Most-Used Hamiltonians

This page is a fast route through the Hamiltonians that appear most often in nonrelativistic quantum mechanics. It is a lookup map, not a derivation page. The full explanations live on the linked canonical pages and reference cards.

The central warning is simple:

$$\text{Hamiltonian expression} \ne \text{complete quantum problem}.$$

A Hamiltonian formula becomes a physical model only after specifying the Hilbert space, domain, boundary conditions, parameters, units, gauge or basis convention, and approximation regime.

Fast Lookup

Situation	Common Hamiltonian	What must be specified	Continue
Free nonrelativistic particle	$\hat H=\hat{\mathbf p}^{\,2}/(2m)$	Dimension, domain, boundary conditions, normalization convention.	Free Particle Hamiltonian and Free Particle
Particle in a scalar potential	$\hat H=\hat{\mathbf p}^{\,2}/(2m)+V(\hat{\mathbf r})$	Potential, coordinate measure, self-adjoint domain, energy zero.	Hamiltonians in Coordinate Space
Infinite square well	$\hat H=\hat p^2/(2m)$ with hard-wall boundary conditions	Interval, wall locations, boundary conditions, quantum-number convention.	Infinite Square Well
Harmonic oscillator	$\hat H=\hat p^2/(2m)+m\omega^2\hat x^2/2$	$m$, $\omega$, oscillator length, whether the model is exact or a local approximation.	Harmonic Oscillator Hamiltonian and Quantum Harmonic Oscillator
General two-level system	$H=c_0I+\mathbf b\cdot\boldsymbol\sigma$	Basis, phase convention, physical meaning of $\mathbf b$, whether $c_0I$ matters.	Two-Level System Hamiltonian and Two-Level Systems
Spin-half in a magnetic field	$\hat H=-\boldsymbol\mu\cdot\mathbf B$	Sign of charge, $g$ factor, magnetic-moment convention, spin basis.	Spin-1/2 as a Canonical System and Spin and Pauli Matrix Conventions
Central potential	$\hat H=\hat{\mathbf p}^{\,2}/(2m)+V(r)$	Radial domain, angular momentum sector, reduced mass when applicable.	Central Potentials
Hydrogenic Coulomb problem	$\hat H=\hat p^2/(2\mu)-e^2/(4\pi\epsilon_0 r)$	Reduced mass, charge convention, nonrelativistic approximation, omitted fine structure.	Hydrogen Atom and Hydrogen Atom Model Card
Rigid rotor	$\hat H=\hat L^2/(2I)$	Configuration space, moment of inertia, angular measure, $J$ versus $\ell$ notation.	Rigid Rotor
Charged particle in electromagnetic potentials	$\hat H=(\hat{\mathbf p}-q\mathbf A)^2/(2m)+q\Phi$	Gauge, charge sign, scalar potential, canonical versus kinetic momentum.	Minimal Coupling
Uniform magnetic field	$\hat H=(\hat{\mathbf p}-q\mathbf A)^2/(2m)$ with uniform $\mathbf B$	Gauge choice, degeneracy convention, whether longitudinal motion is included.	Landau Levels
Composite noninteracting systems	$H=H_A\otimes I_B+I_A\otimes H_B$	Tensor-product order, subsystem identities, interaction terms omitted.	Composite Hamiltonians
Many-particle Hamiltonian	Kinetic terms plus interaction terms	Particle statistics, symmetrization sector, first- or second-quantized notation.	Many-Particle Hamiltonians
Time-dependent drive	$H(t)=H_0+V(t)$	Time-dependence, picture, rotating frame, approximation used.	Time-Dependent Hamiltonians
Effective Hamiltonian	$H_{\text{eff}}$ after projection or scale separation	Subspace, small parameter, eliminated degrees of freedom, error scale.	Schrieffer-Wolff Transformation

For a longer wave-mechanics table with coordinates, spectrum types, and scales, see Common Hamiltonians. For convention-aware continuum, lattice, spin, pairing, and impurity formulas, see Common Many-Body Hamiltonians. For compact cards, see the Operator and Hamiltonian Library.

How to Choose the Right Row

Start with the degrees of freedom. A particle moving on a line, a spin-half system, a rotor, a composite system, and a many-body system do not live in the same Hilbert space, even when their formulas look similar.

Then identify the domain and constraints. The expression $\hat p^2/(2m)$ describes a free particle on the line, an infinite square well after imposing hard-wall boundary conditions, and a particle on a ring after imposing periodic boundary conditions. The operator expression is only part of the model.

Next check whether the Hamiltonian is exact, approximate, or effective. The harmonic oscillator can be an exact idealization or the quadratic approximation to a smooth potential near a stable minimum. A two-level Hamiltonian is usually a projection onto a selected subspace. An effective Hamiltonian can be excellent in its regime and misleading outside it.

Finally check conventions. Magnetic Hamiltonians are especially sign sensitive because charge, magnetic moment, $g$ factor, Pauli-matrix convention, and gauge all matter. Tensor-product Hamiltonians are order sensitive because $H_A\otimes I_B$ and $I_A\otimes H_B$ act on different factors.

Common Scales

Model	Natural scale	Use
Free particle	$E=p^2/(2m)$	Converts momentum or wavenumber into kinetic energy.
Infinite well	$\hbar^2/(2mL^2)$	Sets the spacing scale for confinement in a box of length $L$.
Harmonic oscillator	$\hbar\omega$ and $\ell=\sqrt{\hbar/(m\omega)}$	Sets energy spacing and spatial width.
Two-level system	$2\lvert\mathbf b\rvert$	Sets the level splitting after removing the common shift $c_0I$.
Hydrogen atom	Bohr radius $a_0$ and Hartree scale	Sets atomic length and Coulomb binding scales.
Rigid rotor	$\hbar^2/(2I)$	Sets rotational spacing.
Landau levels	magnetic length $\ell_B$ and cyclotron energy $\hbar\omega_c$	Sets orbital size and magnetic energy spacing.

Scale checks are often the fastest way to catch an error. If the result for a box has no $L$, the oscillator result has no $\omega$, or the Landau-level expression has no magnetic field scale, something has likely been lost.

Assumption Checklist

Before using a Hamiltonian from a table, ask:

1. What Hilbert space does it act on?
2. What is the domain of the differential operator?
3. Which boundary or matching conditions are part of the model?
4. Are the parameters masses, reduced masses, frequencies, charges, or effective parameters?
5. Are constants such as $\hbar$, $c$, $e$, or $\epsilon_0$ explicit or set to one?
6. Is the Hamiltonian time independent?
7. Is the system closed, driven, or open?
8. Is the formula exact, approximate, or effective?
9. Which canonical page maintains the derivation or model assumptions?

If any answer is unclear, use the linked page rather than importing the formula directly.

Common Mistakes

• Treating a Hamiltonian expression as complete without a domain.
• Reusing a spectrum after changing boundary conditions.
• Forgetting the reduced mass in two-body problems.
• Confusing canonical momentum $\hat{\mathbf p}$ with kinetic momentum $\hat{\boldsymbol\pi}=\hat{\mathbf p}-q\mathbf A$.
• Dropping the common shift $c_0I$ in a two-level system without checking whether absolute energies matter.
• Comparing magnetic Hamiltonians without checking charge-sign and spin conventions.
• Treating an effective Hamiltonian as valid outside the subspace or scale range used to derive it.
• Mixing first-quantized and second-quantized notation without stating the particle-number sector.

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.
• D. J. Griffiths and D. F. Schroeter, Introduction to Quantum Mechanics, 3rd ed., Cambridge University Press, 2018.
• C. Cohen-Tannoudji, B. Diu, and F. Laloë, Quantum Mechanics, Wiley, 1977.
• L. D. Landau and E. M. Lifshitz, Quantum Mechanics: Non-Relativistic Theory, 3rd ed., Pergamon, 1977.

Exercises

1. The operator $\hat p^2/(2m)$ appears in both a free-particle page and an infinite-well page. Why do the spectra differ?

Solution

The differential expression is not the whole problem. The free particle on the full line has continuum-normalized momentum eigenstates, while the infinite well imposes hard-wall boundary conditions on a finite interval. The domain and boundary conditions change the spectrum.

2. A calculation for a charged particle in a magnetic field uses $\hat{\mathbf p}$ as the mechanical momentum. What should be checked?

Solution

Check whether the calculation should use the kinetic momentum $\hat{\boldsymbol\pi}=\hat{\mathbf p}-q\mathbf A$. In electromagnetic fields, canonical and kinetic momentum are different, and the distinction affects commutators, velocities, and Landau-level quantization.

3. A two-level Hamiltonian is written as $H=c_0I+\mathbf b\cdot\boldsymbol\sigma$. Which part controls transition frequencies?

Solution

The splitting is controlled by $\mathbf b\cdot\boldsymbol\sigma$, with eigenvalues separated by $2\lvert\mathbf b\rvert$. The term $c_0I$ shifts both energies equally, so it does not affect transition frequencies in a closed two-level model.