# Electron wave function of hydrogen Calculator

## Calculates the electron wave functions of hydrogen-like atoms.

 The wave function Ψ(r,θ,φ) is the solution to the Schrodinger equation.The angle unit of θ and φ are degrees.
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
Magnetic quantum number m
 m=-l,l+1,..,l-1,l
Electron position r
Zenith angle θ
 degree
Azimuth angle φ
 degree

Ψ(r,θ,φ)
rΨ(r,θ,φ)
 $\normal The\ wave\ function\ \psi(r,\theta,\phi)\\\hspace{100}of\ the\ Hydrogen\ atom\\[10pt](1)\ -{\large\frac{\hbar^2}{\2m}}\nabla^2\psi-{\large\frac{Ze^2}{r}}\psi=E\psi\\\hspace{25}E=-{\large\frac{Z^2me^4}{2n^2\hbar^2}},\qquad Z=\{1:H,\ 2:He^+\}\\\vspace{10}(2)\ \psi_{n,l,m}(r,\theta,\phi)=R_{nl}(r)Y_l^{m}(\theta,\phi)\\\hspace{10}{\large\int_{\small 0}^{\hspace{25}\small{\infty}}\int_{\small 0}^{\hspace{25}\small{\pi}}\int_{\small 0}^{\hspace{25}\small{2\pi}}}\psi_{\small{n,l,m}}\psi_{\small{n',l',m'}}\ r^2sin\theta drd\theta d\phi\\\hspace{80}=\delta_{\small{nn'}}\delta_{\small{ll'}}\delta_{\small{mm'}}\\\vspace{10}(3)\ R_{nl}(r)=-\sqrt{({\large\frac{2Z}{na}})^3{\large\frac{(n-l-1)!}{2n(n+l)!}}}e^{-{\normal\frac{Zr}{na}}} \\\hspace{90}\times\ ({\large\frac{2Zr}{na}})^{l}L_{n-l-1}^{2l+1}({\large\frac{2Zr}{na}})\\[15]\hspace{18}Y_l^m(\theta,\phi)=\sqrt{{\large\frac{2l+1}{4\pi}}{\large\frac{(l-m)!}{(l+m)!}}}P_l^m(cos\theta)e^{im\phi}\\$

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