Hydrogen Atom Hamiltonian

Hamiltonian

For a hydrogenic ion with nuclear charge $+Ze$ and reduced mass $\mu$,

$$H = \frac{\hat{\mathbf p}^{\,2}}{2\mu} - \frac{Ze^2}{4\pi\epsilon_0 r}.$$

The ideal nonrelativistic bound-state energies are

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

Assumptions

• Nonrelativistic two-body Coulomb problem reduced to relative motion.
• Spin, fine structure, Lamb shift, hyperfine structure, finite nuclear size, and external fields are omitted.
• The nucleus is represented only through charge and reduced mass.
• SI units are displayed; atomic or natural units change the appearance.

Common Mistakes

• Using electron mass instead of reduced mass when precision matters.
• Treating the $n^2$ degeneracy as generic for all central potentials.
• Forgetting which corrections have been omitted.
• Mixing Gaussian, SI, and atomic-unit Coulomb conventions.

Canonical Links

• Hydrogen Atom
• Hydrogen Atom Model Card
• Hydrogen Spectrum
• Coulomb Potential

References

• H. A. Bethe and E. E. Salpeter, Quantum Mechanics of One- and Two-Electron Atoms, Springer, 1957.
• D. J. Griffiths and D. F. Schroeter, Introduction to Quantum Mechanics, 3rd ed., Cambridge University Press, 2018.