Wave-Mechanics Hamiltonians at a Glance

Compare the Hamiltonians of familiar quantum systems, the coordinates that simplify them, and the scales that set their energies. Follow a model’s link for its derivation and worked examples.

The entries suppress some domain details to stay readable. Boundary conditions, self-adjoint domains, normalization conventions, and approximation assumptions belong on the linked derivations.

Hamiltonians and Physical Scales

System	Hamiltonian	Coordinates	Spectrum type	Key scale	Derivation
Free particle	$\hat p^2/(2m)$	line or $\mathbb R^3$	continuous	momentum $p$, wavenumber $k$	Free Particle
Infinite square well	$\hat p^2/(2m)$ with $\psi=0$ at walls	interval $0\lt x\lt L$	discrete	$L$, $\hbar^2/(2mL^2)$	Infinite Square Well
Finite square well	$\hat p^2/(2m)+V(x)$ with finite depth	one-dimensional regions	discrete bound states plus continuum	well width, depth, decay length	Finite Square Well
Delta potential	$\hat p^2/(2m)-g\delta(x)$	line with matching condition	one bound state plus continuum for $g\gt0$	$\hbar^2/(mg)$	Delta Function Potential
Potential step	$\hat p^2/(2m)+V_0\Theta(x)$	two half-lines	continuous scattering	incident energy, step height	Potential Step
Rectangular barrier	$\hat p^2/(2m)+V(x)$ with finite barrier region	three one-dimensional regions	continuous scattering, resonant structure	barrier width, height, decay constant	Rectangular Barrier Tunneling
Harmonic oscillator	$\hat p^2/(2m)+m\omega^2x^2/2$	line	discrete equally spaced	$\ell=\sqrt{\hbar/(m\omega)}$, $\hbar\omega$	Quantum Harmonic Oscillator
Coupled oscillators	quadratic form in coordinates and momenta	normal coordinates	discrete normal-mode ladders for positive mode frequencies	normal-mode frequencies	Coupled Oscillators
Two-level system	$a_0I+\mathbf b\cdot\boldsymbol\sigma$	two-dimensional Hilbert space	two discrete levels	$\lvert\mathbf b\rvert$	Two-Level Systems
Three-dimensional box	$\hat{\mathbf p}^2/(2m)$ with Dirichlet walls	rectangular box	discrete	side lengths, $\hbar^2/(2mL^2)$	Three-Dimensional Box
Central potential	$\hat{\mathbf p}^2/(2m)+V(r)$	spherical coordinates	model dependent	radial scale, angular momentum barrier	Radial Schrödinger Equation
Hydrogen atom	$\hat p^2/(2\mu)-e^2/(4\pi\epsilon_0r)$	spherical coordinates	discrete bound states plus continuum	$a_0$, Hartree scale	Hydrogen Atom
Rigid rotor	$\hat L^2/(2I)$	sphere $S^2$	discrete rotational levels	$I$, $\hbar^2/(2I)$	Rigid Rotor
Minimal electromagnetic coupling	$(\hat{\mathbf p}-q\mathbf A)^2/(2m)+q\Phi$	chosen gauge and coordinates	field dependent	kinetic momentum, gauge-invariant fields	Minimal Coupling
Landau levels	$(\hat{\mathbf p}-q\mathbf A)^2/(2m)$ in uniform $\mathbf B$	Landau or symmetric gauge	discrete transverse levels with degeneracy	$\ell_B$, $\hbar\omega_c$	Landau Levels
Charged oscillator in magnetic field	$(\hat{\mathbf p}-q\mathbf A)^2/(2m)+m\omega_0^2r^2/2$	symmetric gauge, polar coordinates	discrete Fock–Darwin levels	$\Omega$, $\omega_c$	Charged Harmonic Oscillator in a Magnetic Field

How to Read the Table

The Hamiltonian alone is not the full quantum problem. The same differential expression can define different physics when the domain, boundary conditions, or physical constraints change. A consistent change of coordinates or gauge instead describes the same physics. For example, $\hat p^2/(2m)$ describes a free particle on the line, a particle in a box after boundary conditions are imposed, or angular kinetic energy after a constraint has been applied.

The spectrum column is also idealized. A finite well has discrete bound states below the continuum; a barrier has continuum scattering states but can show resonant structure; a three-dimensional Landau problem has discrete transverse Landau levels plus continuous longitudinal motion.

Use the key scale column as a first diagnostic. If a calculation has no visible length, energy, or frequency scale matching the table, check whether a parameter, boundary condition, or unit convention has been lost.

Common Mistakes

• Treating the displayed Hamiltonian expression as complete without specifying its domain.
• Reusing a spectrum after changing boundary conditions.
• Confusing canonical momentum with kinetic momentum in electromagnetic fields.
• Forgetting reduced mass in two-body problems such as hydrogen.
• Treating a reference table as a derivation substitute.
• Comparing formulas from different unit systems without checking which constants have been set to one.

References

• C. Cohen-Tannoudji, B. Diu, and F. Laloë, Quantum Mechanics, Wiley, 1977.
• D. J. Griffiths and D. F. Schroeter, Introduction to Quantum Mechanics, 3rd ed., Cambridge University Press, 2018.
• J. J. Sakurai and J. Napolitano, Modern Quantum Mechanics, 3rd ed., Cambridge University Press, 2020.
• R. Shankar, Principles of Quantum Mechanics, 2nd ed., Springer, 1994.