By Reinaldo Baretti Machin
reibaretti@yahoo.com
Schrodinger equation is
solved for the harmonic oscillator.
The value of ψ (xlimit)
is plotted vs energy
as an independent variable. The eigenvalues are read from the graph where psi
changes sign.Accurate values can be obtained by writing an interpolating
polynomial with
E =E(ψ (xlimit))
c metodo de
psi final ,osc armonico e=(n+1/2),n=0,2,4,6..
c funcion par
, n=1,3,5.... psi impar
implicit
real*8(a-h,o-z)
dimension e(40),
psifin(40)
v(x)=.5*x**2
c v(x)=x
e(1)=0.8
nstep=5000
kstep=ifix(float(nstep)/20.)
do 10 ie=1,32
c print*,' e(ie)=',
e(ie)
kount=kstep
xlim=1.7*sqrt(2.*e(ie))
c
xlim=2.6*e(ie)
c
print*,'ratio=',xlim/4.28
h=xlim/float(nstep)
c si
n es par initial values psi0=psi1, n impar psi0=0. ,psi1=h
c n =even
number oscillator Initial cond.
psi0=1.
psi1=psi0
c n =odd
number oscillator initial cond.
c psi0=0.
c psi1=h
do 20 i=2,nstep
x=float(i-1)*h
psi2=2.*psi1*(1.-h**2*(e(ie)-v(x)))-psi0
c
if(i.eq.kount)then
c
write(6,110) float(i)*h, psi2
c
print*,'x,psi=',float(i)*h, psi2
c
kount=kount+kstep
c endif
psi0=psi1
psi1=psi2
20 continue
psifin(ie)=psi2
c print*,
e(ie)
print*,'e(ie)
,psifin=',e(ie),psifin(ie)
110 format(1x,'x=',e12.4,3x,'psi=',e12.4)
e(ie+1)=e(ie)+.1
10 continue
stop
end
RUN (has been edited)
Eigenvalues for n=0
e(ie) ,psifin=
0.100000001 0.970023957
e(ie) ,psifin= 0.200000003
0.879767606
e(ie) ,psifin= 0.300000004
0.728298627
e(ie) ,psifin= 0.400000006
0.514237125
e(ie) ,psifin= 0.500000007
0.236014679
e(ie) ,psifin= 0.600000009 -0.107767019
e(ie) ,psifin= 0.70000001 -0.517882569
e(ie) ,psifin= 0.800000012 -0.993941068
*********************************
n=2
e(ie) ,psifin=
2.00000003 -7.13945632
e(ie) ,psifin= 2.10000003 -6.6866482
e(ie) ,psifin= 2.20000003 -5.82269367
e(ie) ,psifin= 2.30000003 -4.47393513
e(ie) ,psifin= 2.40000004 -2.56767288
e(ie) ,psifin= 2.50000004 -0.0356724013
e(ie) ,psifin= 2.60000004
3.18156098
e(ie) ,psifin= 2.70000004
7.1297205
e(ie) ,psifin= 2.80000004
11.8348587
RUN (has been edited)
Eigenvalues fro n=1 n=3
e(ie)
,psifin= 0.800000012 1.71190569
e(ie) ,psifin= 0.900000013
1.66414922
e(ie) ,psifin= 1.00000001
1.56252677
e(ie) ,psifin= 1.10000002
1.40109257
e(ie) ,psifin= 1.20000002
1.17366956
e(ie) ,psifin= 1.30000002
0.874232407
e(ie) ,psifin= 1.40000002
0.497333185
e(ie) ,psifin= 1.50000002
0.0385779025
e(ie) ,psifin= 1.60000002 -0.504843608
e(ie) ,psifin= 1.70000003 -1.13357643
e(ie) ,psifin= 1.80000003 -1.8454741
e(ie) ,psifin= 1.90000003 -2.63493753
e(ie) ,psifin= 2.00000003 -3.49223427
******************************************
e(ie) ,psifin= 3.20000005 -7.55207467
e(ie) ,psifin= 3.30000005 -5.7129254
e(ie) ,psifin= 3.40000005 -3.21640969
e(ie) ,psifin= 3.50000005 -0.00224820672
e(ie) ,psifin= 3.60000005
3.9762797
e(ie) ,psifin= 3.70000006
8.74707378
e(ie) ,psifin= 3.80000006
14.3133793
e(ie) ,psifin= 3.90000006
20.6474739
Psi (asymptotic) changes sign at energy =1.5
and at 3.5