Shan — March 2026
The hydrogen atom is the simplest bound quantum system — a single electron of charge $-e$ bound to a proton by the Coulomb potential:
$$V(r) = -\frac{e^2}{4\pi\epsilon_0 r}$$The time-independent Schrödinger equation in spherical coordinates:
$$\left[ -\frac{\hbar^2}{2m_e}\nabla^2 + V(r) \right] \psi = E\psi$$The wavefunction separates as:
$$\psi_{n\ell m}(r,\theta,\phi) = R_{n\ell}(r)\, Y_\ell^m(\theta,\phi)$$Solving the radial equation gives quantized energies:
$$E_n = -\frac{13.6\text{ eV}}{n^2}, \quad n = 1, 2, 3, \ldots$$The ground state is $E_1 = -13.6\text{ eV}$. The Bohr radius sets the natural length scale:
$$a_0 = \frac{4\pi\epsilon_0\hbar^2}{m_e e^2} \approx 0.529\text{ Å}$$Energy levels for $n = 1$ to $10$, with Lyman series transitions shown:
The first few radial wavefunctions $R_{n\ell}(r)$:
$$R_{10}(r) = 2\left(\frac{1}{a_0}\right)^{3/2} e^{-r/a_0}$$ $$R_{20}(r) = \frac{1}{\sqrt{2}}\left(\frac{1}{2a_0}\right)^{3/2} \left(2 - \frac{r}{a_0}\right)e^{-r/2a_0}$$The Rydberg formula gives photon wavelengths for transitions $n_i \to n_f$:
$$\frac{1}{\lambda} = R_\infty\left(\frac{1}{n_f^2} - \frac{1}{n_i^2}\right)$$where $R_\infty = 1.097 \times 10^7\text{ m}^{-1}$. Key series:
| Series | Region | $n_f$ | $\lambda$ range |
|---|---|---|---|
| Lyman | UV | 1 | 91 – 122 nm |
| Balmer | Visible | 2 | 365 – 656 nm |
| Paschen | IR | 3 | 820 – 1875 nm |