Canonical quantum potential scattering theory Supported in part by the CNPq and FAPESP (Brazil).{}^{{\dagger}}Martin Gutzwiller Fellow, 2007/2008.

# Canonical quantum potential scattering theory ††thanks: Supported in part by the CNPq and FAPESP (Brazil). †Martin Gutzwiller Fellow, 2007/2008.

M. S. Hussein, W. Li and S. Wüster Max-Planck-Institut für Physik komplexer Systeme
Nöthnitzer Strae 38, D-01187 Dresden, Germany
Instituto de Física, Universidade de São Paulo
C.P. 66318, 05315-970 São Paulo, S.P., Brazil
###### Abstract

A new formulation of potential scattering in quantum mechanics is developed using a close structural analogy between partial waves and the classical dynamics of many non-interacting fields. Using a canonical formalism we find non-linear first-order differential equations for the low energy scattering parameters like scattering length and effective range. They significantly simplify typical calculations, as we illustrate for atom-atom and neutron-nucleus scattering systems. A generalization to charged particle scattering is also possible.

###### pacs:
03.65.Nk, 34.50.-s, 67.85.?d, 25.40.Dn

## I Introduction

Low-energy scattering parameters such as the scattering length and the effective range are very important physical quantities that are used to describe a variety of systems. Among these are nuclear reactions of relevance to nucleosynthesis Rolfs () and cold dilute atomic gases Pethik (). To find the low energy scattering behaviour for a given potential, the generic approach is to solve the stationary Schrödinger equation for a range of scattering momenta and to extract the scattering phase shifts Flugge (). In contrast, our formalism allows the computation of the scattering length and effective range directly from the potential.

To reformulate quantum scattering theory, we use methods from classical mechanics. A ”Hamiltonian” function expressed in terms of the coefficients of the Riccati-Bessel and Riccati-Neumann function expansion of the solution of the Schrödinger equation was found amenable to formal canonical manipulations. This Hamiltonian, of a quadratic form, can be rendered constant with an appropriate canonical transformation. From this transformation we obtain a first order, nonlinear differential equation for the scattering problem that generalises Calogero’s equation Calogero (). We demonstrate how this equation can be used to solve the scattering problem for low momenta and several partial waves. Our examples range from nuclear scattering to atomic physics.

In the latter, the basic input into the atom-atom effective interaction is the scattering length , defined as the scattering amplitude evaluated at zero energy , where is determined from the phase shift, .

The effective range expansion for is Teichmann (),

 1Dl(k)=−1al+12rlk2−Plr3lk4. (1)

Our formalism allows the calculation of , and directly from the potential, using first order, non-linear, differential equations, and may allow the development of new approximation schemes for the calculation of these low energy parameters in cases where the usual procedure of solving the second order equation (the Schrödinger equation) may be cumbersome.

This paper is organized as follows: In section II we present our canonical formulation of potential scattering. In section III we derive the Calogero-type equation for the scattering length function and obtain first order non-linear differential equations for the scattering length, effective range and shape parameter. In section IV we apply the theory to the calculation of the low-energy parameters of typical atomic scattering systems. One further application to typical short range potential nuclear scattering is presented in section V. Finally, in section VI several concluding remarks are made. Appendices contain the essentials of potential scattering and generalisations of our work to charged particle scattering.

## Ii Canonical Formulation

In quantum scattering from a central potential one has to solve the radial Schrödinger equation Newton (); CH (),

 [−d2dr2+U(r)+l(l+1)r2]ϕl(k,r)=k2ϕl(k,r), (2)

where is the angular momentum quantum number and the scattering momentum. It is known that the solution can be parametrised as

 ϕl(k,r)=ul(kr)ql(k,r)+vl(kr)pl(k,r), (3)

where and are the Riccati-Bessel and Riccati-Neumann functions.

Following Hara (); Hussein (), we construct a classical ”Hamiltonian” function,

 H(q,p)=U(r)2k(ulql+vlpl)2. (4)

Through a straightforward computation of and from Eq. (2) and Eq. (43), (46)-(48) in the Appendix, we find,

 dqldr=∂H∂pl, (5)
 dpldr=−∂H∂ql. (6)

It is therefore evident that we may regard and as the coordinate and conjugate momentum respectively, with acting as the ”time” variable in a ”classical” dynamics governed by the Hamiltonian above Goldstein (). It should be realized that in this fashion we treat each partial wave independently as there is no coupling between waves for different due to the spherical symmetry of . Thus the scattering problem is reduced to a many- (strictly speaking infinite) non-interacting classical fields problem.

Any function of , , and will evolve in ”time”, , as

 dFldr=[Fl,H]+∂Fl∂r, (7)

where is the Poisson bracket.

In order to identify useful functions for the scattering problem, such as the phase shift, we perform a canonical transformation to a new coordinate, and a new momentum , using the following generating function,

 F2(ql,Pl,r)=12Al(r)q2l+Bl(r)qlPl, (8)

where the functions and are arbitrary to be determined by imposing a condition on the new dynamics, and

 Ql=∂F2∂Pl=qlBl(r), (9)

and

 pl=∂F2∂ql=Al(r)ql+Bl(r)Pl. (10)

The new Hamiltonian function is

 K(Ql,Pl;r)=H(ql,pl;r)+∂F2(ql,Pl;r)∂r. (11)

It is now a simple matter to obtain the explicit form of the new Hamiltonian, , viz

 K(Ql,Pl;r)=Q2l2B2l[U(r)k(ul+vlAl)2+dAldr] +QlPlBl[U(r)kvl(ul+vlAl)Bl+dBldr] +U(r)2kv2lB2lP2l (12)

The goal is now to render the new coordinate cyclic. To this end, we impose the condition that the functions and satisfy the first-order, non-linear differential equations,

 dAl(r)dr+U(r)k(ul(r)+vl(r)Al(r))2=0, (13)

and

 dBl(r)dr+U(r)k(ul(r)+vl(r)Al(r))Bl(r)=0. (14)

To solve these, we need to specify a boundary condition. We choose the values of the functions and , which guarantee that the new coordinate and momentum have the same limiting values at as the original ones, and , namely and as .

With this choice of the coefficients and , the new Hamiltonian function is

 K(Ql,Pl;r)=U(r)2kv2l(r)B2l(r)P2l (15)

and the new equations of motion,

 dPldr=−∂K∂Ql=0 (16)

and

 dQldr=∂K∂Pl=U(r)2kv2l(r)B2lPl. (17)

We choose the solutions, and , which are consistent with the new equations of motion and the boundary conditions. With this choice, we have the solution of the original problem,

 ql(r)=1Bl(r) (18)

and

 pl(r)=Al(r)Bl(r). (19)

By elimination of from the above equations, we obtain for the function the following

 Al(r)=pl(r)ql(r), (20)

which can be identified as the phase shift function , introduced by Calogero Calogero ().

For charged particle scattering, the free solutions , are replaced by the scaled Coulomb wave functions, defined in appendix B. The equation for the tangent function is given by a similar equation as Eq. (13), with the replacement of by the regular scaled Coulomb wave function, and by the irregular scaled Coulomb wave function, . A detailed discussion of the charged particle Calogero equation of the scattering length function is beyond the scope of this paper, but a brief preview is given in appendix B.

## Iii Scattering Length Function

It is convenient to introduce the scattering length function defined as the function,

 al(k,r)=−Al(r)k(2l+1), (21)

which satisfies the Calogero-type equation,

 dal(k,r)dr−U(r)k2l+2(ul(kr)−k(2l+1)vl(kr)al(k,r))2=0. (22)

Note that has the dimension of .

The above equation has to be solved with the boundary condition . The scattering length itself is, by definition, . For the equation becomes,

 da(k,r)dr−U(r)k2(sin(kr)−kcos(kr)a(k,r))2=0 (23)

Further specifying we have

 da(0,r)dr−U(r)[r−a(0,r)]2=0. (24)

A final special case of interest is and integer . Then the asymptotic behaviour of the Bessel functions (see Eqs. 57-58) gives:

 dal(0,r)dr−U(r)[rl+1(2l+1)!!−(2l−1)!!rlal(0,r)]2=0. (25)

Eqs. (22-25) are already sufficient to extract the low energy scattering behaviour that defines Eq. (1) as we shall demonstrate in the next section. However they can also be used to obtain equations that determine the parameters of the effective range expansion directly. This can be done by expanding Eq. (22) in , equating coefficients and using the identification . The resulting Calogero equations for and are

 dr0(r)dr+2rU(r)r0(r)(ra0(r)−1)+2r2U(r)[ra0(r)−1][r3a0(r)−1]=0, (26)
 dp0(r)dr+2rU(r)[ra(r)−1]p0(r)+F(r)=0 (27)

where the function is given by,

 F(r)=r2U(r)12(2r−r0(r))(2r−3r0(r))+r2U(r)45a20(r)[2r4+3a0(r)r2(5r0(r)−4r)] (28)

The above two differential equations can be integrated once the scattering length function, is known. Numerically, Eqs. (24-27) could be solved as coupled system. Notice that and , owing to the boundary condition obeyed by the scattering length function, .

The scattering length equation for any partial wave can be written down, with the aid of Eq. (22) and Eqs. (57-58) of the Appendix

 dal(0,r)dr−U(r)[rl+1(2l+1)!!−(2l−1)!!rlal(0,r)]2=0, (29)

The corresponding version of the Calogero equations for the effective range function, , and shape parameter function, , can be easily worked out using the small form of the Riccati-Bessel and Riccati-Neumann functions, given in appendix A.

## Iv Cold atom scattering

To show how our formalism simplifies typical calculations, we now use it to determine low energy scattering parameters for some exemplary potentials and compare our results with the literature. We shall mostly consider scattering of Cs atoms interacting with the potential Flambaum ()

 V(r)=12βrλe−ηr−(C6r6+C8r8+C10r10)fc(r), (30) fc(r)=θ(r−rc)+θ(rc−r)e−(rc/r−1)2, (31)

where in atomic units , , , , , , and the reduced mass of a cesium pair is .

### iv.1 S-wave scattering length

The s-wave scattering length can be determined according to the Calogero equation (24). It has been recently used to assess corrections to calculations that rely on using the usual Schrödinger equation with long range interactions Ouer (). Here we first summarize their calculations and then extend them to other partial waves and to determining the effective range and shape parameters.

Integrating this equation is numerically non-trivial, owing to poles in that arise from bound states in the potential. This problem can be overcome by using a symplectic integrator or a variable transformation that maps the spatial domain and the range of function values onto a compact interval Ouer (). For the latter one defines and and then solves a nonlinear equation for . Since Eq. (24) is singular at the initial condition is , with chosen sufficiently small. The determination of the Cs-Cs scattering length in this manner has been done in Ref. Ouer (), which also shows as a function of to visualize the above mentioned pole structure. The scattering length following from this potential, is reproduced precisely by Eq. (24).

### iv.2 S-wave effective range expansion

Our formalism allows us to use the transformation of the preceding section to determine quantities beyond the scattering length. For the effective range, we present two methods: the direct integration of Eq. (26) and the integration of Eq. (23) for a range of and subsequent fitting of Eq. (1).

#### iv.2.1 Effective range equation

We use the substitutions , and in Eqs. (24-26). The resulting system of two coupled equations is:

 dθ1(ϕ)dϕ−sec4[ϕ]sin2[θ1(ϕ)−ϕ]U[tan(ϕ)]=0, dθ2(ϕ)dϕ−sec2[ϕ]cos2[θ2(ϕ)]U[tan(ϕ)][2tan[θ2(ϕ)]tan(ϕ)(1−tan(ϕ)tan[θ1(ϕ)]) −2tan(ϕ)2+83tan(ϕ)3tan[θ1(ϕ)]−23tan(ϕ)4tan[θ1(ϕ)]2]=0, (32)

Again we use the initial conditions and , with , small, to avoid an initial singularity. The solution is shown in Fig. 1, similar plots showing the scattering length can be found in Ref. Ouer ().

#### iv.2.2 Low k expansion

Alternatively, in Eq. (23) we define and . This yields

 dθ(k,ϕ)dϕ−sec2[ϕ]cos2[θ(k,ϕ)]V[tan(ϕ)]Ek(sin[ktan(ϕ)]−kcos[ktan(ϕ)]tan[θ(k,ϕ)])2. (33)

We now solve Eq. (33) for several values of , see Fig. 2. Using the effective range expansion as in Eq. (1) we can then extract the low energy scattering parameters from a fit as shown. The values for and agree roughly those of Ref. Flambaum () (which are and ). For the fit we varied the range of values of under consideration until the smallest errors were reported by the fitting routine.

The directly calculated effective range agrees better with Ref. Flambaum () than that obtained from the fit. However the fitting procedure is able to also give an indication of the shape parameter, whereas the direct determination of it, using Eq. (27), was plagued by numerical instabilities.

### iv.3 Higher partial waves

The scattering lengths for higher partial waves (p,d,f,…) can in principle be obtained from the simple Eq. (25). For the potential given at the beginning of this section we find . Higher partial waves for the atom-atom potential cannot be treated with our method, since the centrifugal terms in Eq. (25) combine with the long range tail of the potential to produce a constant slope of the scattering length function. This shortcoming is absent in the case of short range interaction, such as the one encountered in neutron-nucleus scattering, as we discuss in the next section. As additional check of our theory we reproduced the s-wave and p-wave scattering length for He and He given in Gutierrez (), which relate to the HFDHE2 potential of Azis (). This comparison is shown in table 1.

## V Nuclear scattering

For potentials that drop off faster than any polynomial at large all the partial waves can be obtained. To demonstrate this, we apply our method to calculate the low energy scattering parameters for the + C system, where the interaction is short-range and contains a spin-orbit part Baye (). In this latter case, owing to the spin of the neutron, one must specify the total angular momentum, , which can take either one of two values +1/2 or -1/2. The scattering length and the other low-energy parameters acquire two labels . The interaction itself is split into two,

 V(r)=V0(r)+Vso(r)l\textperiodcentereds. (34)

For j = l+1/2, one has

 Vj=l+1/2(r)=V0(r)+l2Vso(r) (35)

and for j = l - 1/2, one has

 Vj=l−1/2(r)=V0(r)−l+12Vso(r). (36)

In the above equations the spherical potential is denoted by and is invariably taken to have a Fermi-type dependence on , the so-called the Woods-Saxon shape

 V0(r)=−V01+exp(r−Rd), (37)

while the spin-orbit potential is given by

 Vso(r)=5.51+exp(r−Rd). (38)

As in section IV.2, we extract the scattering length and effective range from a fit to the low momentum behaviour of . A comparison of our results with those of Ref. Baye () is shown in table 2. As in Baye () we choose fm, and fm, while the strength of the central potential is adjusted for each n-C bound-state energy.

## Vi Conclusion

In this paper we have explored an analogy between quantum potential scattering and the classical dynamics of a conservative system of infinite fields. With the aid of an appropriate canonical transformation of a classical Hamiltonian, we were able to derive the Calogero equation for the tangent of the phase shift function. This allowed us to obtain a first order non-linear differential equation for the so-called scattering length function, whose asymptotic value for large separations supplies the well known low-energy expansion in terms of the scattering length, effective range and shape parameter for any partial wave. We have applied our theory to obtain these parameters for typical atom-atom systems (long-range interaction) and neutron-nucleus systems (short-range interaction). We reached very good agreements with results obtained through conventional methods like a direct solution of the scattering Schrödinger equation.

## Appendix A Essentials of Quantum Potential Scattering Theory

In this appendix we supply the essentials of quantum scattering theory as required in section II. We consider the scattering of a particle from a spherically symmetric potential. The extension to the scattering of a spin-1/2 particle can be easily formulated by adding a spin-orbit interaction term to the potential. The regular solution of the radial Schrödinger equation describing the scattering by a spherical potential, Newton (); CH (),

 [−d2dr2+U(r)+l(l+1)r2]ϕl(k,r)=k2ϕl(k,r) (39)

can be written as,

 ϕl(k,r)=ul(kr)ql(k,r)+vl(kr)pl(k,r), (40)

where and are the Riccati-Bessel and Riccati-Neumann functions, defined in terms of the usual spherical Bessel, , and spherical Neumann functions, , respectively,

 ul(kr)=krjl(kr), (41)
 vl(kr)=krnl(kr). (42)

The functions, and are the regular and irregular solutions of the free radial Schrödinger equation

 [−d2dr2+l(l+1)r2]ωl(kr)=k2ωl(kr), (43)

where is or . Clearly the coefficients and contain all information about the scattering.

Since , we have the following boundary conditions satisfied by and

 ql(k,r)→r→01, (44)
 pl(k,r)→r→00. (45)

Since the functions and are linearly independent in the sense that the Wronskian,

 W[v,u]=vu′−v′u=k, (46)

it follows that and may be expressed as

 ql(k,r)=1kW[vl,ϕl], (47)
 pl(k,r)=−1kW[ul,ϕl]. (48)

The usual method of obtaining the scattering observables, is to integrate the radial equation and adjust the asymptotic form to the following

 ϕl(k,r)→sin(kr−lπ2+δl(k)), (49)

which allows the extraction of the phase shift function . All observables can be written in terms of . For example, the scattering amplitude is just

 f(k,θ)=12ik∞∑l=0(2l+1)(1−exp[2iδl(k)])Pl(cosθ), (50)

with the low-energy limit ( only )

 f(k,0)=1kexp(iδ0(k))sin(δ0(k))=1kcotδ0(k)−ik. (51)

where the function , in the limit of zero energy, is identified with the scattering length.

In the following we enumerate some useful formulae for the Riccati-Bessel and Riccati-Neumann functions used in the last section.

The Riccati-Bessel function is given by

 ul(ρ)=−(−ρ)l+1(1ρddρ)lj0(ρ), (52)

where the spherical Bessel function , and . Similar relation holds for the Riccati-Neumann function

 vl(ρ)=−(−ρ)l+1(1ρddρ)ln0(ρ), (53)

where the spherical Neumann function .
The recursion relation follows

 ωl(ρ)=(2l+1)ωl+1(ρ)ρ−ωl−1(ρ), (54)

where stands for or . For the partial wave, we have

 u1(ρ)=sinρρ−cosρ (55)

and

 v1(ρ)=cosρρ+sinρ. (56)

For higher values of , these functions are easily obtained from the recursion relation above.

Finally, the behaviour of these functions for small values of the argument is

 ul(ρ→0)=ρl+1(2l+1)!! (57)

and

 vl(ρ→0)=(2l−1)!!ρl. (58)

## Appendix B Charged Particle Scattering

When the scattering particles are charged, the Coulomb interaction has to be added to the potential. The free solutions are now Coulomb wave functions and special care must be taken when performing the low-energy expansion to ensure convergent results for the scattering length, effective range and shape parameters. The appropriate low-energy expansion was developed in Teichmann (), and the quantity , becomes in this case

 Dcl(E)=π2exp(2πη)tanδl(E) (59)

where is the Sommerfeld parameter defined in terms of the charges of the two particles and , by , with being the asymptotic relative velocity. The low-energy expansion is then given by

 2ωl(E)(l!)2a2l+1N[2Dcl(E)+h(η)]=−1al+12rlk2−Plr3lk4+O(k6). (60)

In the above, the functions and are, respectively

 ωl(E)=l∏n=1(1+n2η2), (61)
 h(η)=112η2+1120η4+O(1η6), (62)

and is the so-called nuclear Bohr radius, , with being the reduced mass. The above equation does indeed give convergent results as goes to in the zero- limit. Further, the equation reduces to that for neutral particles upon setting .

The corresponding Calogero equation then follows by use of the Wronskian of the scaled Coulomb wave functions

 W[Gl(kr),Fl(kr)]=π2, (63)

where and are related to the regular, , and irregular, , Coulomb wave functions AS ()

 Fl(kr)=k−1/2exp(πη)Fl(kr), (64)

and

 Gl(kr)=π2k−1/2exp(−πη)Gl(kr). (65)

The ”Hamiltonian” function is

 H(ql,pl,r)=U(r)π(Flql+Glpl)2. (66)

The Calogero equation for the tangent function can be derived following the same procedure as the one used for the neutral particle scattering

 (67)

where is just the Coulomb modified tangent function, . Clearly, once this function is calculated, the low energy parameters can be obtained from the expression , above.

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