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Quantum Mechanics

Common Hamiltonians

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”
SystemHamiltonianAssumptionsCanonical Link
Free particleH=p^2/(2m)H=\hat p^2/(2m)nonrelativistic particle, no potentialFree Particle
One-dimensional potentialH=p^2/(2m)+V(x^)H=\hat p^2/(2m)+V(\hat x)stated domain and boundary conditionsHamiltonians in Coordinate Space
Harmonic oscillatorH=p^2/(2m)+mω2x^2/2H=\hat p^2/(2m)+m\omega^2\hat x^2/2quadratic potential, ω>0\omega>0Harmonic Oscillator
Hydrogenic atomH=p^2/(2μ)−Ze2/(4πϵ0r)H=\hat p^2/(2\mu)-Ze^2/(4\pi\epsilon_0 r)nonrelativistic Coulomb problem with reduced massHydrogen Atom
Spin in magnetic fieldH=−μ⋅BH=-\boldsymbol\mu\cdot\mathbf Bmagnetic dipole approximationSpin in Magnetic Field
Two-level systemH=c0I+c⋅σH=c_0I+\mathbf c\cdot\boldsymbol\sigmaarbitrary Hermitian 2×22\times2 HamiltonianTwo-Level System
Charged particle in fieldsH=(p^−qA)2/(2m)+qΦH=(\hat{\mathbf p}-q\mathbf A)^2/(2m)+q\Phinonrelativistic minimal couplingCharged Particle in Magnetic Field
Landau problemH=(p^−qA)2/(2m)H=(\hat{\mathbf p}-q\mathbf A)^2/(2m)uniform magnetic field, gauge specifiedLandau-Level System
Heisenberg chainH=J∑iSi⋅Si+1H=J\sum_i\mathbf S_i\cdot\mathbf S_{i+1}spin lattice and boundary conventionHeisenberg Chain
Hubbard modelH=−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 conventionHubbard 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.