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{SECT 0 {EXCHG {PARA 0 "" 0 "" {TEXT 260 42 "Scattering from a spheric
al potential well" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" 
{TEXT -1 334 "Our interest is in assembling the scattering amplitude f
or a spherical potential barrier or well. We calculate the phase shift
s as a function of energy using the matching condition, and assemble t
he scattering amplitude. We use a simple unit system (h-bar=m=1), and \+
measure length as a multiple of the scale set by the potential well." 
}}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 25 "restart; with(orthopoly);
" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#7(%\"GG%\"HG%\"LG%\"PG%\"TG%\"UG" 
}}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 47 "R:=1; U0:=9/10; # (U0>0 =
 barrier, U0<0 = well)" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"RG\"\"\"
" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#U0G#\"\"*\"#5" }}}{EXCHG {PARA 
0 "> " 0 "" {MPLTEXT 1 0 5 "k:=1;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>
%\"kG\"\"\"" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 20 "kappa:=sqrt(
k^2-U0);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%&kappaG,$*$-%%sqrtG6#\"#
5\"\"\"#F+F*" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 46 "Now we need the t
wo types of Bessel functions." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 
0 51 "jn:=(n,x)->simplify(sqrt(Pi/2/x)*BesselJ(n+1/2,x));" }}{PARA 11 
"" 1 "" {XPPMATH 20 "6#>%#jnGR6$%\"nG%\"xG6\"6$%)operatorG%&arrowGF)-%
)simplifyG6#*&-%%sqrtG6#,$*&%#PiG\"\"\"9%!\"\"#F7\"\"#F7-%(BesselJG6$,
&9$F7F:F7F8F7F)F)F)" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 12 "assu
me(r>0);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 8 "jn(2,r);" }}
{PARA 11 "" 1 "" {XPPMATH 20 "6#,$*&,(*&-%$sinG6#%#r|irG\"\"\")F*\"\"#
F+F+*&\"\"$F+F'F+!\"\"*(F/F+-%$cosGF)F+F*F+F+F+*$)F*F/F+F0F0" }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 63 "nn:=(n,x)->simplify((-1)^(n+
1)*sqrt(Pi/2/x)*BesselJ(-n-1/2,x));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6
#>%#nnGR6$%\"nG%\"xG6\"6$%)operatorG%&arrowGF)-%)simplifyG6#*()!\"\",&
9$\"\"\"F5F5F5-%%sqrtG6#,$*&%#PiGF59%F2#F5\"\"#F5-%(BesselJG6$,&F4F2#F
5F>F2F<F5F)F)F)" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 8 "nn(2,r);
" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#*&,(*&-%$cosG6#%#r|irG\"\"\")F)\"
\"#F*F**&\"\"$F*F&F*!\"\"*(F.F*-%$sinGF(F*F)F*F/F**$)F)F.F*F/" }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 21 "ul:=(l,x)->x*jn(l,x);" }}
{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#ulGR6$%\"lG%\"xG6\"6$%)operatorG%&a
rrowGF)*&9%\"\"\"-%#jnG6$9$F.F/F)F)F)" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 21 "vl:=(l,x)->x*nn(l,x);" }}{PARA 11 "" 1 "" {XPPMATH 
20 "6#>%#vlGR6$%\"lG%\"xG6\"6$%)operatorG%&arrowGF)*&9%\"\"\"-%#nnG6$9
$F.F/F)F)F)" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 121 "The regular and i
rregular solutions to the radial equations in terms of the Ricatti-Bes
sel and Ricatti-Neumann functions." }}{PARA 0 "" 0 "" {TEXT -1 26 "We \+
need their derivatives:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 46 "
ulp:=unapply('simplify(diff(ul(l,x),x))',l,x);" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6#>%$ulpG-%\"@G6$%)simplifyGR6$%\"lG%\"xG6\"6$%)operatorG
%&arrowGF--%%diffG6$-%#ulG6$9$9%F8F-F-F-" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 9 "ulp(2,r);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,$*&,**&-
%$sinG6#%#r|irG\"\"\")F*\"\"#F+!\"$*&\"\"'F+F'F+F+*(F0F+-%$cosGF)F+F*F
+!\"\"*&)F*\"\"$F+F2F+F+F+*$F6F+F4F4" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 46 "vlp:=unapply('simplify(diff(vl(l,x),x))',l,x);" }}
{PARA 11 "" 1 "" {XPPMATH 20 "6#>%$vlpG-%\"@G6$%)simplifyGR6$%\"lG%\"x
G6\"6$%)operatorG%&arrowGF--%%diffG6$-%#vlG6$9$9%F8F-F-F-" }}}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 9 "vlp(2,r);" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6#,$*&,**&-%$cosG6#%#r|irG\"\"\")F*\"\"#F+\"\"$*&\"\"'F+F
'F+!\"\"*(F0F+-%$sinGF)F+F*F+F1*&)F*F.F+F3F+F+F+*$F6F+F1F1" }}}{EXCHG 
{PARA 0 "" 0 "" {TEXT -1 76 "Now we are ready for the matching conditi
on which determines the phaseshift." }}{PARA 0 "" 0 "" {TEXT -1 80 "Th
e derivative functions defined above do not work with composite argume
nts yet:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 11 "vlp(1,k*r);" }}
{PARA 11 "" 1 "" {XPPMATH 20 "6#*&,(*&-%$sinG6#%#r|irG\"\"\"F)F*F*-%$c
osGF(F**&F+F*)F)\"\"#F*!\"\"F**$F.F*F0" }}}{EXCHG {PARA 0 "" 0 "" 
{TEXT -1 135 "We need to specify a simple argument, and then substitut
e. Our hope was that unapply would take care of that, but, of course, \+
with the " }{TEXT 262 1 "l" }{TEXT -1 97 "-value unspecified nothing c
an be done. Therefore, we define the required function for specified \+
" }{TEXT 261 1 "l" }{TEXT -1 58 " inside the loop. The trick is to for
ce the evaluation of " }{TEXT 19 3 "ulp" }{TEXT -1 5 " and " }{TEXT 
19 3 "vlp" }{TEXT -1 85 " before unapplying the answer (this does not \+
work with the simple mapping construct)." }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 9 "Lmax:=10;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%%LmaxG
\"#5" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 24 "for l from 0 to Lma
x do:" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 85 "rho:=kappa*R; Ulp:=unapply
(simplify(ulp(l,r)),r): Vlp:=unapply(simplify(vlp(l,r)),r):" }}{PARA 
0 "> " 0 "" {MPLTEXT 1 0 103 "taneta[l]:=evalf((Ulp(k*R)-Ulp(rho)*ul(l
,k*R)/ul(l,rho))/(-Vlp(k*R)+Ulp(rho)*vl(l,k*R)/ul(l,rho))); od:" }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 14 "print(taneta);" }}{PARA 12 "
" 1 "" {XPPMATH 20 "6#-%&TABLEG6#7-/\"\"!$\"+z^>\\\")!#5/\"\"\"$\"+)3;
sT\"F+/\"\"#$\"+\"4x\\-\"!#6/\"\"$$\"+GqjnI!#8/\"\"%$\"+!)oa3e!#:/\"\"
&$!+%Rh>7\"!#;/\"\"'$!+&*\\rxx!#>/\"\"($F-%*undefinedG/\"\")$!\"!F(/\"
\"*FK/\"#5FK" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 52 "For very small ph
ase shift (which happens for large " }{TEXT 263 1 "l" }{TEXT -1 22 " a
t finite wavenumber " }{TEXT 264 1 "k" }{TEXT -1 69 ") the number can \+
be undefined due to the lack of numerical precision." }}{PARA 0 "" 0 "
" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 394 "For scattering from re
pulsive potential barriers the above matching condition works only whe
n the particle can penetrate the barrier as the solution is assumed to
 be of a simple scattering type there (regular scattering solution). F
or scattering energies below the barrier height the particle would be \+
tunneling into the potential barrier, and an exponentially dying funct
ion would be required." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 61 "s
igma:=4*Pi/k^2*add((2*l+1)*sin(arctan(taneta[l]))^2,l=0..3);" }}{PARA 
11 "" 1 "" {XPPMATH 20 "6#>%&sigmaG,$%#PiG$\"+QvmM=!\"*" }}}{EXCHG 
{PARA 0 "" 0 "" {TEXT -1 31 "The partial-wave contributions:" }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 59 "4*Pi/k^2*[seq((2*l+1)*sin(ar
ctan(taneta[l]))^2,l=0..Lmax)];" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#,$*
&%#PiG\"\"\"7-$\"+MMs!*R!#5$\"+`['o!f!#6$\"+&QQBD&!#8$\"+OsF(e'!#;$\"+
<&Hl.$!#>$\"+2rn%Q\"!#A$\"+q02ky!#F$F&%*undefinedG$\"\"!F@F=F=F&\"\"%
" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 130 "for Lm from 0 to Lmax \+
do: dsdO[Lm]:=1/4/k^2*evalc(abs(add((2*l+1)*(exp(2*I*arctan(taneta[l])
)-1)*P(l,cos(theta)),l=0..Lm))^2); od:" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 77 "plot([seq(dsdO[Lm],Lm=0..4)],theta=0..Pi,color=[red,b
lue,green,brown,black]);" }}{PARA 13 "" 1 "" {GLPLOT2D 694 299 299 
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X8F,7$F^t$\"3w.C1e8xH8F,7$Fat$\"3xUDH'>%)yJ\"F,7$Fdt$\"3?\"fUQkq5J\"F,
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VIEWG6$;F($\"+aEfTJ!\"*%(DEFAULTG" 1 2 0 1 10 0 2 9 1 4 2 1.000000 
45.000000 45.000000 0 0 "Curve 1" "Curve 2" "Curve 3" "Curve 4" "Curve
 5" }}}}{EXCHG {PARA 0 "" 0 "" {TEXT 256 11 "Exercise 1:" }}{PARA 0 "
" 0 "" {TEXT -1 127 "Explore the convergence of the partial-wave expan
sion of the differential cross section for different values of the wav
enumber " }{TEXT 257 1 "k" }{TEXT -1 38 " (variation of the scattering
 energy)." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 
258 11 "Exercise 2:" }}{PARA 0 "" 0 "" {TEXT -1 110 "Compare the diffe
rential cross section at fixed energy for repulsive and attractive pot
ential well scattering." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 
0 "" {TEXT 259 11 "Exercise 3:" }}{PARA 0 "" 0 "" {TEXT -1 402 "Explor
e the dependence of the differential cross section at fixed energy on \+
the size of the potential well (or barrier). Comment in particular on \+
the amount of backscattering. Observe what happens in scattering from \+
barriers of growing size when the scattering energy is just sufficient
 for barrier penetration without tunneling. What is the meaning of the
 structures in the differential cross section?" }}{PARA 0 "" 0 "" 
{TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 265 17 "Energy dependenc
e" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 225 "Let
 us calculate the total scattering cross section as a function of ener
gy in order to demonstrate the convergence of the partial wave expansi
on at low energies. We need to automate the calculation of the matchin
g condition." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 12 "with(plots)
:" }}{PARA 7 "" 1 "" {TEXT -1 50 "Warning, the name changecoords has b
een redefined\n" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 45 "R:=2; U0
:=-4; # (U0>0 = barrier, U0<0 = well)" }}{PARA 11 "" 1 "" {XPPMATH 20 
"6#>%\"RG\"\"#" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#U0G!\"%" }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 9 "Lmax:=10;" }}{PARA 11 "" 1 "
" {XPPMATH 20 "6#>%%LmaxG\"#5" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 
0 69 "for ik from 1 to 10 do: k:=ik/10; kappa:=sqrt(k^2-U0); Ev[ik]:=k
^2/2;" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 24 "for l from 0 to Lmax do:" 
}}{PARA 0 "> " 0 "" {MPLTEXT 1 0 85 "rho:=kappa*R; Ulp:=unapply(simpli
fy(ulp(l,r)),r): Vlp:=unapply(simplify(vlp(l,r)),r):" }}{PARA 0 "> " 
0 "" {MPLTEXT 1 0 102 "taneta[ik,l]:=evalf((Ulp(k*R)-Ulp(rho)*ul(l,k*R
)/ul(l,rho))/(-Vlp(k*R)+Ulp(rho)*vl(l,k*R)/ul(l,rho)));" }}{PARA 0 "> \+
" 0 "" {MPLTEXT 1 0 58 "sigma[ik,l]:=4*Pi/k^2*(2*l+1)*sin(arctan(tanet
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display(P1);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%%LmaxG\"\"!" }}
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{MPLTEXT 1 0 108 "P2:=plot([seq([Ev[ik],log10(add(sigma[ik,l],l=0..Lma
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x))],ik=1..10)],style=point,color=black): display(P1,P2,P3,P4);" }
{TEXT -1 0 "" }{MPLTEXT 1 0 0 "" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%%
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e 4" }}}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 216 "We see that at low scatt
ering energies the cross section is dominated by isotropic (s-wave) sc
attering. As the energy increases the higher partial waves begin to co
ntribute more to the total scattering cross section." }}}{EXCHG {PARA 
0 "" 0 "" {TEXT 266 11 "Exercise 4:" }}{PARA 0 "" 0 "" {TEXT -1 278 "E
xplore the energy dependence of the total scattering cross section for
 attractive potential barriers of increasing depth and size. What is t
he interpretation of the minimum in the total cross section at low ene
rgies (which appears for potential wells of sufficient depth/size)?" }
}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}}{MARK "0 0 0" 2 }
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