Numerical solution of the Schroedinger equation for partial waves: scattering from a spherical potential [we choose the attractive exponential potential below, but any potential can be put in] restart; Digits:=14: L:=0; # partial wave number

IiIh

k:=0.5; # wavenumber

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V0:=-2; # strength parameter

ISIj

V:=V0*exp(-0.25*r);

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SE:=diff(u(r),r$2)+(k^2-2*V-L*(L+1)/r^2)*u(r)=0;

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eta:=1E-3; # we can't start the integration at r=0!

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IC:=u(eta)=eta^(L+1),D(u)(eta)=(L+1)*eta^L;

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sol:=dsolve({SE,IC},numeric,output=listprocedure): uS:=eval(u(r),sol): u1S:=eval(diff(u(r),r),sol): R:=25;

IiNE

uR:=uS(R);

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u1R:=u1S(R);

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plot([1*V,uS(r),u1S(r)/k],r=0..3*R,color=[black,red,blue],thickness=2,numpoints=1000);

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L=0: In the r>R region we see a sine/cosine wave (derivative is divided by k!)] How do we extract the phase shift? [we could fit to A*F(k*r) + B*G(k*r) at r=R, then tan(delta)=B/A] Here F and G are the spherical Bessel functions (Bessel-Neumann and Bessel-Ricatti) Is there a simpler strategy? Can we use the asymptotic large-r formula for the spherical Bessel functions? u'/u should go like k*cos(k*r-0.5*L*pi+delta_L)/sin(cos(k*r-0.5*L*pi+delta_L)) for sufficiently large r>R (asymptotic form of the spherical Bessel functions!) a simpler strategy? u'/u should go like k*cos(k*r-0.5*L*pi+delta0)/sin(k*r-0.5*L*pi+delta0) for sufficiently large r>R (asymptotic form of the spherical Bessel functions!) pi:=evalf(Pi);

JCIvKSplYEVmVEohIzg=

d0:=arctan(k*uR/u1R)-k*R+0.5*L*pi;

JCEvYyZbVGY3TCIhIzc=

Does this make sense? How accurate is the answer? R-dependence? possibly eta-dependence tan(d0);

JCEvYmY9P25YIyohIzk=

uR:=uS(2*R);

JCIzZFBCIilvdy1MdSEjPQ==

u1R:=u1S(2*R);

JCEzOXBmQjNJNklLISM9

d0:=arctan(k*uR/u1R)-2*k*R+0.5*L*pi;

JCEvd0l5LmAmZSMhIzc=

tan(d0);

JCEvOCczPD5oIikpISM5

uR:=uS(3*R);

JCIzenQiKnB1bipbJXkhIz0=

u1R:=u1S(3*R);

JCEzdzRUYGxkcndIISM9

d0:=arctan(k*uR/u1R)-3*k*R+0.5*L*pi;

JCEvKilHVEc7VVEhIzc=

tan(d0);

JCEvJ1IlWz5JOikpISM5

uR:=uS(4*R);

JCIzZVphNSoqKnpDQSkhIz0=

u1R:=u1S(4*R);

JCEzIW9gSlkmNC01RiEjPQ==

d0:=arctan(k*uR/u1R)-4*k*R+0.5*L*pi;

JCEvX0A8Kip6KTQmISM3

tan(d0);

JCEvJD5OXy9gIikpISM5

R;

IiNE

uR:=uS(5*R);

JCIzM3kyVSEzZFFjKSEjPQ==

u1R:=u1S(5*R);

JCEzJ2VYR2QjPVJKQyEjPQ==

d0:=arctan(k*uR/u1R)-5*k*R+0.5*L*pi;

JCEvKik9JipwVmJqISM3

tan(d0);

JCEvQldpdUk6KSkhIzk=

We see that the tangent of the phase shift does converge, but the phase shift itself does not when we increase the extraction point (in multiples of R). The arctan function is not able to keep track of the multiples of 2*pi, it answers in the range (-pi/2,pi/2). PhSh:=proc(k,L) local SE,IC,sol,uS,u1S,uR,u1R,d0A1,LL,j; global V,R,eta; SE:=diff(u(r),r$2)+(k^2-2*V-L*(L+1)/r^2)*u(r)=0; IC:=u(eta)=eta^(L+1),D(u)(eta)=(L+1)*eta^L; sol:=dsolve({SE,IC},numeric,output=listprocedure,method=dverk78,maxfun=0): uS:=eval(u(r),sol): u1S:=eval(diff(u(r),r),sol): LL:=[]: for j from 1 to 100 do: uR:=uS(j*R); u1R:=u1S(j*R); d0A1:=arctan(k*uR/u1R)-j*k*R+0.5*L*pi; LL:=[op(LL),[j*R,(d0A1)]]: od: LL; end: PhSh(0.5,0): plot(%);

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# repeat this with tan(delta_0): this takes care of the arctan problem: PhSh:=proc(k,L) local SE,IC,sol,uS,u1S,uR,u1R,d0A1,LL,j; global V,R,eta; SE:=diff(u(r),r$2)+(k^2-2*V-L*(L+1)/r^2)*u(r)=0; IC:=u(eta)=eta^(L+1),D(u)(eta)=(L+1)*eta^L; sol:=dsolve({SE,IC},numeric,output=listprocedure,method=dverk78,maxfun=0): uS:=eval(u(r),sol): u1S:=eval(diff(u(r),r),sol): LL:=[]: for j from 2 to 100 do: uR:=uS(j*R); u1R:=u1S(j*R); d0A1:=arctan(k*uR/u1R)-j*k*R+0.5*L*pi; LL:=[op(LL),[j*R,tan(d0A1)]]: od: LL; end: PhSh(0.5,0): plot(%,style=point);

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# does it really converge over this range of j*R extraction radii? # why do the results seem to slip? # we show that the default solver is doing even worse: PhSh:=proc(k,L) local SE,IC,sol,uS,u1S,uR,u1R,d0A1,LL,j; global V,R,eta; SE:=diff(u(r),r$2)+(k^2-2*V-L*(L+1)/r^2)*u(r)=0; IC:=u(eta)=eta^(L+1),D(u)(eta)=(L+1)*eta^L; sol:=dsolve({SE,IC},numeric,output=listprocedure,method=rkf45,maxfun=0): uS:=eval(u(r),sol): u1S:=eval(diff(u(r),r),sol): LL:=[]: for j from 2 to 100 do: uR:=uS(j*R); u1R:=u1S(j*R); d0A1:=arctan(k*uR/u1R)-j*k*R+0.5*L*pi; LL:=[op(LL),[j*R,tan(d0A1)]]: od: LL; end: PhSh(0.5,0): plot(%,style=point);

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# aha, we shouldn't be too anxious to move to very large extraction radii: the numerical integration to very large r may introduce more error than is healthy. # # special-purpose integrators have been developed to deal with this! (Numerov, DeVogelaere methods are examples) # Now generate the L=0 phase shift as a function of wavenumber. PhSh:=proc(k,L) local SE,IC,sol,uS,u1S,uR,u1R,d0A1,LL,j; global V,R,eta; SE:=diff(u(r),r$2)+(k^2-2*V-L*(L+1)/r^2)*u(r)=0; IC:=u(eta)=eta^(L+1),D(u)(eta)=(L+1)*eta^L; sol:=dsolve({SE,IC},numeric,output=listprocedure,method=dverk78,maxfun=0): uS:=eval(u(r),sol): u1S:=eval(diff(u(r),r),sol): j:=4: uR:=uS(j*R); u1R:=u1S(j*R); d0A1:=arctan(k*uR/u1R)-j*k*R+0.5*L*pi; tan(d0A1): end: dk:=0.005;

JCIiJiEiJA==

PS:=[]: for i from 1 to 100 do: PS:=[op(PS),[i*dk,arctan(PhSh(i*dk,0))]]: od: PS: plot(PS);

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

# what is the corresponding partial L=0 cross section? PCSN:=[seq([PS[j,1],(4*pi/PS[j,1]^2*sin(PS[j,2])^2)],j=1..nops(PS))]: PCSN_u:=[seq([PS[j,1],(4*pi/PS[j,1]^2)],j=1..nops(PS))]: plot([PCSN,PCSN_u],color=[red,blue],thickness=2,view=[0..0.5,0..10000]);

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VyIhIzY3JCQiIiQhIiIkIi9iZixNRSdSIiEjNjckJCIkMCQhIiQkIi95VTcmZjNOIiEjNjckJCIjSiEiIyQiL0FSPltqMjghIzY3JCQiJDokISIkJCIvI1JcMV9rRSIhIzY3JCQiI0shIiMkIi8mMy5qJT1GNyEjNjckJCIkRCQhIiQkIi9HN2hicio9IiEjNjckJCIjTCEiIyQiL01gJnBPUjoiISM2NyQkIiROJCEiJCQiL3d3Iyl5dT42ISM2NyQkIiNNISIjJCIuPjpHY3EzIiEjNTckJCIkWCQhIiQkIi9aKipHZHhiNSEjNjckJCIjTiEiIyQiLzNnc2gjZS0iISM2NyQkIiRiJCEiJCQiLiNmczpMcioqISM2NyQkIiNPISIjJCIvMz5BT0YncCohIzc3JCQiJGwkISIkJCIuWyg+PVdLJSohIzY3JCQiI1AhIiMkIi9jY2xNQnoiKiEjNzckJCIkdiQhIiQkIi8zQHFkM08qKSEjNzckJCIjUSEiIyQiL2NpQ0ZaLSgpISM3NyQkIiQmUSEiJCQiLykpKkdELXpaKSEjNzckJCIjUiEiIyQiLjMqPU8iPkUpISM2NyQkIiQmUiEiJCQiLnk3MXZTMCkhIzY3JCQiIiUhIiIkIi9XKFJqIilSJnkhIzc3JCQiJDAlISIkJCIvJVtCQmA3bSghIzc3JCQiI1QhIiMkIi8vbCd5S2JaKCEjNzckJCIkOiUhIiQkIi9HKHBnJVsnSCghIzc3JCQiI1UhIiMkIi9HR2xHekJyISM3NyQkIiREJSEiJCQiLzdzLC07ZHAhIzc3JCQiI1YhIiMkIi9zblZrSSd6JyEjNzckJCIkTiUhIiQkIi9vS0Z1JzRrJyEjNzckJCIjVyEiIyQiL2NQN1IqM1wnISM3NyQkIiRYJSEiJCQiL09uImVdZU0nISM3NyQkIiNYISIjJCIuLSM9XmgwaSEjNjckJCIkYiUhIiQkIi9zbV90KCpwZyEjNzckJCIjWSEiIyQiL19mdiVRKFFmISM3NyQkIiRsJSEiJCQiLVQzLnI2ZSEjNTckJCIjWiEiIyQiLzddR1lyKW8mISM3NyQkIiR2JSEiJCQiLy93VkRlcGIhIzc3JCQiI1shIiMkIi9DWzdSOmFhISM3NyQkIiQmWyEiJCQiLjlcIW9GVWAhIzY3JCQiI1whIiMkIi9fMS5xIVFCJiEjNzckJCIkJlwhIiQkIi4vZGAyJ0deISM2NyQkIiImISIiJCIvT3VYI1tsLSYhIzctJSZDT0xPUkc2JiUkUkdCRyQiIiEhIiIkIiIhISIiJCIjNSEiIi0lJVZJRVdHNiQ7JCIiISEiIiQiIiYhIiI7JCIiISEiIiQiJysrNSEiIi0lK0FYRVNMQUJFTFNHNidRITYiUSE2Ii0lJUZPTlRHNiUlKERFRkFVTFRHJShERUZBVUxURyIjNSUrSE9SSVpPTlRBTEclK0hPUklaT05UQUxHLSUqVEhJQ0tORVNTRzYjIiIjLSUlUk9PVEc2Jy0lKUJPVU5EU19YRzYjJCIkSSchIiItJSlCT1VORFNfWUc2IyQiJD8iISIiLSUtQk9VTkRTX1dJRFRIRzYjJCIlIVIoISIiLSUuQk9VTkRTX0hFSUdIVEc2IyQiJT9NISIiLSUpQ0hJTERSRU5HNiI=

# Now we see how the unitarity limit is reached when the phase shift hits \134pm pi. # the s-wave cross section starts with a maximum at k=0. (delta0 vanishes, or goes to a multiple of pi, and the division by k^2 results in a constant) # For L>0 we need to match the Bessel functions. Otherwise we run into a slow-convergence problem, since the centrifugal potential fall slowly with r. F:=(k,r,L)->sqrt(0.5*pi*k*r)*BesselJ(L+1/2,k*r);

Zio2JUkia0c2IkkickdGJUkiTEdGJUYlNiRJKW9wZXJhdG9yR0YlSSZhcnJvd0dGJUYlKiYtSSVzcXJ0R0YlNiMqKiQiIiYhIiIiIiJJI3BpR0YlRjNGJEYzRiZGM0YzLUkoQmVzc2VsSkdGJTYkLCZGJ0YzI0YzIiIjRjMqJkYkRjNGJkYzRjNGJUYlRiU=

G:=(k,r,L)->sqrt(0.5*pi*k*r)*(-1)^L*BesselJ(-L-1/2,k*r);

Zio2JUkia0c2IkkickdGJUkiTEdGJUYlNiRJKW9wZXJhdG9yR0YlSSZhcnJvd0dGJUYlKigtSSVzcXJ0R0YlNiMqKiQiIiYhIiIiIiJJI3BpR0YlRjNGJEYzRiZGM0YzKUYyRidGMy1JKEJlc3NlbEpHRiU2JCwmRidGMiNGMyIiI0YyKiZGJEYzRiZGM0YzRiVGJUYl

Fp:=(K,r,L)->evalf(eval(diff(F(K,s,L),s),s=r));

Zio2JUkiS0c2IkkickdGJUkiTEdGJUYlNiRJKW9wZXJhdG9yR0YlSSZhcnJvd0dGJUYlLUkmZXZhbGZHJSpwcm90ZWN0ZWRHNiMtSSVldmFsR0YtNiQtSSVkaWZmR0YtNiQtSSJGR0YlNiVGJEkic0dGJUYnRjgvRjhGJkYlRiVGJQ==

Fp(k,5.0,0);

JCEvWnR4IT1kKyUhIzk=

Gp:=(K,r,L)->evalf(eval(diff(G(K,s,L),s),s=r));

Zio2JUkiS0c2IkkickdGJUkiTEdGJUYlNiRJKW9wZXJhdG9yR0YlSSZhcnJvd0dGJUYlLUkmZXZhbGZHJSpwcm90ZWN0ZWRHNiMtSSVldmFsR0YtNiQtSSVkaWZmR0YtNiQtSSJHR0YlNiVGJEkic0dGJUYnRjgvRjhGJkYlRiVGJQ==

Gp(k,5.0,0);

JCEvKT4wc2dCKkghIzk=

uR:=uS(R);

JCIzb1gqKnBuJjRbOCghIz0=

u1R:=u1S(R);

JCEzeUpxNywnZSV5TCEjPQ==

MatchEq:=u1R/uR=(Fp(k,R,0)+BoA*Gp(k,R,0))/(F(k,R,0)+BoA*G(k,R,0));

LyQhL1ZheXA8TlohIzkqJiwmJCIvSSplUiIqKikpXEYlIiIiKiYkIi8sY25bNDtMISM6RipJJEJvQUc2IkYqRipGKiwmJCIvKj5edCo9S21GLiEiIiomJCIvZHkieiMpeigqKkYlRipGL0YqRipGNA==

arctan(solve(MatchEq,BoA));

JCEvMSg+TU5BWSghIzk=

uR:=uS(2*R);

JCIzKSpRTCIpb3ctTHUhIz0=

u1R:=u1S(2*R);

JCEzKyFITyMzSTZJSyEjPQ==

MatchEq:=u1R/uR=(Fp(k,2*R,0)+BoA*Gp(k,2*R,0))/(F(k,2*R,0)+BoA*G(k,2*R,0));

LyQhL15DL0FpWFYhIzkqJiwmJCIvdUpmUyxjXEYlIiIiKiYkIi8nKSlbXShlPG0hIzpGKkkkQm9BRzYiRipGKkYqLCYkIi94KDRdPE5LIkYlISIiKiYkIi9aaj0iRz8iKipGJUYqRi9GKkYqRjQ=

arctan(solve(MatchEq,BoA));

JCEvWWRWYmlEcyEjOQ==

PhSh2:=proc(k,L) local SE,IC,sol,uS,u1S,uR,u1R,MatchEq,d0A1,LL,j; global V,R,eta; SE:=diff(u(r),r$2)+(k^2-2*V-L*(L+1)/r^2)*u(r)=0; IC:=u(eta)=eta^(L+1),D(u)(eta)=(L+1)*eta^L; sol:=dsolve({SE,IC},numeric,output=listprocedure,method=dverk78,maxfun=0): uS:=eval(u(r),sol): u1S:=eval(diff(u(r),r),sol): LL:=[]: for j from 2 to 100 do: uR:=uS(j*R); u1R:=u1S(j*R); MatchEq:=u1R/uR=(Fp(k,j*R,L)+BoA*Gp(k,j*R,L))/(F(k,j*R,L)+BoA*G(k,j*R,L)); d0A1:=arctan(solve(MatchEq,BoA)); LL:=[op(LL),[j*R,(d0A1)]]: od: end: R:=20;

IiM/

(PhSh2(0.5,1)): plot(%,style=point);

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

# looks OK ? PhSh2:=proc(k,L) local SE,IC,sol,uS,u1S,uR,u1R,MatchEq,d0A1,LL,j; global V,R,eta; SE:=diff(u(r),r$2)+(k^2-2*V-L*(L+1)/r^2)*u(r)=0; IC:=u(eta)=eta^(L+1),D(u)(eta)=(L+1)*eta^L; sol:=dsolve({SE,IC},numeric,output=listprocedure,method=dverk78,maxfun=0): uS:=eval(u(r),sol): u1S:=eval(diff(u(r),r),sol): j:=25; uR:=uS(j*R); u1R:=u1S(j*R); MatchEq:=u1R/uR=(Fp(k,j*R,L)+BoA*Gp(k,j*R,L))/(F(k,j*R,L)+BoA*G(k,j*R,L)); d0A1:=arctan(solve(MatchEq,BoA)); tan(d0A1); end: PS1:=[]: for i from 1 to 100 do: PS1:=[op(PS1),[i*dk,arctan(PhSh2(i*dk,1))]]: od: #PS1; plot(PS1,style=point);

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

PCSN1:=[seq([PS1[j,1],(4*pi/PS1[j,1]^2*sin(PS1[j,2])^2)],j=1..nops(PS1))]: PCSN1_u:=[seq([PS1[j,1],(4*pi/PS1[j,1]^2)],j=1..nops(PS1))]: plot([PCSN1,PCSN1_u],color=[red,blue],thickness=2,view=[0..0.5,0..2000],style=point);

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

# what do we observe? # the L=1 partial cross section starts low at k=0. Very low wavenumbers are dominated by spherical scattering. # # for the attractive potential shown there are resonances in the L=1 partial wave (at k>=0.2 and at k>=0.5).