Hydrogen Atom

The nonrelativistic hydrogen atom is the Coulomb two-body bound-state problem reduced to a one-body central-potential problem with reduced mass $\mu$. It is the canonical model for radial equations, spherical harmonics, atomic orbitals, and Coulomb degeneracy.

Formula Hook

The ideal Hamiltonian is

$$H = \frac{p^2}{2\mu} - \frac{e^2}{4\pi\epsilon_0 r}.$$

The bound-state energies in this model are

$$E_n = - \frac{\mu e^4} {2(4\pi\epsilon_0)^2\hbar^2} \frac{1}{n^2}, \qquad n=1,2,\ldots .$$

The state labels are usually $n,\ell,m$, with angular dependence carried by spherical harmonics.

Canonical Home

See Hydrogen Atom for the teaching page, Radial Schrödinger Equation for the separation workflow, and Hydrogen Atom for the model card.

Common Confusions

• The nonrelativistic Coulomb model is not the full precision spectrum of real hydrogen.
• The reduced mass matters for quantitative spectra.
• Orbitals are wavefunctions or probability amplitudes, not classical electron orbits.
• Fine structure, hyperfine structure, Lamb shifts, external-field shifts, and finite nuclear size are additional corrections.

Related Entries

• Hamiltonian
• Eigenstate
• Stark Effect

References

• D. J. Griffiths and D. F. Schroeter, Introduction to Quantum Mechanics, 3rd ed., Cambridge University Press, 2018.
• L. D. Landau and E. M. Lifshitz, Quantum Mechanics: Non-Relativistic Theory, 3rd ed., Pergamon, 1977.
• R. Shankar, Principles of Quantum Mechanics, 2nd ed., Springer, 1994.