restart; Digits:=14: with(LinearAlgebra): with(orthopoly): Laguerre basis set for central potentials. L(2,1,x);

LCgiIiQiIiJJInhHNiIhIiQqJEYlIiIjI0YkRik=

Radial orbitals (multiplied by r, leaving out Y_Lm): look almost like hydrogenic orbitals, but form an enumerably infinite set. uL:=(n,l,beta)->r^(l+1)*exp(-beta*r/2)*L(n-1,2*l+2,beta*r);

Zio2JUkibkc2IkkibEdGJUklYmV0YUdGJUYlNiRJKW9wZXJhdG9yR0YlSSZhcnJvd0dGJUYlKigpSSJyR0YlLCY5JSIiIkYwRjBGMC1JJGV4cEdGJTYjLCQqJjkmRjBGLUYwIyEiIiIiI0YwLUkiTEdGJTYlLCY5JEYwRjhGMCwmRi9GOUY5RjBGNUYwRiVGJUYl

plot([seq(uL(n,0,1.5),n=1..4)],r=0..20,color=[red,blue,green,black]);

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#:7$$"2lm;a8ABO"!#<$"1P0mEFI+k!#;7$$",Dc'yM;!#6$"1%)Hxn-)=I(!#;7$$"2ML$e*)4D2>!#<$"18>%G>bU4)!#;7$$"2mmmmT:(z@!#<$"0(4(*pR0%y)!#:7$$"+DJdpK!#5$"2brc2!*yR1"!#;7$$"1LLLL3VfV!#;$"2=&*3l.I68"!#;7$$"2k;zWn+lf%!#<$"1tOXv[3L6!#:7$$".Dc^qN$[!#8$"2#)=PKbR68"!#;7$$"1K3xc.kq]!#;$"2)Q)Q;u`b7"!#;7$$"1mm"z>5xI&!#;$"1TDPBfd;6!#:7$$"1L$3-))\=y&!#;$"2Ggi2e%Q*3"!#;7$$",Dc*)fD'!#6$"2GH^=8W80"!#;7$$"1LL3F*oU?(!#;$"14>$>8[1\*!#;7$$"1nmm"H[D:)!#;$"1=vz,*>0@)!#;7$$"1MLe9w)*=#*!#;$"1<+Js]myl!#;7$$"-v$pU&G5!#6$"113!RZ$QX[!#;7$$"2lmTgi'=N6!#;$"2;BoB`/U4$!#<7$$"2LLL$e0$=C"!#;$"1tz!Rb[#*Q"!#;7$$"2MLLeR"=\8!#;$!2t**Hh`[!=B!#=7$$"2NLLLBKlX"!#;$!2:=lwr'>A<!#<7$$"2OLL32$)Qc"!#;$!2`"o3&))[z0$!#<7$$"2OLL$3RBr;!#;$!2&eM;;ACCU!#<7$$"2-+]P9:\)=!#;$!13GXNaj<g!#;7$$"2mmm"zjf)4#!#;$!1.$p%=1&)=r!#;7$$"2OLL$38l(>#!#;$!10K!3(pt4u!#;7$$",vB1nH#!#5$!1^)3*Ge!Gd(!#;7$$"2NL$3-PBYB!#;$!1)[ga5Z$4w!#;7$$"2lmmm;hdR#!#;$!1xrrdla<w!#;7$$"-DJ')GXC!#6$!1=<Lz8o)f(!#;7$$"2MLLe4;[\#!#;$!1w3j2"[Sb(!#;7$$"1nm;zt%**p#!#:$!1">=/4pL7(!#;7$$",Dmy]!H!#5$!1`;#)QYZlj!#;7$$"1nm;HdA<J!#:$!1R.pFa"3L&!#;7$$"2PLLezs$HL!#;$!0([QlR!y7%!#:7$$"2nmmT]R3a$!#;$!2EwyL&=hOG!#<7$$",D@1Bv$!#5$!2oTa7qDj^"!#<7$$"1MLe9d#)pR!#:$!1fPq,5R4=!#<7$$"1nmm;_M(=%!#:$"2;1'fvBk*3"!#<7$$"1,+]7l$*yV!#:$"0=in8X.8#!#:7$$"1MLL3y_qX!#:$"1il1&o5@3$!#;7$$"1nm;HU@'y%!#:$"0lmi")*\NS!#:7$$"20+++l+>+&!#;$"2mh-6%G>d[!#<7$$"*vW]V&!")$"1v-H!f9x5'!#;7$$"*0_Pk&!")$"1&*fZ5Y7Gl!#;7$$"*NfC&e!")$"1(Q(>(QD&Ro!#;7$$"1nm"Hd&)>/'!#:$"1C#=>jp_.(!#;7$$"1ML$ez6:B'!#:$"1#\v;x/c:(!#;7$$"1n;/,V>Wj!#:$"0W$oD`Y%>(!#:7$$"-D1o(oX'!#6$"0D'[:!)*3@(!#:7$$"1M$e9Jf&pl!#:$"1:0=(R"[1s!#;7$$"1nmm;=C#o'!#:$"1<EnEZx#=(!#;7$$"1ommTb:to!#:$"1:'*QnU'G5(!#;7$$"1nmmm#pS1(!#:$"1o5U_p8zp!#;7$$",DOD#3v!#5$"1F,ae%\(el!#;7$$"1mmmm(y8!z!#:$"1ySr%pq%zg!#;7$$"1,+]i.tK$)!#:$"1*\P=cx0\&!#;7$$",v3zMu)!#5$"1.))Q;O72\!#;7$$"1omm"H_?<*!#:$"2NuHf(*[_I%!#<7$$"1nm;zihl&*!#:$"2$>rVU>FxP!#<7$$"1LLL3#G,***!#:$"1M:D^$4tC$!#;7$$"2ML$ezw5V5!#:$"2YFz!e')R[F!#<7$$"-v$Q#\"3"!#5$"11W2")>yfB!#;7$$"2NLLe"*[H7"!#:$"0_%)phgz)>!#:7$$"*dxd;"!"($"2#4f(yz!3a;!#<7$$",D0xw?"!"*$"2)zt#>j)\t8!#<7$$"2,+]i&p@[7!#:$"2r">6*Q+:9"!#<7$$",vgHKH"!"*$"1Mt6%QyYC*!#<7$$"2lmmmZvOL"!#:$"1BUc5Nb9w!#<7$$"2,++]2goP"!#:$"2ac%3IMCjh!#=7$$"2KL$eR<*fT"!#:$"2&3yv*H..2&!#=7$$"2-++])Hxe9!#:$"2O7.Q.!3"3%!#=7$$"2mm;H!o-*\"!#:$"1A^q4K];L!#<7$$"2,+]7k.6a"!#:$"2&*z"\;[hhE!#=7$$"2mmm;WTAe"!#:$"2$>A$[XN.9#!#=7$$"-D1*3`i"!#5$"1pp`ZZt)p"!#<7$$"2NLLL*zym;!#:$"2J!y*3SCiN"!#=7$$"2OLL3N1#4<!#:$"2U2o_&*pX2"!#=7$$"1nm"HYt7v"!#9$"1bC=eVQ5&)!#=7$$"2-+++xG**y"!#:$"1$yz*p0Hbo!#=7$$"1nm;9@BM=!#9$"1C*)HS'3'Q`!#=7$$"2OLLLbdQ(=!#:$"1y)\J"yogU!#=7$$"-DOl5;>!#5$"2Z!e]r@zVL!#>7$$"2-+]P?Wl&>!#:$"1%4rH:&3ZE!#=7$$"$+#!""$"2$pAx$fjc0#!#>-%&COLORG6&%$RGBG$""!!""$"#5!""$""!!""-%'CURVESG6$7^r7$$""!!""$""!!""7$$"2mm;a8ABO"!#=$"2x%ohV$e5K"!#<7$$"2KLL3FWYs#!#=$"1R[a*\S:c#!#;7$$"-D1k'p3%!#8$"1<B9(*\JCP!#;7$$"2kmm;a)G\a!#=$"1**zTqO<7[!#;7$$",D"G$R<)!#7$"1;h&)Gw$Rx'!#;7$$"2LLL$3x&)*3"!#<$"1WN&*yspn%)!#;7$$"2lm;a8ABO"!#<$"1(*)HgQ'>8**!#;7$$",Dc'yM;!#6$"2G!48h2"H6"!#;7$$"2ML$e*)4D2>!#<$"2-5u!e!3L@"!#;7$$"2mmmmT:(z@!#<$"2A:Pe!)yTH"!#;7$$"1LL$3FWYs#!#;$"1)f4*fxa.9!#:7$$"+DJdpK!#5$"2Zth?FmDX"!#;7$$"2OL$3_v.UN!#<$"2d()ej8&pd9!#;7$$"1nm;z>]9Q!#;$"2$pO4cg[^9!#;7$$"-D1k'p3%!#7$"2k&zBTs.N9!#;7$$"1LLLL3VfV!#;$"1M<E,,Q49!#:7$$"1mm"z>5xI&!#;$"2$\2[vhah7!#;7$$",Dc*)fD'!#6$"2#[]:v;j[5!#;7$$"1m;zW#H,t'!#;$"1`wAx"QvE*!#;7$$"1LL3F*oU?(!#;$"1&yq9-Ec)z!#;7$$".v$4'3%yw!#8$"1ENAh6!em'!#;7$$"1nmm"H[D:)!#;$"1j*y$RW/I`!#;7$$"1,]7`zw&o)!#;$"1/;"y)=RKQ!#;7$$"1MLe9w)*=#*!#;$"2-MUP>W;O#!#<7$$"1n;/ws?_(*!#;$"1R-'p2m-P*!#<7$$"-v$pU&G5!#6$!2NM,:tYlD%!#=7$$"2K$e*)fY'=3"!#;$!1r3^0oj8<!#;7$$"2lmTgi'=N6!#;$!1XI#*eu&o"H!#;7$$"/v=#f3&)="!#8$!2O$e"fLUw-%!#<7$$"2LLL$e0$=C"!#;$!169_G2US]!#;7$$"2MLLeR"=\8!#;$!1dc9LZmon!#;7$$"2NLLLBKlX"!#;$!1iN"R8?V2)!#;7$$"2OLL32$)Qc"!#;$!1x%p3'HRl*)!#;7$$"2OLL$3RBr;!#;$!0&=4lMKk%*!#:7$$"1</wiS%zp"!#:$!1F5EwN:J&*!#;7$$"2,](=<UlC<!#;$!1M#y"[!eid*!#;7$$"2Ne9;Pk8v"!#;$!0*p>@8B+'*!#:7$$"2omTg_u!y<!#;$!1AkL6lo.'*!#;7$$"2-vo/o%y/=!#;$!0$e>MGD(e*!#:7$$"2N$e*[$[\J=!#;$!1&f&4"=t:b*!#;7$$"1<HK*)\?e=!#:$!1f5jS=I(\*!#;7$$"2-+]P9:\)=!#;$!1t')3fB5D%*!#;7$$"2MLe9wb<*>!#;$!1$Q8)p!40(*)!#;7$$"2mmm"zjf)4#!#;$!1&QT]&oH$G)!#;7$$",vB1nH#!#5$!1$\SW))3$Rl!#;7$$"2MLLe4;[\#!#;$!2')Rx"\X7-W!#<7$$",vt"Q(f#!#5$!1NO#\zqt@$!#;7$$"1nm;zt%**p#!#:$!2'=(p+#=u:?!#<7$$"2NLL3-8D!G!#;$!1"fkU;,L@)!#<7$$",Dmy]!H!#5$"1N#3O***p[M!#<7$$"1nm;HdA<J!#:$"1,5HF@e$f#!#;7$$"2PLLezs$HL!#;$"2ckIMG*pFX!#<7$$"2nmmT]R3a$!#;$"1Ad.IWApg!#;7$$",D@1Bv$!#5$"1De4eU`!>(!#;7$$"1n;ajf1hQ!#:$"1=RI%yv8g(!#;7$$"1MLe9d#)pR!#:$"0d(R,Xu,z!#:7$$"1nT5!f0U-%!#:$"17K+N'=:,)!#;7$$".DcY&eyS!#7$"14t'R$Q/&4)!#;7$$"1Me9T`'H8%!#:$"1/i6a">H:)!#;7$$"1nmm;_M(=%!#:$"0"z^t,!e=)!#:7$$"1,]iSICNU!#:$"1(4Gu=>Y>)!#;7$$"1MLek39$G%!#:$"1cFY3_5&=)!#;7$$"1o;a)oQ5L%!#:$"1Z]M,Fzd")!#;7$$"1,+]7l$*yV!#:$"1d7>(yOK6)!#;7$$"1omTg@tuW!#:$"1Y1$o2-Z(z!#;7$$"1MLL3y_qX!#:$"0'fapODux!#:7$$"1nm;HU@'y%!#:$"1"=-`e')>7(!#;7$$"20+++l+>+&!#;$"1TidV*o#Qi!#;7$$")FZ=_!"($"1yJ_CN9v^!#;7$$"*vW]V&!")$"1p^P%y3,*R!#;7$$"*0_Pk&!")$"2W0JVnY"zF!#<7$$"*NfC&e!")$"2`Qz3`j.a"!#<7$$"1n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int(uL(1,0,1.5)^2,r=0..infinity);

JCIvZiNmI2YjZiNmISM5

This basis is orthogonal, but not yet normalized. Orthogonality is demonstrated below. int(uL(1,0,1.5)*uL(2,0,1.5),r=0..infinity);

JCIiIUYj

# Define a basis with fixed parameter beta: phi:=n->uL(n,0,3/2);

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OL:=(m,n)->int(phi(n)*phi(m),r=0..infinity);

Zio2JEkibUc2IkkibkdGJUYlNiRJKW9wZXJhdG9yR0YlSSZhcnJvd0dGJUYlLUkkaW50RzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliR0YlNiQqJi1JJHBoaUdGJTYjOSUiIiItRjI2IzkkRjUvSSJyR0YlOyIiIUkpaW5maW5pdHlHRi1GJUYlRiU=

OL(2,2);

IyIjOyIiKg==

# a few orthogonality checks: OL(1,2);

IiIh

OL(2,4);

IiIh

# Now define the exponential potential and its matrix elements: V0:=-2;

ISIj

r0:=4.0; # to be consistent with the potential used in PH5000L5.mw where phase shifts are calculated.

JCIjUyEiIg==

V:=V0*exp(-r/r0);

LCQtSSRleHBHNiQlKnByb3RlY3RlZEdJKF9zeXNsaWJHNiI2IywkSSJyR0YoJCEvKysrKysrRCEjOSEiIw==

VME:=(m,n)->int(phi(n)*V*phi(m),r=0..infinity);

Zio2JEkibUc2IkkibkdGJUYlNiRJKW9wZXJhdG9yR0YlSSZhcnJvd0dGJUYlLUkkaW50RzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliR0YlNiQqKC1JJHBoaUdGJTYjOSUiIiJJIlZHRiVGNS1GMjYjOSRGNS9JInJHRiU7IiIhSSlpbmZpbml0eUdGLUYlRiVGJQ==

VME(1,1);

JCEvP0peb2JqdSEjOQ==

Had we defined the basis with a non-float (using 15/10 instead of 1.5) the numbers would be computed symbolically exactly. VME(2,1);

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VME(3,1);

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VME(1,6);

JCEvdidSVGdiSyohIzw=

TME:=(m,n)->-1/2*int(phi(n)*diff(phi(m),r$2),r=0..infinity);

Zio2JEkibUc2IkkibkdGJUYlNiRJKW9wZXJhdG9yR0YlSSZhcnJvd0dGJUYlLCQtSSRpbnRHNiQlKnByb3RlY3RlZEdJKF9zeXNsaWJHRiU2JComLUkkcGhpR0YlNiM5JSIiIi1JJWRpZmZHRi42JC1GMzYjOSQtSSIkR0YuNiRJInJHRiUiIiNGNi9GQDsiIiFJKWluZmluaXR5R0YuIyEiIkZBRiVGJUYl

TME(1,1);

IyIiIiIiJw==

TME(1,3);

IyIiIiIiJA==

TME(1,2),TME(2,1);

NiQjIiIiIiIkRiM=

TME(10,10);

IyIkOigiIic=

TME(10,11);

IyIkJlEiIiQ=

The kinetic energy is not diagonal, but symmetric! Nb:=10;

IiM1

VM:=Matrix(Nb,Nb,shape=symmetric): HM:=Matrix(Nb,Nb,shape=symmetric): OM:=Matrix(Nb,Nb,shape=symmetric): for m from 1 to Nb do: for n from 1 to m do: VM[m,n]:=VME(m,n): OM[m,n]:=OL(m,n): od: od: #print(map(evalf,VM)); for m from 1 to Nb do: for n from 1 to m do: HM[m,n]:=evalf(VM[m,n])+evalf(TME(m,n)): od: od: E10:=(convert(sort(Re(Eigenvalues(HM,OM))),list));

NywkITMqKio0L3ZzeCk0NSEjPCQhM3c+dFBqTCM0J1whIz0kITMsXUdXXktQTkBGKCQhMyIqNHAmbzQqKltJJyEjPiQhM0hJNnMjSDoncCkpISNAJCIzKikqXGEkeWhDSFhGLSQiMygpelBGLE1LXjhGKCQiMykpNDZAeCN6SkMlRigkIjMtIT11XE0iR0U+RiUkIjMkKio0KCpbeSIqeT4iISM7

Nb:=20;

IiM/

VM:=Matrix(Nb,Nb,shape=symmetric): HM:=Matrix(Nb,Nb,shape=symmetric): OM:=Matrix(Nb,Nb,shape=symmetric): for m from 1 to Nb do: for n from 1 to m do: VM[m,n]:=VME(m,n): OM[m,n]:=OL(m,n): od: od: #print(map(evalf,VM)); for m from 1 to Nb do: for n from 1 to m do: HM[m,n]:=evalf(VM[m,n])+evalf(TME(m,n)): od: od: E20:=(convert(sort(Re(Eigenvalues(HM,OM))),list));

NzYkITMoKlI1KlJbeSk0NSEjPCQhMykqZmo4ZngvaFwhIz0kITMuSUFNUHlyTkBGKCQhMzNTaSo0KVFxM2ohIz4kITM1XW11XF0lUnEkISM/JCIzJypmWyQpWyx1eVxGMCQiMyoqZnMlXE5nc24iRi0kIjNIcVsvZz5aJ1wkRi0kIjMkKjRLWTF1O3BnRi0kIjNwPytmJUhDLWcqRi0kIjMtNVxkJ2U9PVciRigkIjMxK1MzYVk2MUBGKCQiMyIqNEUkKSo0QkYvJEYoJCIzN2cpZSU9SEc9V0YoJCIzXUltJkdjaSp6bEYoJCIzLiFSdmE/LChcNUYlJCIzJypwOl1YMjcwPkYlJCIzQyEpUS84Y0BtU0YlJCIzM11IYDU/VCEzIiEjOyQiMy1dejJPMnlMW0ZN

# We see that the first 4 negative-E eigenvalues are confirmed in their lower significant digits. # Problem 10.8 in Quantum Mechanics by Ghatak and Lokanathan, QC174.12 G485 2004 asks to transform the SE for the s-states of an exp. potential to the Bessel equation. # The exact s-state eigenvalues are obtained from: g:=sqrt(r0^2*abs(8*V0));

JCIvKysrKysrOyEjNw==

plot(BesselJ(nu,g),nu=0..30);

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

# Five bound states exist in this case: the roots wrt nu are to be found: nu0:=fsolve(BesselJ(nu,g),nu=4..5);

LUknZnNvbHZlRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiNiUtSShCZXNzZWxKR0YkNiRJI251R0YnJCIvKysrKysrOyEjN0YsOyIiJSIiJg==

E0:=-nu0^2/(8.)/r0^2; # this is the energy for given nu.

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nu1:=fsolve(BesselJ(nu,g),nu=1.5..2);

LUknZnNvbHZlRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiNiUtSShCZXNzZWxKR0YkNiRJI251R0YnJCIvKysrKysrOyEjN0YsOyQiIzohIiIiIiM=

E1:=-evalf(nu1^2/(8.)/r0^2);

LCQqJC1JJ2Zzb2x2ZUc2JCUqcHJvdGVjdGVkR0koX3N5c2xpYkc2IjYlLUkoQmVzc2VsSkdGJjYkSSNudUdGKSQiLysrKysrKzshIzdGLjskIiM6ISIiIiIjRjYkIS8rKysrXTd5ISM7

nu2:=fsolve(BesselJ(nu,g),nu=5..6);

JCIvPE45eFxHXyEjOA==

E2:=-evalf(nu2^2/(8.)/r0^2);

JCEvVHdSeXJOQCEjOQ==

nu3:=fsolve(BesselJ(nu,g),nu=2..4);

JCIvQzpmI3o7JUchIzg=

E3:=-evalf(nu3^2/(8.)/r0^2);

JCEvWGE6UnEzaiEjOg==

nu4:=fsolve(BesselJ(nu,g),nu=0.5..1.5);

JCIvPzVsYjwvcCEjOQ==

E4:=-evalf(nu4^2/(8.)/r0^2);

JCEvPkUkUk1TcyQhIzs=

# We can study systematically for each of the 5 eigenvalues how well the numerical calculation is doing. # We optimize the calculation wrt the basis parameter. phi:=n->uL(n,0,1.0);

Zio2I0kibkc2IkYlNiRJKW9wZXJhdG9yR0YlSSZhcnJvd0dGJUYlLUkjdUxHRiU2JTkkIiIhJCIjNSEiIkYlRiVGJQ==

Diag:=proc(Nb,beta,N_E) local phi,VME,TME,OL,VM,HM,OM,m,n,res; phi:=n->uL(n,0,beta); VME:=(m,n)->int(phi(n)*V*phi(m),r=0..infinity); TME:=(m,n)->-1/2*int(phi(n)*diff(phi(m),r$2),r=0..infinity); OL:=(m,n)->int(phi(n)*phi(m),r=0..infinity); VM:=Matrix(Nb,Nb,shape=symmetric): HM:=Matrix(Nb,Nb,shape=symmetric): OM:=Matrix(Nb,Nb,shape=symmetric): for m from 1 to Nb do: for n from 1 to m do: VM[m,n]:=VME(m,n): OM[m,n]:=OL(m,n): od: od: for m from 1 to Nb do: for n from 1 to m do: HM[m,n]:=evalf(VM[m,n])+evalf(TME(m,n)): od: od: res:=(convert(sort(Re(Eigenvalues(HM,OM))),list)); [seq(res[j],j=1..N_E)]; end: Diag(5,1,5);

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Diag(5,1/2,5);

NyckITNxUlZZIj1WaCEqKiEjPSQhMyIpSE82LiZlWkUlRiUkITMxU3M7ckopPkEiRiUkITMpKnB1YzMoUkZUIiEjPiQiMyUpcCJ5QCYpPSEzRyEjPw==

# A smaller beta-value results in worse (higher) energies for all states, except the last one (which is still coming out as unbound). Note this 5-by-5 calculation has only 5 eigenvalues! j:=4; # pick the state for which the analysis is carried out:

IiIl

LL5:=[]: for i from 1 to 20 do: beta:=3/10*i; LL5:=[op(LL5),[beta,Diag(5,beta,5)[j]]]: od: PL1:=plot(LL5): beta:='beta': PL0:=plot(E||(j-1),beta=0..2,color=green): plots[display](PL0,PL1,view=[0..2,-0.1..0.2]);

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

j;#

IiIl

#LL5; # Exercise: find the minimum value from the 5-by-5 calculation by optimizing wrt beta; how big is the relative error? # Try 10-by-10 calculations: LL10:=[]: for i from 1 to 5 do: beta:=1+1/10*i; LL10:=[op(LL10),[beta,Diag(10,beta,5)[j]]]: od: PL2:=plot(LL10,color=blue,style=point,symbol=cross): plots[display](PL0,PL1,PL2,view=[0..2,-0.065..-0.035]);

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

# we see that the 10-by-10 diagonalization is less sensitive to the basis parameter beta. # Exercise: track the relative error (compared to the exact answer) as a function of beta. # Now ask the questions: is there a bound state in the L=1 sector (p-wave)? Diag:=proc(Nb,beta,L,N_E) local phi,VME,TME,OL,VM,HM,OM,m,n,res; phi:=n->uL(n,L,beta); VME:=(m,n)->int(phi(n)*(V+L*(L+1)/2/r^2)*phi(m),r=0..infinity); TME:=(m,n)->-1/2*int(phi(n)*diff(phi(m),r$2),r=0..infinity); OL:=(m,n)->int(phi(n)*phi(m),r=0..infinity); VM:=Matrix(Nb,Nb,shape=symmetric): HM:=Matrix(Nb,Nb,shape=symmetric): OM:=Matrix(Nb,Nb,shape=symmetric): for m from 1 to Nb do: for n from 1 to m do: VM[m,n]:=VME(m,n): OM[m,n]:=OL(m,n): od: od: for m from 1 to Nb do: for n from 1 to m do: HM[m,n]:=evalf(VM[m,n])+evalf(TME(m,n)): od: od: res:=(convert(sort(Re(Eigenvalues(HM,OM))),list)); [seq(res[j],j=1..N_E)]; end: Diag(5,1,1,1);

NyMkITNsUkhZKHB6ISlmJyEjPQ==

Diag(5,2,1,1);

JCEzKikqXGlIVlhObCchIz0=

Diag(5,3,1,1);

JCEzJSpSb104LS9hbSEjPQ==

Diag(5,4,1,3);

NyUkITNUU0FwaUU0Vm0hIz0kITM1NXg6YTlkVj5GJSQiM19IaSp6OyxALSlGJQ==

Diag(10,4,1,3);

NyUkITNNZ0w8RUYvYW0hIz0kITNGU1ApKXpqNSY0JEYlJCEzTj5meTFLWiZlKSEjPg==

Diag(20,4,1,3);

NyUkITNwPnZgdUYvYW0hIz0kITN5PmY3LF8zKDQkRiUkITMrK3RaQjM7NTZGJQ==

Diag(30,4,1,4);

NyYkITNwUkFkdUYvYW0hIz0kITMkKkgibzg/JjMoNCRGJSQhMzAhKTRGPTlANTZGJSQhMy4hWyVvUDhUIXAiISM+

Diag(30,5,1,5);

NyckITNvejg1dUYvYW0hIz0kITNDPz8+LF8zKDQkRiUkITMwUyNlJmUlNC02IkYlJCEzKikqZTMqNCdIJiplIiEjPiQiM0tdWi1fNTJCTkYs

Diag(30,3,1,5);

NyckITNJSTRrdUYvYW0hIz0kITMyPyNlNz8mMyg0JEYlJCEzJSpmXCM9VTYtNiJGJSQhMykqKkg9ckVtLXEiISM+JCIzRCFlIW88XXJRbiEjPw==

Diag(30,2,1,5);

NyckITNCNU5ddUYvYW0hIz0kITMjKSoqPkosXzMoNCRGJSQhMyw/dyU9VTYtNiJGJSQhMzs1cXB6Q1crPCEjPiQiMy0/aUAoSCpcN0QhIz8=

# For L>0 we do not have an exact result to compare against. Thus, we need to be diligent when determining the best results from matrix diagonalization.