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{SECT 0 {EXCHG {PARA 0 "" 0 "" {TEXT 256 55 "Eigenstates of a one-dime
nsional Coulomb-type potential" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}
{PARA 0 "" 0 "" {TEXT -1 302 "A number of problems involving atom-lase
r interactions have been modeled revcently by a one-dimensional potent
ial. Such an approach may be justified when the linearly polarized las
er field affects the bound electrons so strongly that the spherical na
ture of the atomic potential becomes less important." }}{PARA 0 "" 0 "
" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 60 "We use this potential t
o demonstrate matrix diagonalization." }}{PARA 0 "" 0 "" {TEXT -1 0 "
" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 20 "restart; Digits:=15:" }
}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 33 "with(LinearAlgebra): with(
plots):" }}{PARA 7 "" 1 "" {TEXT -1 50 "Warning, the name changecoords
 has been redefined\n" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 244 "The mat
rix representation of the problem is obtained by choosing a basis and \+
calculating matrix elements as integrals. The ladder approach is very \+
powerful for simple polynomial-type potentials, but not for our potent
ial given in Bohr units as:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 
18 "V:=-1/sqrt(1+x^2);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"VG,$*&\"
\"\"F'*$,&F'F'*$)%\"xG\"\"#F'F'#F'F-!\"\"F/" }}}{EXCHG {PARA 0 "> " 0 
"" {MPLTEXT 1 0 66 "PLV:=plot(V,x=-5..5,view=[-5..5,-1..0],thickness=2
): display(PLV);" }}{PARA 13 "" 1 "" {GLPLOT2D 618 206 206 {PLOTDATA 
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45.000000 0 0 "Curve 1" }}}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 189 "The p
otential has a long-range tail. It is bounded from below at x=0. The u
ncertainty principle is not strong enough to prevent collapse in the 1
d case, i.e., V(x)=-1/|x| is not acceptable." }}{PARA 0 "" 0 "" {TEXT 
-1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 112 "The harmonic oscillator basis \+
with some choice of oscillator constant should work for the bound stat
es at least." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 
-1 22 "We pick a matrix size:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 
0 6 "N:=25;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"NG\"#D" }}}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 32 "H0:=Matrix(N,N,shape=symmetric):" }
}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 40 "The matrix is initialized to zer
o, e.g.:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 8 "H0[2,3];" }}
{PARA 11 "" 1 "" {XPPMATH 20 "6#\"\"!" }}}{EXCHG {PARA 0 "" 0 "" 
{TEXT -1 38 "We pick the Bohr unit system in which " }{TEXT 19 16 "hba
r=1, m=1, e=1" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 159 "The ha
rmonic oscillator basis depends on a scale parameter. The spring const
ant is chosen in Bohr units, or we simply set the circular frequency i
n Bohr units." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 12 "omega:=0.2
5:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 55 "lambda:=sqrt(omega); \+
# sqrt(m*omega/hbar) is the scale!" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#
>%'lambdaG$\"0+++++++&!#:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 
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{MPLTEXT 1 0 22 "plot(phi(24),x=-X..X);" }}{PARA 13 "" 1 "" {GLPLOT2D 
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%+AXESLABELSG6$Q\"x6\"Q!F_bt-%%VIEWG6$;F(Faat%(DEFAULTG" 1 2 0 1 10 0 
2 9 1 4 2 1.000000 45.000000 45.000000 0 0 "Curve 1" }}}}{EXCHG {PARA 
0 "> " 0 "" {MPLTEXT 1 0 60 "for i from 1 to N do: n:=i-1: for j from \+
1 to i do: np:=j-1:" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 96 "H0[i,j]:=eva
lf[10](Int(phi(np)*(-1/2*diff(phi(n),x$2)+V*phi(n)),x=-X..X,method=_Gq
uad)): od: od:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 53 "EV:=conve
rt(Eigenvalues(H0),list): seq(EV[n],n=1..5);" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6'$!0\\4JRgvp'!#:$!0=,>`!z[FF%$!0rYOi%)G^\"F%$!0E1iq!*Q$)
)!#;$!06'*R/')QL%F," }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 143 "Now assem
ble the eigenfunctions for these states. For this we need to have acce
ss to the eigenvectors which contain the expansion coefficients." }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 23 "EVEC:=Eigenvectors(H0):" }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 22 "convert(EVEC[1],list);" }}
{PARA 12 "" 1 "" {XPPMATH 20 "6#7;$!0e4JRgvp'!#:$!0;,>`!z[FF&$!0QYOi%)
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\"R.;FB$\"0sWWbBH'>FB$\"0UW<3,o8#FB$\"0<_fjC3d#FB$\"0C7h5O8w#FB$\"0%*f
i@p)pLFB$\"0(3a)*R9yNFB$\"0$\\Z=H5IVFB$\"0uo0Mbub%FB" }}}{EXCHG {PARA 
0 "> " 0 "" {MPLTEXT 1 0 14 "nops(EVEC[2]);" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6#\"\"$" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 13 "EVE
C[2][1,2];" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$\"%(p\"!#?" }}}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 57 "?Eigenvectors # explains how the ei
genvectors are stored." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 63 "V
1:=convert(convert(SubMatrix(EVEC[2],1..N,1..1),Vector),list);" }}
{PARA 12 "" 1 "" {XPPMATH 20 "6#>%#V1G7;$\"0z;$RA&f#)*!#:$\"0q&)zW9Kp
\"!#J$!0U[*4gs(f\"F($\"0\"\\\"pQ?=A&F+$\"0ZTfC8]?)!#;$\"0%R#Gbyxq#F+$!
0<B+*=*=t$F2$!0zjRl5'z:!#I$\"0#p+&o#ypAF2$!0J')e#4<9dF+$!0@XD8$\\\\8F2
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FD$!#DF2$!0\")RmhA@T\"FD$\"$3*FD$\"0c$4'R#)3(=FD" }}}{EXCHG {PARA 0 "
" 0 "" {TEXT -1 166 "Why are there many zeros? The antisymmetric harmo
nic oscillator eigenstates are not needed to construct the ground stat
e which is of even parity (symmetric function)." }}}{EXCHG {PARA 0 "> \+
" 0 "" {MPLTEXT 1 0 33 "psi1:=add(V1[i]*phi(i-1),i=1..N):" }}}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 99 "PL1:=plot([EVEC[1][1],EVEC[1][1]+ps
i1^2],x=-5..5,color=[black,blue],thickness=2): display(PLV,PL1);" }}
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" {TEXT -1 264 "This graph shows the turning points (intercepts of the
 ground-state energy and the potential), and the density of the state \+
in such a way that the relative probability for the quantum particle t
o spend time in the classically forbidden region is displayed visibly.
" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 260 11 "Exer
cise 1:" }}{PARA 0 "" 0 "" {TEXT -1 304 "Explore some of the other sta
tes in a similar fashion. Verify the accuracy (or rather inaccuracy) o
f the highest negative-energy state of the calculation by: (i) increas
ing the matrix size and tracking the energy for this state; (ii) findi
ng the corresponding state in the numerical integration approach." }}
{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 261 11 "Exercise
 2:" }}{PARA 0 "" 0 "" {TEXT -1 158 "What is the meaning of the positi
ve-energy solutions? Can they represent true scattering states? Look a
t the wavefunction for one of them to find your answer." }}}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}}{MARK "32" 0 }{VIEWOPTS 1 1 0 
1 1 1803 1 1 1 1 }{PAGENUMBERS 0 1 2 33 1 1 }