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Hydrogen Atom & Energy Levels

Interactive quantum mechanics — computed live in the browser via Pyodide + scipy.

The Hydrogen Atom

The hydrogen atom is the simplest bound quantum system: a single electron of charge $-e$ orbiting a proton of charge $+e$ via the Coulomb potential.

$$V(r) = -\frac{e^2}{4\pi\epsilon_0 r}$$

The time-independent Schrödinger equation in spherical coordinates is:

$$\left[ -\frac{\hbar^2}{2m_e}\nabla^2 + V(r) \right] \psi(r,\theta,\phi) = E\,\psi(r,\theta,\phi)$$

Separation of variables gives $\psi_{n\ell m} = R_{n\ell}(r)\,Y_\ell^m(\theta,\phi)$, where $R_{n\ell}$ are radial wavefunctions and $Y_\ell^m$ are spherical harmonics.

Energy Eigenvalues

Solving the radial equation yields quantised energy levels:

$$E_n = -\frac{m_e e^4}{2\hbar^2(4\pi\epsilon_0)^2}\cdot\frac{1}{n^2} = -\frac{13.6\text{ eV}}{n^2}, \quad n = 1,2,3,\ldots$$

The Bohr radius $a_0 = 4\pi\epsilon_0\hbar^2/m_e e^2 \approx 0.529\text{ Å}$ sets the natural length scale. The ionisation energy from the ground state is $|E_1| = 13.6\text{ eV}$.

Interactive Energy Levels

Drag the slider to include more levels. Click a level line to read off its exact energy.

Click an energy level line to inspect it.

Radial Wavefunctions

The exact radial wavefunctions use associated Laguerre polynomials $L_{n-\ell-1}^{2\ell+1}$, computed here via scipy.special.genlaguerre running inside Pyodide:

$$R_{n\ell}(r) = -\sqrt{\left(\frac{2}{na_0}\right)^3 \frac{(n-\ell-1)!}{2n\,[(n+\ell)!]^3}}\; e^{-r/na_0}\left(\frac{2r}{na_0}\right)^\ell L_{n-\ell-1}^{2\ell+1}\!\left(\frac{2r}{na_0}\right)$$
▶ Show Python code (runs in browser via Pyodide) 
import numpy as np
from scipy.special import genlaguerre, factorial

def R_nl(n, l, r):
    """Radial wavefunction, r in units of a0."""
    rho  = 2 * r / n
    norm = np.sqrt(
        (2/n)**3 * factorial(n-l-1) /
        (2 * n * factorial(n+l)**3)
    )
    L = genlaguerre(n-l-1, 2*l+1)(rho)
    return norm * np.exp(-rho/2) * rho**l * L

r = np.linspace(0, 40, 1000)

orbitals = [(1,0,'1s'),(2,0,'2s'),(2,1,'2p'),
            (3,0,'3s'),(3,1,'3p'),(3,2,'3d')]

results = {}
for n, l, label in orbitals:
    psi  = R_nl(n, l, r)
    prob = r**2 * psi**2
    results[label] = prob.tolist()

Orbital Explorer

Select any valid $(n, \ell)$ combination. The plot shows the radial probability density $r^2|R_{n\ell}|^2$ and marks the most probable radius.

Select an orbital and click Compute.

Spectral Series

Photon energy for a transition $n_i \to n_f$:

$$\Delta E = 13.6\left(\frac{1}{n_f^2} - \frac{1}{n_i^2}\right)\text{ eV}, \qquad \lambda = \frac{hc}{\Delta E}$$

Loading transition table...

Custom Transition Calculator

Enter any initial and final quantum numbers to compute the transition energy and wavelength, and identify which spectral series it belongs to.

Enter values and click Calculate.