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Hydrogen Solution Calculator


Shodor > CSERD > Resources > Models > Hydrogen Solution Calculator

  Software  •  Instructions  •  Theory


Instructions - Hydrogen Solutions Calculator

Purpose

This applet creates an equation which is the solution to Schrodinger's equation for Hydrogen with a single electron.

What is returned to the user is


\begin{displaymath}
\Psi(l\vert m=0\vert n, \vec{r})^*\Psi(l\vert m=0\vert n, \vec{r})
\end{displaymath}

where $\Psi$ solves the equation


\begin{displaymath}
\mathbf{H} \Psi(\vec{r}) \equiv
\left[
-\frac{\hbar^2}{2m}\Delta
+V(\vec{r})
\right]
=E \Psi(\vec{r})
\end{displaymath}

for a potential of $V(r)=e^2/r$.

The magnitude of the wave is not normalized, and the radial variable is normalized such that r in this problem has been multiplied by $2 \sqrt{E}$. The energy level is related to the actual energy of the electron by


\begin{displaymath}
\nu = - \frac{m e^2}{\sqrt{e} \hbar^2}
\end{displaymath}

where


\begin{displaymath}
\nu = l + 1 + n
\end{displaymath}

and the scaled radial variable $x = 2 \sqrt{E}$ solves the equation


\begin{displaymath}
\left[
\frac{d^2}{dx^2}-
\frac{l(l+1)}{x^2}+
\frac{\nu}{x}-
\frac{1}{4}
\right]
y_l(x)
=0
\end{displaymath}

In addition, the user can select to see the real or imaginary portions of the solution, as well as the Associated Legendre function or the radial solution for the given energy levels.

Fundamentals

Select the desired solution from the wavefunction, the Legendre functions, or the radial solution.

Select the value of l, m, and n for which you would like to see a solution.


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