Hydrogenic one dimensional atom 

by Reinaldo Baretti Machin
reibaretti2004@yahoo.com
url www.oocities.org/reibaretti2004
url www.oocities.org/serienumerica

The energy eigenvalues of the one dimensional hydrogenic atom are also given by 
 E= - (Z/n)2 where n== 1,2,3… just as in the three dimensional case.


c23456 example of particle in well of  infinite depth
c initial width=0.2
      dimension psi(0:10000)
      u(x)=-z/x
      g(e,x)= e- u(x)
      pi=2.*asin(1.)
      nstep=7000
      kprint=int(float(nstep)/30.)
      z=1.
      e1=-.6
      e2=-.4
      iter=20
      e=e1
      deltae=(e2-e1)/float(iter)
      do 40 it=1,iter
      xfin=-2.9/e
      dx=xfin/float(nstep)
c     initial conditions
      psi(0)=0.
      psi(1)=+dx+psi(0)
      do 30 i=2,nstep
      x=dx*float(i)
      psi(i)=2.*psi(i-1)-psi(i-2) -2.*dx**2*g(e,x-dx)*psi(i-1)
30    continue
      write(6,110)e,psi(nstep)
c      call anorm(pi,dx,psi,nstep,alfac)
c      Print*,'  '
c      call prtpsi(psi,pi,alfa2,nstep,kprint,dx)
c       Print*,'  '
      e=e+deltae
40    continue
110   format(2x,'e,psif=',1x,2(3x,e11.4))
      stop
      end



      subroutine anorm(pi,dx,psi,nstep,alfac)
      dimension psi(0:10000)
      pi=2.*asin(1.)
      sum=0.
      do 10 i=1,nstep
      x=dx*float(i)
      sum=sum+psi(i-1)**2+ psi(i)**2
10    continue
      sum=sum*dx/2.
      cnorm=1./sqrt(sum)
      do 20 i=0,nstep
      psi(i)=psi(i)*cnorm
20    continue
      alfac=(pi*psi(0)**2)**(1./3.)
      print*,'alfacorrected=',alfac
      return
      end

      subroutine prtpsi(psi,pi,alfa2,nstep,kprint,dx)
      dimension psi(0:10000)
      psi2s(x)=sqrt(alfa2**3/(32.*pi))*(2.-alfa2*x)*exp(-alfa2*x/2.)
      do 10 i=0,nstep,kprint
      x=dx*float(i)
      print 100, x, psi(i)**2
10    continue
100   format(2x,'x, psi=',2(3x,e11.4))
      return
      end

RUN
ground state
  e,psif=    -0.6000E+00    0.1218E+01
  e,psif=    -0.5900E+00    0.1135E+01
  e,psif=    -0.5800E+00    0.1047E+01
  e,psif=    -0.5700E+00    0.9530E+00
  e,psif=    -0.5600E+00    0.8526E+00
  e,psif=    -0.5500E+00    0.7413E+00
  e,psif=    -0.5400E+00    0.6318E+00
  e,psif=    -0.5300E+00    0.4845E+00
  e,psif=    -0.5200E+00    0.3489E+00
  e,psif=    -0.5100E+00    0.2122E+00
  e,psif=    -0.5000E+00   -0.6610E-02
  e,psif=    -0.4900E+00   -0.1535E+00
  e,psif=    -0.4800E+00   -0.3593E+00
  e,psif=    -0.4700E+00   -0.6127E+00
  e,psif=    -0.4600E+00   -0.8344E+00
  e,psif=    -0.4500E+00   -0.1084E+01
  e,psif=    -0.4400E+00   -0.1408E+01
  e,psif=    -0.4300E+00   -0.1703E+01
  e,psif=    -0.4200E+00   -0.2051E+01
  e,psif=    -0.4100E+00   -0.2439E+01

 
************************************************************

First excited state
  e,psif=    -0.1400E+00   -0.2215E+02
  e,psif=    -0.1385E+00   -0.2081E+02
  e,psif=    -0.1370E+00   -0.1913E+02
  e,psif=    -0.1355E+00   -0.1744E+02
  e,psif=    -0.1340E+00   -0.1567E+02
  e,psif=    -0.1325E+00   -0.1348E+02
  e,psif=    -0.1310E+00   -0.1129E+02
  e,psif=    -0.1295E+00   -0.8778E+01
  e,psif=    -0.1280E+00   -0.6084E+01
  e,psif=    -0.1265E+00   -0.3122E+01
  e,psif=    -0.1250E+00   -0.4762E-01
  e,psif=    -0.1235E+00    0.3441E+01
  e,psif=    -0.1220E+00    0.7104E+01
  e,psif=    -0.1205E+00    0.1125E+02
  e,psif=    -0.1190E+00    0.1560E+02
  e,psif=    -0.1175E+00    0.2036E+02
  e,psif=    -0.1160E+00    0.2549E+02
  e,psif=    -0.1145E+00    0.3095E+02
  e,psif=    -0.1130E+00    0.3695E+02
  e,psif=    -0.1115E+00    0.4333E+02