Electronic distribution of hydrogen (chart)

Calculates a table of the electron radial distribution of hydrogen-like atoms and draws the chart.

The electron position r with the Bohr radius a = 1 unit is the distance from the nucleus.

Electronic distribution of hydrogen (chart)

Atomic number Z

H(1) He+(2)

Principal quantum number n

n=1,2,3,...

Angular quantum number l

l=0,1,2,..,n-1

[ Initial position r

of electron

Increment

Repetition

]

r	R(r)R(r)rr

$\normal Probability\ density\ of\ finding\\ \hspace{50}\ electron\ in\ a\ Hydrogen\ atom\\[10pt] (1)\ -{\large\frac{\hbar^2}{\2m}}\nabla^2\psi-{\large\frac{Ze^2}{r}}\psi=E\psi\\ \hspace{30}E=-{\large\frac{Z^2me^4}{2n^2\hbar^2}},\qquad Z=\{1:H,\ 2:He^+\}\\ \hspace{30}\psi_{n,l,m}(r,\theta,\phi)=R_{nl}(r)Y_l^{m}(\theta,\phi)\\\vspace{10} (2)\ R_{nl}(r)=-\sqrt{({\large\frac{2Z}{na}})^3{\large\frac{(n-l-1)!}{2n(n+l)!}}}e^{-{\normal\frac{Zr}{na}}} \\ \hspace{90}\times\ ({\large\frac{2Zr}{na}})^{l} L_{\small{n-l-1}}^{\small{2l+1}}({\large\frac{2Zr}{na}})\\ \hspace{30}a={\large\frac{\hbar^2}{me^2}}\ :Bohr\ radius\\\vspace{10} (3)\ Probability\ density:\ R_{nl}(r)R_{nl}^*(r)r^2\\ \hspace{20}{\large\int_{\small 0}^{\hspace{25}\small{\infty}}}R_{nl}R_{nl}^*(r)r^2dr=1\\$

Related links

Hydrogen-like atom (Wikipedia)

Electronic distribution of hydrogen (chart)

Calculates a table of the electron radial distribution of hydrogen-like atoms and draws the chart.

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