More Scattering: the Partia Wave Expansion Michae Fower /7/8 Pane Waves and Partia Waves We are considering the soution to Schrödinger s equation for scattering of an incoming pane wave in the z-direction by a potentia ocaized in a region near the origin, so that the tota wave function beyond the range of the potentia has the form ir ir cosθ e ψ ( r, θϕ, ) = e + f ( θϕ, ) r The overa normaization is of no concern, we are ony interested in the fraction of the ingoing wave that is scattered Ceary the outgoing current generated by scattering into a soid ange dω at ange θ, φ is f ( θϕ, ) dω mutipied by a veocity factor that aso appears in the incoming wave Many potentias in nature are sphericay symmetric, or neary so, and from a theorist s point of view it woud be nice if the experimentaists coud expoit this symmetry by arranging to send in spherica waves corresponding to different anguar momenta rather than breaing the symmetry by choosing a particuar direction Unfortunatey, this is difficut to arrange, and we must be satisfied with the remaining azimutha symmetry of rotations about the ingoing beam direction In fact, though, a fu anaysis of the outgoing scattered waves from an ingoing pane wave yieds the same information as woud spherica wave scattering This is because a pane wave can actuay be written as a sum over spherica waves: i r ir cos e = e θ = i + j r P ( ) (cos θ ) Visuaizing this pane wave fowing past the origin, it is cear that in spherica terms the pane wave contains both incoming and outgoing spherica waves As we sha discuss in more detai in the next few pages, the rea function j ( r) is a standing wave, made up of incoming and outgoing waves of equa ampitude We are, obviousy, interested ony in the outgoing spherica waves that originate by scattering from the potentia, so we must be carefu not to confuse the pre-existing outgoing wave components of the pane wave with the new outgoing waves generated by the potentia The radia functions j ( r) appearing in the above expansion of a pane wave in its spherica components are the spherica Besse functions, discussed beow The azimutha rotationa symmetry of pane wave + spherica potentia around the direction of the ingoing wave ensures that the anguar dependence of the wave function is just P (cos θ ), not Ym ( θ, ϕ ) The coefficient
i ( + ) is derived in Landau and Lifshitz, 34, by comparing the coefficient of ( cos ) n r θ on the two sides of the equation: as we sha see beow, ( r ) n does not appear in j ( r ) for greater than n, and ( cos ) n θ does not appear in P (cos θ ) for ess than n, so the combination n ( r cosθ ) ony occurs once in the n th term, and the coefficients on both sides of the equation can be matched (To get the coefficient right, we must of course specify the normaizations for the Besse function see beow and the Legendre poynomia) Mathematica Interva: The Spherica Besse and Neumann Functions e i r The pane wave is a trivia soution of Schrödinger s equation with zero potentia, and therefore, since the P (cos θ ) form a ineary independent set, each term j( r) P ( cosθ ) in the pane wave series must be itsef a soution to the zero-potentia Schrödinger s equation It foows that j ( r) satisfies the zero-potentia radia Schrödinger equation: The standard substitution / ( + ) d d R ( r) + R( r) + R r dr r dr r R r = u r r yieds ( + ) du r dr + u( r) = r = u r = sin r, cos r The corresponding radia functions R (r) are (apart from overa constants) the zeroth-order Besse and Neumann functions respectivey For the simpe case = the two soutions are The standard normaization for the zeroth-order Besse function is and the zeroth-order Neumann function j ( r) sin r =, r n ( r) cos r = r Note that the Besse function is the one we-behaved at the origin: it coud be generated by integrating out from the origin with initia boundary conditions of vaue one, sope zero
3 Here is a pot of j r and n ( r) from r = to : For nonzero, near the origin is R () r r or r + The we-behaved r soution is the Besse function, the singuar function the Neumann function The standard normaizations of these functions are given beow Here are j 5 ( r) and j 5 ( r ) :
4 Detaied Derivation of Besse and Neumann Functions This subsection is just here for competeness We use the dimensioness variabe ρ = r To find the higher soutions, we foow a cever tric given in Landau and Lifshitz ( 33) Factor out the ρ behavior near the origin by writing
5 The function χ ρ satisfies d d R = ρ χ ρ ( + ) d χ ( ρ) + χ( ρ) + χ( ρ) = d ρ ρ ρ The tric is to differentiate this equation with respect to ρ : ( ) ( ) 3 d + d + d χ 3 ( ρ) + χ ( ρ) + χ ( ρ) = dρ ρ dρ ρ dρ d d ρ Writing purey formay χ ( ρ) ρχ ( ρ) d d But this is the equation that χ ( ρ ) =, the equation becomes + ( + ) d χ ( ρ) + χ ( ρ) + χ ( ρ) = d + + + ρ ρ ρ + must obey! So we have a recursion formua for generating d a the j ( ρ ) from the zeroth one: χ ( ρ) = χ, + ρ d ρ ρ and j ρ = ρ χ ( ρ), up to a normaization constant fixed by convention In fact, the standard normaization is Now j d sinρ = ρ d ρ ρ ( ρ) ( ρ) n n ( ρ) ρ ρ ( sin / = / n +! This is a sum of ony even powers of ρ It is easiy checed that operating on this series with d can never generate any negative powers of ρ It foows that j ( ρ ), written as a power ρ d ρ series in ρ, has eading term proportiona to ρ The coefficient of this eading term can be sin ρ / ρ, found by appying the differentia operator to the series for )
6 j ( ρ) ρ!! ( + ) as ρ This r behavior near the origin is the usua we-behaved soution to Schrödinger s equation in the region where the centrifuga term dominates Note that the sma ρ behavior is not immediatey evident from the usua presentation of the j ( ρ ) s, written as a mix of powers and trigonometric functions, for exampe j sin ρ cos ρ 3 3cos ρ =, ρ = sinρ ρ, etc ρ ρ ρ ρ ρ j 3 Turning now to the behavior of the j j ρ s for arge ρ, from d sinρ = ρ d ρ ρ ( ρ) ( ρ) it is evident that the dominant term in the arge ρ regime (the one of order /ρ) is generated by differentiating ony the trigonometric function at each step Each such differentiation can be seen to be equivaent to mutipying by (-) and subtracting p/ from the argument, so These j j π ρ sin ρ as ρ ρ ρ, then, are the physica partia-wave soutions to the Schrödinger equation with zero potentia When a potentia is turned on, the wave function near the origin is sti ρ (assuming, as we aways do, that the potentia is negigibe compared with the ( + / ) ρ term sufficienty cose to the origin) The wave function beyond the range of the potentia can be found numericay in principe by integrating out from the origin, and in fact wi be ie j ( ρ ) above except that there wi be an extra phase factor, caed the phase shift and denoted by δ) in the sine The significance of this is that in the far region, the wave function is a inear combination of the Besse function and the Neumann function (the soution to the zero-potentia Schrödinger equation singuar at the origin) It is therefore necessary to review the Neumann functions as we As stated above, the = Neumann function is cos ρ n ( ρ ) =, ρ the minus sign being the standard convention
7 An argument parae to the one above for the Besse functions estabishes that the higher-order Neumann functions are given by: Near the origin n d cosρ = ρ d ρ ρ ( ρ) ( ρ) ( )!! n ρ as ρ ρ + and for arge ρ n π ρ cos ρ as ρ, ρ so a function of the form π sin ρ + δ asymptoticay can be written as a inear ρ combination of Besse and Neumann functions in that region Finay, the spherica Hane functions are just the combinations of Besse and Neumann functions that oo ie outgoing or incoming pane waves in the asymptotic region: so for arge ρ, ( ρ ) = ( ρ) + ( ρ), *( ρ) = ( ρ) ( ρ ) h j in h j in, h ( ρ) ( ρ π /) ( ρ π /) i i e e, h* ( ρ) iρ iρ The Partia Wave Scattering Matrix Let us imagine for a moment that we coud just send in a (time-independent) spherica wave, with θ variation given by P (cosθ) For this th partia wave (dropping overa normaization constants as usua) the radia function far from the origin for zero potentia is ( π /) ir ( π /) ir + π i e e j ( r) sin r = r r r If now the (sphericay symmetric) potentia is turned on, the ony possibe change to this standing wave soution in the faraway region (where the potentia is zero) is a phase shift δ:
8 π π sin r sin r + δ ( ) This is what we woud find on integrating the Schrödinger equation out from nonsinguar behavior at the origin But in practice, the ingoing wave is given, and its phase cannot be affected by switching on the potentia Yet we must sti have the soution to the same Schrödinger equation, so to match with i ( ) the resut above we mutipy the whoe partia wave function by the phase factor e δ The resut is to put twice the phase change onto the outgoing wave, so that when the potentia is switched on the change in the asymptotic wave function must be ( /) ( /) ( / ) S ( π /) i r π + i r π i r π + ir i e e i e e r r r r This equation introduces the scattering matrix i S = e δ which must ie on the unit circe S = to conserve probabiity the outgoing current must equa the ingoing current If there is no scattering, that is, zero phase shift, the scattering matrix is unity It shoud be noted that when the radia Schrödinger s equation is soved for a nonzero potentia by integrating out from the origin, with ψ = and ψ = initiay, the rea function thus i ( ) generated differs from the wave function given above by an overa phase factor e δ Scattering of a Pane Wave We re now ready to tae the ingoing pane wave, brea it into its partia wave components corresponding to different anguar momenta, have the partia waves individuay phase shifted by -dependent phases, and add it a bac together to get the origina pane wave pus the scattered wave We are ony interested here in the wave function far away from the potentia In this region, the origina pane wave is ( ), ( π /) ir ( π /) ir + i r ir cos i e e θ e = e = i (+ ) j( r) P(cos θ ) = i (+ ) P (cos θ ) r r Switching on the potentia phase shifts factor the outgoing wave:
9 e S e ( π /) + ir π / + ir r The actua scattering by the potentia is the difference between these two terms The compete wave function in the far region (incuding the incoming pane wave) is therefore: r ( ) ir S ir cosθ e ψ ( r, θϕ, ) = e + ( + ) P ( cos θ) i r The i i factor canceed the e π / The - in S is there because zero scattering means S = Aternativey, it coud be regarded as subtracting off the outgoing waves aready present in the pane wave, as discussed above There is no ϕ-dependence since with the potentia being sphericay-symmetric the whoe probem is azimuthay-symmetric about the direction of the incoming wave It is perhaps worth mentioning that for scattering in just one partia wave, the outgoing current is equa to the ingoing current, whether there is a phase shift or not So, if switching on the potentia does not affect the tota current scattered in any partia wave, how can it cause any scattering? The point is that for an ingoing pane wave with zero potentia, the ingoing and outgoing components have the right reative phase to add to a component of a pane wave a tautoogy, perhaps But if an extra phase is introduced into the outgoing wave ony, the ingoing + outgoing wi no onger give a pane wave there wi be an extra outgoing part proportiona to S ( ) Reca that the scattering ampitude f ( θ, ϕ ) was defined in terms of the soution to Schrödinger s equation having an ingoing pane wave by ir ir cosθ e ψ ( r, θϕ, ) = e + f ( θϕ, ) r We re now ready to express the scattering ampitude in terms of the partia wave phase shifts (for a sphericay symmetric potentia, of course): ( S ( ) ) θ ( cos ) f θ, ϕ = f θ = + P cosθ = + f P i where iδ ( ) f( ) = e sinδ is caed the partia wave scattering ampitude, or just the partia wave ampitude
So the tota scattering ampitude is the sum of these partia wave ampitudes: The tota scattering cross-section σ iδ ( ) f ( θ ) = ( + ) e sinδ( ) P( cos θ) ( θ) = f dω = π f sin π θ θdθ π iδ ( ) = π ( + ) e sinδ( ) P( cosθ) sinθdθ and the normaization of the Legendre poynomias gives π P ( cos ) θ sinθdθ = + 4π σ = 4π + = + sin δ ( ) f( ) ( ) = = So the tota cross-section is the sum of the cross-sections for each vaue This does not mean, though, that the differentia cross-section for scattering into a given soid ange is a sum over separate vaues the different components interfere It is ony when a anges are integrated over that the orthogonaity of the Legendre poynomias guarantees that the cross-terms vanish Notice that the maximum possibe scattering cross-section for partices in anguar momentum state is ( 4 π / )( + ), which is four times the cassica cross section for that partia wave impinging on, say, a hard sphere! (Imagine semicassicay partices in an annuar area: anguar momentum L = rp, say, but L= and p = so = r Therefore the annuar area corresponding to anguar momentum between and + has inner and outer radii / and ( + / ) and therefore area π ( + ) / ) The quantum resut is essentiay a diffractive effect, we discuss it more ater It s easy to prove the optica theorem for a sphericay-symmetric potentia: just tae the imaginary part of each side of the equation iδ ( ) f ( θ ) = ( + ) e sinδ( ) P( cosθ)
P =, at θ =, using from which the optica theorem Im sin f ( θ = ) = ( + ) δ ( ) Im f = σ / 4π foows immediatey It s aso worth noting what the unitarity of the th partia wave scattering matrix S S = impies for the partia wave ampitude f e iδ ( ) = sinδ ( ) Since = + S if S e δ ( ) i =, it foows that From this, S S = gives: = Im f f This can be put more simpy: In fact, f ( ) Im f = = ( cotδ ( ) i) Phase Shifts and Potentias: Some Exampes We assume in this section that the potentia can be taen to be zero beyond some boundary radius b This is an adequate approximation for a potentias found in practice except the Couomb potentia, which wi be discussed separatey ater Asymptoticay, then, ψ ( r) ( π /) δ ( π /) ir i + ir i e e e = r r iδ ( ) e = sin ( r + δ ( ) π / ) r iδ ( ) e = ( sin ( r π / ) cosδ( ) + cos ( r π / ) sin δ( )) r
This expression is ony exact in the imit r, but since the potentia can be taen zero beyond r = b, the wave function must have the form for r > b iδ ( ) ( cos sin ) ψ r = e δ j r δ n r (The - sign comes from the standard convention for Besse and Neumann functions see earier) The Hard Sphere The simpest exampe of a scattering potentia: V r = for r < R, V r = for r R The wave function must equa zero at r = R, so from the above form of ( r) ( R) ( R) j tan δ ( ) = n ψ, For =, ( R) ( R) sin / tan δ R ( ) = tan, cos / R = R so δ = R This amounts to the wave function being effectivey moved over to begin at R instead of at the origin: for r > R, of course ψ = for r < R ( + δ ) sin ( ) sin r sin r r R = r r r For higher anguar momentum states at ow energies (R << ), ( + ) + ( ) j R R /!! R tan δ ( ) = = + n R!!/ R +!! Therefore at ow enough energy, ony = scattering is important as is obvious, since an incoming partice with momentum p = and anguar momentum is most iey at a distance / from the center of the potentia at cosest approach, so if this is much greater than R, the phase shift wi be sma
3 The Born Approximation for Partia Waves From the definition of f ( θ, ϕ ) and i r e ψ ( r) = e + f( θϕ, ) r ir ψ m e = π r ir i r 3 if r ( r) e d re V( r ) ψ ( r ) reca the Born approximation amounts to repacing the wave function ψ ( r ) in the integra on the right by the incoming pane wave, therefore ignoring rescattering To transate this into a partia wave approximation, we first tae the incoming to be in the z- direction, so in the integrand we repace ψ ( r ) by Labeing the ange between f and r ( ) ( ) (cos θ ) ir cosθ e = i + j r P by γ, if r ( ) ( ) (cos ) e = i + j r P γ Now f θ, ϕ and r θ, ϕ, and γ is the ange between them For this situation, there is an addition theorem for spherica harmonics: is in the direction in the direction 4π * P( cos γ ) = Ym( θ, ϕ ) Ym( θ, ϕ) + m= On inserting this expression and integrating over θ, ϕ, the nonzero m terms give zero, in fact the ony nonzero term is that with the same as the term in the ψ ( r ) expansion, giving and remembering m f P r drv r j r ( θ) = ( + ) ( cos ) ( ) θ = iδ ( ) f ( θ ) = ( + ) e sinδ( ) P( cosθ) it foows that for sma phase shifts (the ony pace it s vaid) the partia-wave Born approximation reads
4 δ m ( ) r drv ( r) ( j ) r Low Energy Scattering: the Scattering Length From f ( ) = cotδ i ( ), the = cross section is σ 4π = = cotδ ( ) i At energy E, the radia Schrödinger equation for u = rψ away from the potentia becomes du/ dr =, with a straight ine soution u( r) = C( r a) This must be the imit of u( r) = C sin ( r+ δ ( ) ), which can ony be correct if δ is itsef inear in for sufficienty sma, and then it must be δ ( ) = a, a being the point at which the extrapoated externa wavefunction intersects the axis (maybe at negative r!) So, as goes to zero, the cot term dominates in the denominator and The quantity a is caed the scattering ength π σ = = 4 a Integrating the zero-energy radia Schrödinger equation out from u(r) = at the origin for a wea (spherica) square we potentia, it is easy to chec that a is positive for a repusive potentia, negative for an attractive potentia Repusive potentia, zero-energy wave function (so it s a straight ine outside of the we!):
5 Scattering Length for Square We -5 5 - - Potentia Tangent Wavefunction Attractive potentia: Scattering Length for Square We -5 5 - - Potentia Tangent Wavefunction On increasing the strength of the repusive potentia, sti soving for the zero-energy wave function, a tends to the potentia wa here s the zero-energy wavefunction for a barrier of height 6:
6 Scattering Length for Square We -5 5 - Potentia Tangent Wavefunction For an infinitey high barrier, the wave function is pushed out of the barrier competey, and the hard sphere resut is recovered: scattering ength a, cross-section 4π a On increasing the strength of the attractive we, if there is a phase change greater that π/ within the we, a wi become positive In fact, right at π/, a is infinite! Scattering Length for Square We -5 5 - - Potentia Tangent Wavefunction And a itte more depth to the we gives a positive scattering ength:
7 Scattering Length for Square We -5 5 - - Potentia Tangent Wavefunction In fact, a we deep enough to have a positive scattering ength wi aso have a bound state This becomes evident when one considers that the depth at which the scattering ength becomes infinite can be thought of as formay having a zero energy bound state, in that athough the wave function outside is not normaizabe, it is equivaent to an exponentiay decaying function with infinite decay ength If one now deepens the we a itte, the zero-energy wave function inside the we curves a itte more rapidy, so the sope of the wave function at the edge of the we becomes negative, as in the picture above With this sighty deeper we, we can now ower the energy sighty to negative vaues This wi have itte effect on the wave function inside the we, but mae possibe a fit at the we edge to an exponentia decay outside a genuine bound state, with wave function e κr outside the we 5 5 5 5 3-5 Potentia Wavefunction
8 If the binding energy of the state is reay ow, the zero-energy scattering wave function inside the we is amost identica to that of this very ow energy bound state, and in particuar the ogarithmic derivative at the wa wi be very cose, so κ /a, taing a to be much arger than the radius of the we This connects the arge scattering ength to the energy of the weay bound state, (Saurai, p 44) B E = / m= / ma Wigner was the first to use this to estimate the binding energy of the deuteron from the observed cross section for ow energy neutron-proton scattering