticle exists nd where clssicl prticle cnnot exist. This is why the WKB pproximtion is not relly clssicl limit. It pplies lso where physics is truly quntum mechnicl. In the clssiclly forbidden region, the solution Eq. 3 needs to be modified to ψx, t = e is 0x+ hs x/ h e iet/ h = c ± mv x E exp h / mv x Edx e iet/ h,8 by following the sme steps s in the clssiclly llowed region.. Mtching WKB pproximtion cn be good both in the region E > V x nd the region E < V x but cnnot be good in between the regions close to the clssicl turning point E = V x c. In order to utilize the WKB pproximtion to work out wve functions, we need to somehow overcome this limittion. The stndrd method is to expnd round x c nd solve for the wve function exctly. Then you cn mtch on to the WKB solutions wy from x c to determine the entire wve function.
The common method is to pproximte the potentil round the clssicl turning point x c by liner one: V x = V x c + V x c x x c + Ox x c, 9 nd ignore the second order term. By definition V x c = E. Schrödinger eqution round this point is therefore h d m dx + V x E ψ = h d m dx + V x c x x c ψ = 0. 0 Using the new vrible u = /3 m dv h dx x c x x c, the differentil eqution simplifes drsticlly to d du u ψ = 0. The solution to this eqution is known s the Airy function Aiu = dt cos π 0 3 t3 + ut. 3 This cn be checked s follows. By cting the differentil opertor in Eq. on the definition of the Airy function, we find d du u Aiu = dt d π 0 dt sin 3 t3 + ut. The boundry term t t = 0 obviously vnishes. The behvior t t = is trickier. The point is tht the rgument of the sin grows s t 3 nd oscilltes more nd more rpidly s t. Therefore for ny infinitesiml intervl of lrge t, the oscilltion bsiclly cncels the integrnd except for left-over tht goes down s /t. Therefore the boundry term for t cn lso be dropped. It cn be shown using the steepest descent method tht the symptotic behvior of the Airy function smoothly mtches to the WKB solutions. The symptotic behvior is / π Aiu u exp 3 u3/ u 0 / π u cos u. 5 u + π 3 u 0 5
Note first tht, for u 0, the symptotic behvior is Aiu = hmv / h /3 π me V / cos h x c me V dx + π. 6 Here we used the liner expnsion V x = E +V x c x x c to relte powers of u to me V. This expression is consistent with the WKB solution for prticulr choice of the overll constnt. Similrly, for u 0, Aiu = hmv / h /3 π mv E.3 Bound Sttes / exp h x c mv x Edx. 7 Consider the following sitution. In the region I x <, E < V nd it is clssiclly forbidden. In the region II < x < b, E > V nd the the prticle is clssiclly llowed. But gin in the region III x > b it is clssiclly forbidden. In this cse, we expect bound sttes with discrete energy levels. From the mtching using the Airy function t x = b, we wnt the wve function in the clssicly llowed region II to be s in Eq. 6, ψx = c hmv b/ h /3 / b cos me V x π me V h dx + π. b 8 On the other hnd from the mtching t x =, the wve function in the region II must be ψx = c hmv / h /3 / cos me V x π me V h dx π. 9 c b nd c re normliztion constnts. The minus sign in fron of π/ is becuse the clssiclly llowed region is to the right of the turning point nd hence we need to use Ai-u insted of Aiu for mtching. The two behviors obtined from mtching t both sides must be the sme in order to hve 6
consistent wve function for the entire region II. Becuse we do not know the coefficients c,b, which cn in prticulr differ in signs, we hve to require h b me V x dx + π = h Further simplifying the eqution we obtin me V x dx π nπ. 30 me V xdx = n + π h. 3 This is reminiscent of Bohr Sommerfeld quntiztion condition tht ppered t the erly stge of quntum mechnics, pdq = nh 3 with h = π h except tht there is n extr contribution / here. Remember tht the Bohr Sommerfeld quntiztion condition ws n d hoc requirement which hppened to give correct energy spectr for the hydrogen-like toms. Nonetheless, this is n importnt eqution becuse it puts constrints on the llowed vlues of E, nd hence we find discrete energy levels. Becuse the WKB pproximtion is better for lrger E, the constrint Eq. 3 is expected to give correct energy levels t lest for highly excited sttes. Let us pply this constrint Eq. 3 for hrmonic oscilltor. The condition then is mx me m ω x dx = n + π h. 33 x min The l.h.s. is n elementry integrl me mω E = π nd we find n + hω. 3 This is precisely the energy levels of hrmonic oscilltor. Of course, obtining the exct energy levels is n ccident for hrmonic oscilltor. In generl, we expect to get pproximtely correct energy levels for high n. Indeed, if we pply the sme constrint for the hydrogen-like toms for l = 0 nd pplying the WKB method to the rdil wve function, we would obtin the correct results if we drop / this is wht Bohr Sommerfeld did, but don t if we keep / s obtined by the WKB method. 7
Wht we obtin from the WKB method is not just the energy levels, but more detils such s the pproximte wve function. To simplify the nottion, let us choose unit system such tht m = ω = h =. Then the wve function in the region II for the hrmonic oscilltor is ψ E x / cos Doing the stndrd integrl, we find ψ cos E sin E x / E E x dx π x E + π + x x E E. 35 π. 36 Becuse E = n + / in this unit, it further simplifies to [ ] cos E sin x E + x E x E ψ [ E x / ] sin E sin x E + x E x E n even n odd 37. Tunneling To study tunneling process, we gin hve three regions, clssiclly llowed region I x <, where the prticle initilly exist, clssiclly forbidden region II < x < b, nd clssiclly llowed region III x > b where the prticle tunnels to. We follow the sme mtching procedure s in the bound stte exmple. A difference is tht the mtching t x = b uses different version of the Airy function which rises exponentilly wy from x = b. This version of the Airy function Bi is explined in the note on steepest descent method. The bottom line is tht the mtching t x = is done s ψ hmv / h /3 π me V ψ hmv / h /3 π mv E / cos h / x exp h me V dx + π, x < The mtching t x = b is, on the other hnd, ψ C i hmv b/ h /3 / + exp π mv E h 8 x mv Edx. x > mv Edx, x < b
ψ C i hmv b/ h /3 π me V / sin h b me V dx + π. x > b The overll normliztion fctor C is determined by the requirement tht the behvior of the wve function is consistent for < x < b between two mtching procedures. We therefore find C = i V /6 exp h V b mv x Edx. 38 Compring two clssiclly llowed regions, nd tking dvntge of further normliztion chnge, the mtching reduces to ψ me V cos / h x ψ exp h mv x Edx me V dx + π, x < me V / sin h In other words, the mplitude in the region x > b due to tunneling from the region x < is suppressed by exp h mv x Edx. 39 b me V dx + π. Usully people refer to Gmov s trnsmission coefficient exp h mv x Edx 0 s suppression fctor for the tunneling rte squre of the mplitude. 9