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Quantum Mechanics

Common Hamiltonians: Reference Table

This table lists common Hamiltonians and the assumptions that make them meaningful. Use model pages for derivations and spectra.

For the dedicated convention sheet covering continuum gases, Hubbard and spin models, BCS pairing, and the Kondo impurity model, see Common Many-Body Hamiltonians.

Hamiltonian Table

Section titled “Hamiltonian Table”
System Hamiltonian Assumptions Canonical Link
Free particle H=p^2/(2m)H=\hat p^2/(2m) nonrelativistic particle, no potential Free Particle
One-dimensional potential H=p^2/(2m)+V(x^)H=\hat p^2/(2m)+V(\hat x) stated domain and boundary conditions Hamiltonians in Coordinate Space
Harmonic oscillator H=p^2/(2m)+mω2x^2/2H=\hat p^2/(2m)+m\omega^2\hat x^2/2 quadratic potential, ω>0\omega>0 Harmonic Oscillator
Hydrogenic atom H=p^2/(2μ)−Ze2/(4πϵ0r)H=\hat p^2/(2\mu)-Ze^2/(4\pi\epsilon_0 r) nonrelativistic Coulomb problem with reduced mass Hydrogen Atom
Spin in magnetic field H=−μ⋅BH=-\boldsymbol\mu\cdot\mathbf B magnetic dipole approximation Spin in Magnetic Field
Two-level system H=c0I+c⋅σH=c_0I+\mathbf c\cdot\boldsymbol\sigma arbitrary Hermitian 2×22\times2 Hamiltonian Two-Level System
Charged particle in fields H=(p^−qA)2/(2m)+qΦH=(\hat{\mathbf p}-q\mathbf A)^2/(2m)+q\Phi nonrelativistic minimal coupling Charged Particle in Magnetic Field
Landau problem H=(p^−qA)2/(2m)H=(\hat{\mathbf p}-q\mathbf A)^2/(2m) uniform magnetic field, gauge specified Landau-Level System
Heisenberg chain H=J∑iSi⋅Si+1H=J\sum_i\mathbf S_i\cdot\mathbf S_{i+1} spin lattice and boundary convention Heisenberg Chain
Hubbard model H=−t∑⟨ij⟩,σ(ciσ†cjσ+h.c.)+U∑ini↑ni↓H=-t\sum_{\langle ij\rangle,\sigma}(c_{i\sigma}^\dagger c_{j\sigma}+\mathrm{h.c.})+U\sum_i n_{i\uparrow}n_{i\downarrow} lattice, hopping, and interaction convention Hubbard Model

Common Mistakes

Section titled “Common Mistakes”
  • Copying a Hamiltonian without its domain, boundary conditions, or gauge.
  • Using electron charge sign inconsistently in minimal coupling.
  • Confusing a model Hamiltonian with a first-principles microscopic description.
  • Treating an effective two-level Hamiltonian as valid outside its truncation regime.
Section titled “Related Tables”

References

Section titled “References”
  • R. Shankar, Principles of Quantum Mechanics, 2nd ed., Springer, 1994.
  • J. J. Sakurai and J. Napolitano, Modern Quantum Mechanics, 3rd ed., Cambridge University Press, 2020.
  • A. Altland and B. Simons, Condensed Matter Field Theory, 2nd ed., Cambridge University Press, 2010.