Relating Scattering Amplitudes to Bound States

Transcription

1 Reating Scatteing Ampitudes to Bound States Michae Fowe UVa. 1/17/8 Low Enegy Appoximations fo the S Matix In this section we examine the popeties of the patia-wave scatteing matix ( ) = 1+ ( ) S k ikf k fo compex vaues of the momentum vaiabe k. Of couse genea compex vaues of k do not coespond to physica scatteing but it tuns out that the scatteing of physica waves can often be most simpy undestood in tems of dominant singuaities in the compex k pane. We begin with the compex k connection between (positive enegy) scatteing and (negative enegy) bound states. The asymptotic fom of the = patia wave function in a scatteing expeiment is (fom the pevious ectue) ( ) ik i e S k e k An = bound state has asymptotic wave function + ik. whee C is a nomaization constant. Ce Notice that this esembes an outgoing wave with imaginay momentum k = iκ. If we anayticay continue the scatteing wave function fom ea k into the compex k-pane we get both exponentiay inceasing and deceasing wave functions making no physica sense. But thee is an exception to this genea obsevation: if the scatteing matix S becomes infinite at some compex vaue of k the exponentiay deceasing tem wi be infinitey age than the exponentiay inceasing tem. In othe wods we ony have a deceasing wave function a bound state. We know that the enegy of a bound state has to be ea and negative and is aso equa to k /m so this can ony happen fo k pue imaginay k = iκ. Now the existence of a ow enegy bound state means that the S matix has a poe (on the imaginay axis) cose to the oigin so this wi stongy affect ow enegy (nea the oigin but ea k) scatteing. Let s see how that woks using the ow-enegy appoximation discussed peviousy. Reca that the = patia wave ampitude f = k κ 1 ( cotδ i)

2 and at ow enegy δ and = ka f so Note that S 1as as it shoud since = 1 1 k ka i = ik + 1/ a (( 1/ ) ) ( i/ a) ( i/ a) k + S = 1+ ikf =. k k δ = ka and ( ) that this appoximation coecty gives S ( k ) = 1. ( ) i S k e δ =. Note aso This S k has a poe in the compex pane at k = i/ a and if this coesponds to a bound state having κ =1/a then the binding enegy κ / m = / ma. In fact though we un into a pobem hee: we get the same fom of S k at ow enegies even fo a epusive potentia ( ) which cetainy doesn t have a bound state! The poe in S ( ) k ony means that we can have an asymptotic wave function of the ight fom but it does not guaantee that this asymptotic fom wi go smoothy to nonsingua behavio at the oigin. Fo a epusive potentia it s easy to see that the zeo (o negative) enegy wave function on integating out fom the oigin sopes moe and moe steepy upwads so coud neve with inceasing go ove to asymptotic decay. Effective Range The ow enegy appoximation above can be witten We sha now deive a bette appoximation kcotδ 1/ a. 1 kcotδ k 1/ a + k whee caed the effective ange gives some measue of the extent of the potentia (in contast to a which as we have seen can be abitaiy age even fo a shot-ange potentia). A usefu mathematica too needed at this point is the Wonskian. Fo two functions f ( x) g( x ) the Wonskian is defined as ( ) W f g = fg fg

3 3 the pime denoting diffeentiation as usua. Fom this W ( f g) = fg f g and if f ( x) g( x) satisfy the same second-ode diffeentia equation (ike the Schödinge equation with the same enegy) then W = so W( f g ) is constant independent of x. Fo the adia Schödinge equation asymptoticay ( ) sin( + δ ) ( ) u k C k v k whee we now show k expicity. This asymptotic function v(k ) satisfies the Schödinge equation fo zeo potentia but does not have the coect physica bounday behavio at =. Since in the ow-enegy imit ( ) (taking C = 1/sinδ). Fom the Schödinge equation δ k = ka the k = asymptotic wave function ( ) v = 1 / a ( ) u k + mv / u k = k u k it is easy to check that the Wonskian of u( k ) with the coesponding zeo enegy function u( ) satisfies: d W uk ( ) u ( ) = kuku ( ) ( ) d. (The tem invoving the potentia has canceed out: dw/d is nonzeo hee because these two functions don t satisfy the same diffeentia equation the enegy tems ae diffeent.) The coesponding functions v( k ) v( ) satisfy the same Wonskian equation: d W vk ( ) v ( ) = kvkv ( ) ( ) d. We can find a fomua fo the effective ange by integating the diffeence between these two equations fom = to infinity: the two soutions u v diffe ony within the ange of the potentia and appopiatey nomaizing them then taking the diffeence gives a measue of this ange. So

4 4 = { ( ) ( ) ( ) ( ) } = ( ) ( ) ( ) ( ) W vk v W uk u k vkv uku d ( ) ( ) = Fo age u k v k so thee is zeo contibution fom the uppe end. Fo the popey-behaved u functions go to zeo the v functions ae fom which with it foows immediatey that and theefoe with a ow-enegy imit whee ( ) = ( + ) = ( + ) v k Csin k sin k /sin δ δ δ δ k = ka v = 1 / a. W 1 v k v k = a ( ) ( ) = cotδ 1 cot δ a k = + k v k v u k u d 1 1 kcotδ = + k a = v u d. Now by definition u( ) v( ) coincide outside the ange of the potentia but moving fom that egion towads the oigin they pat company when the potentia kicks in with u v 1 as. Theefoe the intega above is a ough measue of the actua ange of the potentia about haf of it (hence the facto of in defining ). Note again the contast with a which can be infinite fo a shot ange potentia. Couomb Scatteing and the Hydogen Atom Bound States One paticua set of bound states in a potentia we ve spent a good dea of time on ae the states of the hydogen atom and it is inteesting to see how they eate to scatteing. Reca that the asymptotic fom of the bound state wave function is: / na n e Rn ().

5 5 But this doesn t have the bound-state fom we found above fom the anaytic continuation n agument thee s an exta! What s going on? The pobem is that in a ou pevious wok we assumed that if we ooked fa enough away fom the cente of the potentia the adia Schödinge equation coud be taken to be that fo zeo potentia to any desied accuacy. The Couomb potentia though does not decay fast enough with distance fo this to be tue. Fo instance it has bound states having abitaiy age adii. Witing we have κ = 1 me na = n 1 κ+ ( me / κ)n Rn () e. Note that the exta tem in the exponent keeps on gowing without imit! We ae neve fee of the potentia. But how does this anaysis of hydogen atom wave functions eate to positive-enegy scatteing states? We can just anayticay continue this esut back to ea k to find out. Repacing κ by ik gives: 1 i( k+ ( me / k)n ) Rn () e. So we have scatteing states that ae not of the standad fom eithe the phase shift is infinite and not we-defined. But we found this esut be anayticay continuing fom the hydogen atom bound states. Let s check it: et us ook at the adia Schödinge equation fo positive enegies at age. Witing R() = u()/ as usua et us aso put u() = e ik v() fo age and we can aso ignoe the centifuga baie tem so e k u u = Eu = m m u. The equation fo v() is e ( ikv + v ) v = m and since v() is sowy vaying the second deivative can be ignoed so ime v v k

6 6 This eads immediatey to the same fom we found by anaytic continuation. Resonances and Associated Zeos Reca in the fist semeste we discussed α-decay: an α patice in a heavy nuceus can be thought of as tapped in a potentia we geneated by the attactive nucea foces. A spheica squae we is a wokabe appoximation except that this squae we is at the top of a hi outside the nuceus the epusive eectostatic potentia is ( Z ) e / soping down fom the we edge to zeo as. This means that fo a adioactive nuceus athough the enegy eve woud be at negative enegy fo the squae we on eve gound actuay it s above the bottom of the eectostatic hi and the wave function wi not be decaying but osciating. This asymptotic wave is of couse vey tiny since typicay the chances of detecting the α we outside the nuceus that is of decay is one in miions of yeas. Now conside the evese pocess: imagine we bombad a decayed nuceus with α patices. If we sent in one at exacty the ight enegy (vey difficut this is a thought expeiment!) the wave function woud be exacty the same as that fo α decay. The wave function inside the nuceus woud be huge compaed with that outside we d neve see ou α again. Less damaticay if we sent in one cose to that enegy the wave function woud sti be vey age inside the nuceus meaning that the patice woud spend a ong time inside befoe coming out again. (Reca fo a patice in a oe coaste potentia in one dimension the wave function is age whee the patice spends a ot of time that s whee you e most ikey to find it.) This is a esonance: at just the ight enegy the ampitude of the wave function within the potentia becomes vey age anaogous to the ampitude of a cassica diven osciato as the diving fequency is adjusted to the natua osciato fequency. Can we undestand this in tems of poes in the S matix? Consideed as a function of enegy the S matix has poes at negative enegies coesponding to bound states. But this is a positive enegy and S( k ) = 1fo positive enegies: that s the egime of ea physica scatteing. What it can have is a poe nea a positive enegy in the compex pane. To keep S( k ) = 1 it woud then have to have a zeo at the mio image point that is be ocay of the fom Fom this iδ ( ) / E E E Γ i S ( E) = e = E E +Γ i / tan δ E = Γ/ E E so ( ) δ E = π / and the scatteing coss section eaches its maximum possibe vaue eca

7 7 so 4π σ = 4π + 1 = + 1 sin δ k ( ) f ( ) = = max 4π σ = ( + 1). k Fo a naow esonance (sma ) the phase shift δ E goes apidy fom to π as the enegy is inceased though E. Most of the vaiation occus within an enegy ange Γ of E Γ is caed the width of the esonance. If the esonance is supeimposed on a sowy vaying backgound phase shift δ then it causes an incease fom δ to δ + π. This wi pass though o π depending on the initia sign of δ so the maximum scatteing at phase shift π / wi have associated with it an enegy at which thee is zeo scatteing. Fo substantia backgound δ the zeo coud be cose to the peak as iustated beow: Γ ( ) Resonance Scatteing Phase Shift Coss Section Unitay Limit Pi