- #1
torehan
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Hi,
If the normalized 1D wave-function of hydrogen atom for n=1, l=0, m_l=0;
[tex]\psi_{1s}(x)=\frac{1}{\sqrt{\pi} a_{0}^{3/2}}e^{-x/a_{0}}[/tex]
and probability distribution of wave-function,
[tex]\mid\psi_{1s}(x)^2\mid[/tex]
so integration of rho over all x should give the number of electrons which is equal to 1
[tex]\int{\mid\psi_{1s}(x)^2\mid}dx=1[/tex]
theory is simple and understandable. But when I apply a numerical procedure to this mathematical aspect I couldn't get the right result.
With a very simple computer program,hoping to find number of electrons, I use the easiest integration technique as converting integral to a discrete sum of each value of the charge density function multiplied by dx which is increment of variable x,
But this doesn't return the correct value of nelect.
Am I making a fundamental mistake which I couldn't see right now?
Thanks
If the normalized 1D wave-function of hydrogen atom for n=1, l=0, m_l=0;
[tex]\psi_{1s}(x)=\frac{1}{\sqrt{\pi} a_{0}^{3/2}}e^{-x/a_{0}}[/tex]
and probability distribution of wave-function,
[tex]\mid\psi_{1s}(x)^2\mid[/tex]
so integration of rho over all x should give the number of electrons which is equal to 1
[tex]\int{\mid\psi_{1s}(x)^2\mid}dx=1[/tex]
theory is simple and understandable. But when I apply a numerical procedure to this mathematical aspect I couldn't get the right result.
With a very simple computer program,hoping to find number of electrons, I use the easiest integration technique as converting integral to a discrete sum of each value of the charge density function multiplied by dx which is increment of variable x,
Code:
a0=0.53;
x=-10*a0:0.001:10*a0;
R=-0.36;
NORM=1/sqrt(%pi*a0^3);
dx=20*a0/size(x,2);
nelect=0;
psi1s=NORM*exp(-abs((x-R))/a0);
rho1s=psi1s^2;
for i=1:size(x,2)
nelect=nelect+rho(i)*dx;
end
return nelect;
Code:
> nelect= 1.1331234
But this doesn't return the correct value of nelect.
Am I making a fundamental mistake which I couldn't see right now?
Thanks
Last edited: