Canonical Systems Formulas

Canonical systems are reusable test problems: their spectra and states anchor perturbation theory, semiclassical limits, numerical benchmarks, and local approximations to more complicated Hamiltonians. These cards make their most used formulas available without detaching them from boundary conditions, normalization conventions, degeneracies, or energy-zero choices.

Each card is a lookup layer. Follow its canonical link for the derivation and full physical interpretation.

Find the right card

Task	Card
Quantize a particle between hard walls	Particle-in-a-Box Spectrum
Solve a finite attractive square well by parity	Finite Square-Well Equations
Look up oscillator energies, eigenfunctions, or moments	Harmonic Oscillator Spectrum
Evaluate oscillator matrix elements with creation and annihilation operators	Harmonic Oscillator Ladder Operators
Expand, overlap, displace, or evolve a coherent state	Coherent-State Expansion
Compute Coulomb bound energies, transitions, or degeneracy	Hydrogen Spectrum
Look up normalized Coulomb radial functions, nodes, or radial moments	Hydrogen Radial Wavefunctions
Convert molecular inertia into ideal rotational levels or lines	Rigid-Rotor Spectrum
Compute magnetic scales, orbital levels, or flux degeneracy	Landau Levels

Model comparison

System	Core Hamiltonian or condition	Discrete label	Spectrum or level equation
Infinite square well	$\psi(0)=\psi(L)=0$	$n=1,2,\ldots$	$E_n=\pi^2\hbar^2n^2/(2mL^2)$
Symmetric finite well	$V=-V_0$ for $\lvert x\rvert<a$	parity and bound-state root	$z\tan z=\sqrt{z_0^2-z^2}$ or $-z\cot z=\sqrt{z_0^2-z^2}$
Harmonic oscillator	$P^2/(2m)+m\omega^2X^2/2$	$n=0,1,2,\ldots$	$E_n=\hbar\omega(n+1/2)$
Hydrogenic Coulomb problem	$p^2/(2\mu)-Ze^2/(4\pi\epsilon_0r)$	$n,\ell,m$	$E_n=-\mu c^2(Z\alpha)^2/(2n^2)$
Linear rigid rotor	$L^2/(2I)$	$J,M$	$E_J=\hbar^2J(J+1)/(2I)$
Spinless Landau problem	$(\mathbf p-q\mathbf A)^2/(2m)$	$n$ plus guiding center	$E_n=\hbar\omega_c(n+1/2)$

The labels in the third column do not all play the same role. The box label $n$ identifies a nondegenerate one-dimensional level. Hydrogenic $\ell,m$ labels resolve a Coulomb degeneracy. The Landau guiding-center label changes the orbital state while leaving the cyclotron energy fixed.

Bound systems

Square wells

The Particle-in-a-Box Spectrum uses impenetrable Dirichlet walls. It includes normalized eigenfunctions, matrix elements, parity in a centered interval, multidimensional products, and revival scales. Its familiar $n^2$ spectrum does not survive unchanged when a wall becomes finite.

The Finite Square-Well Equations use a symmetric attractive well of width $2a$, with exterior potential chosen as zero. Bound energies are roots of parity-resolved transcendental equations, not a closed polynomial spectrum. Check the width and energy-zero conventions before importing equations from a source that uses a well from $0$ to $L$ or sets the interior potential to zero.

Hydrogenic states

The Hydrogen Spectrum collects reduced-mass scales, $Z^2/n^2$ energies, transition wavenumbers, degeneracy, virial identities, and the limits of the ideal Coulomb model.

The Hydrogen Radial Wavefunctions separate three objects that are often conflated:

$$R_{n\ell}(r), \qquad u_{n\ell}(r)=rR_{n\ell}(r), \qquad P_{n\ell}(r)=r^2\lvert R_{n\ell}(r)\rvert^2.$$

Use the spectrum card for energy and transition calculations; use the radial card for normalization, shell probabilities, nodes, moments, and radial matrix elements.

Oscillator toolkit

Three cards form one connected toolkit:

1. Harmonic Oscillator Spectrum gives the exact levels, coordinate-space states, natural scales, moments, and multidimensional extensions.
2. Harmonic Oscillator Ladder Operators gives operator definitions, repeated actions, matrix elements, commutators, and Heisenberg evolution.
3. Coherent-State Expansion gives the number expansion, Poisson statistics, displacement identities, overlaps, resolution of identity, quadrature moments, and exact ideal evolution.

Keep the oscillator length convention visible. This compendium uses

$$\ell = \sqrt{\frac{\hbar}{m\omega}},$$

so the ground-state Gaussian is proportional to $e^{-x^2/(2\ell^2)}$. Some sources call $\ell/\sqrt2$ the oscillator length, which shifts factors of $\sqrt2$ in ladder and variance formulas without changing the physics.

Angular and magnetic systems

The Rigid-Rotor Spectrum uses scalar wavefunctions on $S^2$ and integer $J$. It distinguishes the rotational constant in energy, frequency, and wavenumber units, and it explains why ideal adjacent dipole lines are equally spaced even though the energy levels are not.

The Landau Levels card separates cyclotron excitation from guiding-center degeneracy. It gives both Landau-gauge wavefunctions and gauge-independent algebra, bulk flux counting, the three-dimensional dispersion, and a signed-charge Zeeman extension.

These two systems both organize states into degenerate multiplets, but for different reasons:

• rotor $M$ degeneracy follows ordinary rotational invariance;
• Landau guiding-center degeneracy follows magnetic translation structure and flux density.

Characteristic scales

System	Energy scale	Length or inertia scale	Main scaling
Infinite well	$\hbar^2/(2mL^2)$	$L$	$E_n\propto n^2/(mL^2)$
Finite well	$\hbar^2/(2ma^2)$ and $V_0$	$a$	roots depend on $z_0=a\sqrt{2mV_0}/\hbar$
Oscillator	$\hbar\omega$	$\ell=\sqrt{\hbar/(m\omega)}$	equally spaced levels
Hydrogenic ion	$\mu c^2(Z\alpha)^2/2$	$a_Z=\hbar/(\mu cZ\alpha)$	$E\propto\mu Z^2$, $r\propto1/(\mu Z)$
Linear rotor	$\hbar^2/(2I)$	$I=\mu R_e^2$	$E\propto1/(\mu R_e^2)$
Landau level	$\hbar\lvert q\rvert B/m$	$\ell_B=\sqrt{\hbar/(\lvert q\rvert B)}$	gap $\propto B$, area per state $\propto1/B$

Dimensionless control parameters make comparisons safer. Examples are the finite-well strength $z_0$, the hydrogenic coupling $Z\alpha$, and ratios such as a perturbation energy to $\hbar\omega$ or $B_E$.

Convention checks

Before applying a canonical-system formula, identify:

1. Configuration space: line, interval, half-line, sphere, plane, or three- dimensional space.
2. Boundary conditions: hard wall, decaying tail, periodic identification, regular radial endpoint, or magnetic boundary condition.
3. Energy zero: finite-well depths and ionization energies depend on it.
4. Mass parameter: particle mass, reduced mass, effective mass, or moment of inertia.
5. Charge convention: $q$ is signed, while $\omega_c$ and $\ell_B$ use $\lvert q\rvert$.
6. Normalization measure: $dx$, $r^2dr$, $d\Omega$, box normalization, or continuum delta normalization.
7. Degeneracy count: orbital only, spin resolved, or including additional symmetry and exchange weights.
8. Approximation boundary: infinite wall, rigid bond, point nucleus, quadratic potential, or uniform field.

How the models connect

Several formulas recur because one canonical system reduces locally or algebraically to another:

• a smooth potential near a stable minimum becomes a harmonic oscillator to quadratic order;
• Landau cyclotron motion is an oscillator built from noncommuting kinetic momenta;
• the rigid rotor is the angular part of a fixed-radius particle on a sphere;
• hydrogenic states combine a Coulomb radial equation with spherical-harmonic angular states;
• the deep finite well approaches hard-wall behavior for low-lying states, but retains exponentially decaying tails at every finite depth.

These connections justify reusing methods. They do not erase differences in domain, measure, degeneracy, or boundary conditions.

Related reference layers

• Exactly Solvable Model Cards summarize Hamiltonians, observables, and model failure modes.
• Spin and Angular Momentum Formulas provide $J_\pm$, coupling, spherical harmonics, and tensor rules.
• Dynamics Formulas provide evolution operators, propagators, Green functions, and path integrals.
• Derivation Index locates the canonical route from assumptions to result.
• Examples Index locates calculations that apply these formulas.
• Common Convention Translations compares notation, units, transforms, and phase choices across sources.

Reliability rules

• A familiar spectrum is not portable without its Hamiltonian and domain.
• Degeneracy is part of a level specification, not an optional annotation.
• Gauge-dependent labels may organize states without being observables.
• A normalized radial factor need not be normalized with ordinary $dr$.
• Spectroscopic constants must carry units until the final conversion.
• Reduced and effective masses must be stated rather than silently replaced by a bare mass.
• The ideal model should be named before any correction is appended.

References

• R. Shankar, Principles of Quantum Mechanics, 2nd ed., Springer, 1994.
• D. J. Griffiths and D. F. Schroeter, Introduction to Quantum Mechanics, 3rd ed., Cambridge University Press, 2018.
• C. Cohen-Tannoudji, B. Diu, and F. Laloe, Quantum Mechanics, Wiley, 1977.
• L. D. Landau and E. M. Lifshitz, Quantum Mechanics: Non-Relativistic Theory, 3rd ed., Pergamon, 1977.
• R. N. Zare, Angular Momentum: Understanding Spatial Aspects in Chemistry and Physics, Wiley, 1988.