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e ODE as an initial value problem" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}
{PARA 0 "" 0 "" {TEXT -1 345 "We solve the Schroedinger equation numer
ically for eigenvalues/eigenfunctions by imposing symmetry at x=0, and
 using the energy value as a trial value. It is giving results for arb
itrary symmetric potential shapes. The eigenenergy is selected by dema
nding wavefunctions that vanish as they approach infinity so that they
 are square-normalizable." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "
" 0 "" {TEXT -1 182 "We choose for a potential a shape that shares som
e features with the 3d Coulomb potential. It has a long-range tail, bo
und states for E<0 and continuous scattering solutions for E>0." }}
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{MPLTEXT 1 0 47 "EVeven:=proc(E) local sol,U,IC; global Xmax,SE;" }}
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ph the ground state energy superimposed on the potential energy plot t
o realize the amount of zero-point fluctuations." }}{PARA 0 "" 0 "" 
{TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 47 "Now find the next
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0 "" 0 "" {TEXT -1 143 "This looks a little suspicious. The tail of th
e wavefunction does not approach the axis smoothly (with negligible sl
ope). We ought to increase " }{TEXT 19 4 "Xmax" }{TEXT -1 39 " and see
 how it affects the eigenvalue." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}
{PARA 0 "" 0 "" {TEXT 256 11 "Exercise 1:" }}{PARA 0 "" 0 "" {TEXT -1 
113 "Find the first four eigenenergies in the even-symmetry sector to \+
four significant digits. Be careful to increase " }{TEXT 19 4 "Xmax" }
{TEXT -1 91 " sufficiently for the higher states by looking at the wav
efunctions. Report your choice of " }{TEXT 19 4 "Xmax" }{TEXT -1 136 "
, and observe which way the eigenvalue shifted. Can you explain why it
 changes and how it changes when a node is forced at too small an " }
{TEXT 19 1 "x" }{TEXT -1 7 "-value?" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }
}{PARA 0 "" 0 "" {TEXT 257 11 "Exercise 2:" }}{PARA 0 "" 0 "" {TEXT 
-1 260 "Write a modified procedure to calculate the four lowest eigens
tates in the odd-symmetry sector. Ensure that you know the eigenvalues
 to four significant digits. Again explore what happens to the eigenva
lues when the wavefunction is cut off to zero at a finite " }{TEXT 19 
1 "x" }{TEXT -1 7 "-value." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 
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