Homework: Expectation Value of Radius in the Hydrogen Atom

In class, we numerically solved the radial Schrödinger equation for the hydrogen atom and obtained the first 14 bound eigenstates:

🔗 Hydrogen Atom Numerical Solution (Rutgers link)

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🎯 Your Task

For each of the 14 eigenstates $u_\ell(r)$ of the hydrogen atom computed in class:

1. Normalize the wavefunction:

   \begin{equation} \int_0^\infty u_\ell^2(r)\,dr = 1 \end{equation}

2. Compute the expectation value of the electron's position:

   \begin{equation} \langle r \rangle = \int_0^\infty u_\ell^2(r) \cdot r\,dr \end{equation}

3. Compare it to the Bohr model prediction:

   \begin{equation} r_{\text{Bohr}} = n^2 \cdot r_B \end{equation}

   where $r_B = 1$ in atomic units.

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✅ Output Format

For each state, print a line like:

l = <int>   E = <float>   <r> = <float>   r_Bohr = <float>

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📈 Visualization

Also generate and include a 2D plot showing all 14 wavefunctions $u_\ell(r)$ vs. $r$.

Use labels or colors to distinguish them.

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Optional Bonus

Try computing the most probable radius (where $u_\ell^2(r)$ is maximal) and compare that with $r_{\text{Bohr}}$.