Model Encyclopedia

The Model Encyclopedia is the quick-reference layer for canonical wave mechanics. It helps identify a model, recover its standard result, and check whether a derivation or numerical calculation has used the right domain, normalization, scales, and limiting behavior.

These pages are not alternate derivations. Every compact row links to the canonical article where the Hamiltonian, assumptions, spectrum, eigenfunctions, observables, and caveats are developed. Use the tables to orient and verify; use the linked model pages to understand and cite.

Scope and ownership

The encyclopedia owns seven kinds of lookup:

• standard Hamiltonian forms and their natural scales;
• standard spectra and eigenfunction shapes;
• boundary, matching, and asymptotic conditions;
• limiting-case diagnostics;
• dimensionless control parameters;
• normalization conventions and measures;
• visual signatures of canonical systems.

It does not own the derivation of the infinite well, oscillator, hydrogen atom, Landau levels, or any other listed system. It also does not replace the broader Model Encyclopedia or Operator and Hamiltonian Library in the Reference. This chapter is specifically organized around solving and checking coordinate-space model problems in this volume.

The central editorial rule is simple: a table row may state a standard result, but it must preserve enough assumptions to prevent misuse and must point to one canonical derivation.

A model is more than a Hamiltonian formula

A complete wave-mechanics model can be summarized schematically as

$$\mathfrak M = \left( \mathcal Q, d\mu, \mathcal H, H, D(H), \boldsymbol\lambda \right).$$

Here

• $\mathcal Q$ is configuration space;
• $d\mu$ is the integration measure;
• $\mathcal H=L^2(\mathcal Q,d\mu)$ is the Hilbert space;
• $H$ is the Hamiltonian action;
• $D(H)$ is its operator domain, including boundary conditions;
• $\boldsymbol\lambda$ collects physical parameters and conventions.

The same differential expression can describe different systems. For example,

$$H = -\frac{\hbar^2}{2m} \frac{d^2}{dx^2}$$

has a continuous spectrum on the line, a discrete sine spectrum on a Dirichlet interval, and a discrete plane-wave spectrum on a periodic circle. The expression alone does not determine the physics.

Likewise, an eigenfunction formula is incomplete until its coordinate range, measure, state label, phase convention, and normalization are known. A limit is incomplete until the fixed quantities are stated. A plot is incomplete until the displayed quantity and scale are identified.

The verification workflow

For a new derivation, code result, or formula encountered in a paper, use the encyclopedia in this order:

Step	Question	Reference page
1. Identify	What ideal system and Hamiltonian are being used?	Common Hamiltonians
2. Complete	What configuration space, domain, and matching rules complete the operator?	Boundary Conditions Table
3. Normalize	What measure and state-label convention define the amplitudes?	Normalization Table
4. Compare	What exact spectrum or eigenfunction form should be recovered?	Spectra and Eigenfunctions Table
5. Scale	Which dimensionless combinations control the answer?	Dimensionless Parameters Table
6. Stress-test	What happens in weak, strong, large, small, or singular limits?	Limiting Cases Table
7. Inspect	Does the result have the expected nodes, tails, symmetry, and asymptotics?	Canonical Plots Gallery

This order is not arbitrary. Looking up a spectrum before identifying the domain encourages memorized-formula errors. Comparing plots before checking normalization can make two equivalent states look inconsistent. Taking a limit before nondimensionalizing can hide which physical ratio actually tends to zero or infinity.

Common Hamiltonians

Common Hamiltonians is the starting lookup. Each row records

• the ideal Hamiltonian;
• natural coordinates;
• discrete, continuous, or mixed spectrum type;
• the controlling length or energy scale;
• the canonical derivation page.

The table includes free particles, wells, point interactions, steps, barriers, oscillators, finite-dimensional two-level systems, boxes, central potentials, hydrogen, rotors, minimal electromagnetic coupling, Landau levels, and the Fock–Darwin oscillator.

Read its Hamiltonian column as a compact identifier, not a complete specification. For instance,

$$\frac{p^2}{2m}+V(x)$$

does not say whether $V$ is finite, singular, periodic, confining, or asymptotically constant. Those distinctions determine matching rules and spectral type.

The key-scale column is a dimensional checksum. An infinite well of width $L$ must produce energies proportional to

$$E_L = \frac{\hbar^2}{2mL^2}.$$

A harmonic oscillator must expose $\hbar\omega$ and $\ell=\sqrt{\hbar/(m\omega)}$. A uniform magnetic field must expose $\hbar\omega_c$ and $\ell_B$. If the natural scales have disappeared from a result without a stated unit convention, stop and inspect the setup.

Spectra and eigenfunctions

Spectra and Eigenfunctions Table records standard exact forms while keeping the canonical derivation one link away. It is most useful for checking

• energy dependence on quantum numbers;
• expected parity, angular, or product structure;
• discrete versus continuum labels;
• ideal degeneracy counts;
• the broad functional family of an eigenstate.

The table distinguishes formulas such as

$$E_n \propto n^2$$

for a box,

$$E_n = \hbar\omega \left( n+\frac12 \right)$$

for an oscillator, and

$$E_n \propto -\frac{1}{n^2}$$

for ideal hydrogenic bound states. The symbol $n$ does not mean the same physical label in all three rows, so one should never compare formulas by label name alone.

Continuum eigenfunctions require particular caution. A plane wave, scattering state, or positive-energy Coulomb state is a generalized eigenfunction, not an ordinary square-integrable vector. Its displayed amplitude depends on whether the state is normalized to a delta function in $p$, $k$, energy, or flux.

Boundary and matching conditions

Boundary Conditions Table completes the differential equation. Its main cases are

Situation	Essential condition
Infinite wall	Wavefunction vanishes at the wall
Finite regular interface	$\psi$ and $\psi'$ are continuous for constant mass
Delta interaction	$\psi$ is continuous and $\psi'$ has a prescribed jump
Periodic coordinate	State and operator-domain derivatives match after one period
Bound-state infinity	Growing asymptotic branch is excluded and the state is square-integrable
Scattering infinity	Incoming channel and outgoing response are specified
Radial origin	The physical regular branch is selected, usually with $u(0)=0$

For $V(x)=\lambda\delta(x)$, integration across the singular point gives

$$\psi'(0^+)-\psi'(0^-) = \frac{2m\lambda}{\hbar^2} \psi(0).$$

Imposing derivative continuity would erase the interaction. Conversely, importing this jump into a finite step would invent a singular force that is not present.

Boundary conditions also encode the experiment. A scattering solution with an incoming wave from the left is not the same state as one with incidence from the right, even though both solve the same local equation. A periodic box used to regularize continuum states is not automatically a physical ring.

Normalization and measure

Normalization Table begins with the question

$$\int\lvert\psi\rvert^2,d\mu=1: \qquad \text{what is }d\mu?$$

Common answers include

Object	Measure or convention
Bound state on a line	$dx$ and unit norm
Position wavefunction in three dimensions	$d^3r$ and unit norm
Radial function $R(r)$	$r^2dr$
Reduced radial function $u(r)=rR(r)$	$dr$
Angular wavefunction	$d\Omega=\sin\theta\,d\theta\,d\phi$
Momentum continuum state	Delta normalization in the chosen label
Scattering channel	Unit incident amplitude or unit flux, stated explicitly
Periodic-box plane wave	Inverse square root of box length or volume
Numerical grid vector	Quadrature-weighted norm

If $p=\hbar k$, then

$$\delta(p-p') = \frac{1}{\hbar} \delta(k-k').$$

States normalized in $p$ and $k$ therefore carry different prefactors. This is a label conversion, not a physical disagreement.

For a uniform numerical grid, a library vector satisfying

$$\sum_i\lvert v_i\rvert^2=1$$

corresponds approximately to sampled continuum values

$$\psi(x_i) \simeq \frac{v_i}{\sqrt{\Delta x}}.$$

Checking only the Euclidean vector norm can conceal a missing grid measure.

Dimensionless controls

Dimensionless Parameters Table identifies the combinations that actually control behavior. For a one-dimensional length $L$,

$$E_L = \frac{\hbar^2}{2mL^2}, \qquad \xi=\frac{x}{L}, \qquad \varepsilon=\frac{E}{E_L}.$$

A potential scale $V_0$ produces

$$\lambda = \frac{V_0}{E_L} = \frac{2mV_0L^2}{\hbar^2}.$$

Other recurring families are

• phase accumulation $kL$;
• barrier opacity $\kappa a$;
• oscillator coordinate $x/\ell$;
• Coulomb coordinate $Zr/a_0$;
• detuning-to-coupling ratio $\delta/\Omega$;
• magnetic area $A/(2\pi\ell_B^2)$;
• semiclassical action $S/\hbar$.

Numerical factors depend on conventions. A finite well parameter based on full width $L$ differs from one based on half-width $a=L/2$. The correct practice is to state the definition, not to insist that one convention is uniquely standard.

Dimensionless form is also the bridge between analytic and numerical work. A grid spacing is meaningful only relative to the smallest physical length, and a propagation time step is meaningful only relative to the fastest energy or frequency scale.

Limiting cases

Limiting Cases Table is the main error-detection tool. A correct result should recover simpler physics when parameters are removed or made extreme, but the limit must state what remains fixed.

For a rectangular barrier of height $V_0$ and width $a$:

• $a\to0$ at fixed $V_0$ removes the barrier;
• $a\to0$ and $V_0\to\infty$ at fixed $V_0a$ produces a delta interaction;
• $\kappa a\to\infty$ at sub-barrier energy produces exponential suppression.

These are different limits of the same family. Saying only “take the thin-barrier limit” is insufficient.

Several other checks require combined behavior. As $B\to0$, Landau-level spacing tends to zero, but the degeneracy density of each level also tends to zero. The two-dimensional free continuum is recovered from many increasingly dense levels, not by following one fixed Landau level.

Classical correspondence also needs care. A highly excited oscillator eigenfunction does not converge pointwise to a classical orbit. After coarse graining, its probability density approaches the classical time-spent distribution; a localized coherent state provides a different correspondence through expectation-value motion.

Reading canonical plots

Canonical Plots Gallery is a recognition index for nodes, tails, tunneling, packet spreading, oscillator structure, angular lobes, and magnetic guiding centers. It should answer “does this look structurally plausible?” rather than replace a quantitative check.

Before comparing a plot, identify

1. whether it displays $\psi$, $\operatorname{Re}\psi$, $\lvert\psi\rvert$, or $\lvert\psi\rvert^2$;
2. whether vertical offsets or rescaling are schematic;
3. which coordinate measure turns the plotted quantity into probability;
4. which parameters and units set the axes;
5. which gauge, basis, or phase convention affects the appearance;
6. which boundaries, turning points, or asymptotic regions are included.

A radial plot of $\lvert R(r)\rvert^2$ is not the radial probability density; the latter contains $r^2$. A real spherical-harmonic lobe plot is a basis-dependent visualization inside a degenerate angular subspace. One Landau-gauge orbital is gauge dependent even though the Landau spectrum and flux degeneracy are physical.

Two compact audits

Finite square well

A finite-well result should survive the following sequence:

Check	Expected statement
Hamiltonian	Piecewise finite attractive potential plus kinetic energy
Domain	$\psi$ and $\psi'$ match at finite interfaces
Bound-state normalization	Unit norm on the whole line, including evanescent tails
Control parameter	Depth relative to $\hbar^2/(2mL^2)$
Spectrum	Finite number of discrete bound states plus continuum
Deep-well limit	Low levels approach hard-wall values at fixed width
Plot	Oscillatory interior and decaying exterior tails

If a proposed finite-well eigenfunction vanishes exactly at each finite interface, it has probably imported infinite-wall boundary conditions.

Landau level

A Landau-level result needs a different audit:

Check	Expected statement
Hamiltonian	$(\mathbf p-q\mathbf A)^2/(2m)$ in uniform $\mathbf B$
Gauge	Vector potential and wavefunction labels are stated
Scales	$\omega_c=\lvert q\rvert B/m$ and $\ell_B=\sqrt{\hbar/(\lvert q\rvert B)}$
Spectrum	$E_n=\hbar\omega_c(n+1/2)$ for spinless transverse motion
Degeneracy	$A/(2\pi\ell_B^2)$ in the ideal bulk
Three-dimensional extension	Free longitudinal kinetic energy remains
Gauge check	Energies and physical currents agree across gauges

Getting the level spacing right does not automatically get the state count, boundary physics, or normalization right.

Reading route

For quick problem setup, use Common Hamiltonians, then Boundary Conditions Table and Normalization Table.

For analytic or numerical verification, continue with Dimensionless Parameters Table, Spectra and Eigenfunctions Table, and Limiting Cases Table. Use Canonical Plots Gallery last, after the plotted quantity and normalization are known.

Page map

Page	Canonical role
Common Hamiltonians	Model identification, coordinates, spectrum type, and key scales
Spectra and Eigenfunctions Table	Standard exact results with canonical derivation links
Boundary Conditions Table	Endpoint, interface, singular, periodic, radial, and scattering domains
Limiting Cases Table	Weak, strong, singular, classical, continuum, and order-of-limits checks
Dimensionless Parameters Table	Scale-free control parameters and convention warnings
Normalization Table	Measures, square and delta norms, flux, radial, angular, and grid conventions
Canonical Plots Gallery	Visual recognition and plotting standards

Common mistakes

• Treating the Hamiltonian differential expression as a complete model.
• Looking up a spectrum before checking the operator domain.
• Copying an eigenfunction without its coordinate range and normalization convention.
• Using square normalization for a continuum state.
• Confusing $R(r)$ with $u(r)=rR(r)$ or omitting the radial measure.
• Using amplitude ratios instead of current ratios in scattering.
• Mixing full-width and half-width definitions of a dimensionless well parameter.
• Taking a singular limit without naming the fixed product of parameters.
• Expecting pointwise quantum-to-classical convergence where coarse graining is required.
• Comparing schematic plot heights as physical amplitudes.
• Treating a gauge-dependent orbital picture as a gauge-invariant observable.
• Citing a reference table where the canonical derivation should be cited.

References

• 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.
• 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.
• L. D. Landau and E. M. Lifshitz, Quantum Mechanics: Non-Relativistic Theory, 3rd ed., Pergamon, 1977.
• M. Reed and B. Simon, Methods of Modern Mathematical Physics II: Fourier Analysis, Self-Adjointness, Academic Press, 1975.
• N. J. Higham, Accuracy and Stability of Numerical Algorithms, 2nd ed., SIAM, 2002.

Exercises

1. Explain why $-\hbar^2d^2/(2m\,dx^2)$ does not determine whether the spectrum is continuous or discrete. Give the line, hard interval, and periodic interval as examples.

Solution

The differential expression must be paired with a configuration space and operator domain. On the full line, generalized plane waves have

$$E(k)=\frac{\hbar^2k^2}{2m},$$

with continuous $k$. On an interval of length $L$ with Dirichlet conditions, the eigenfunctions are sines and

$$E_n = \frac{n^2\pi^2\hbar^2}{2mL^2}, \qquad n=1,2,\ldots .$$

On a periodic interval, the eigenfunctions are $e^{i2\pi nx/L}$ and

$$E_n = \frac{2\pi^2\hbar^2n^2}{mL^2}, \qquad n\in\mathbb Z.$$

The local kinetic expression is the same, while the global domains produce different spectra and degeneracies.

2. A narrow rectangular barrier has width $a$ and height $V_0$. Compare the limits $a\to0$ at fixed $V_0$ and $a\to0$, $V_0\to\infty$ at fixed $V_0a$.

Solution

At fixed height, the integrated barrier strength $V_0a$ tends to zero. The interaction disappears, so reflection tends to zero and transmission tends to one.

At fixed nonzero $V_0a$, the barrier does not disappear. It becomes a delta interaction. The limiting wavefunction is continuous, but its derivative has a finite jump proportional to the fixed area. The second limit is singular and cannot be inferred from width alone.

3. A radial calculation returns a function $R(r)$ satisfying $\int_0^\infty\lvert R(r)\rvert^2dr=1$. Why is this generally not the physical radial normalization, and how can the reduced function repair it?

Solution

For a separated three-dimensional state, the spatial measure is

$$d^3r=r^2dr\,d\Omega.$$

If the angular function is normalized, the radial condition is

$$\int_0^\infty \lvert R(r)\rvert^2r^2dr = 1.$$

Defining

$$u(r)=rR(r)$$

gives

$$\int_0^\infty\lvert u(r)\rvert^2dr=1.$$

Thus ordinary $dr$ normalization belongs to the reduced radial function $u$, not generally to $R$.

4. In the limit $B\to0$, a spinless Landau level has spacing $\hbar\omega_c\to0$. Why is this fact alone insufficient to demonstrate recovery of the two-dimensional free-particle continuum?

Solution

Each Landau level also has bulk degeneracy density

$$\frac{1}{2\pi\ell_B^2} = \frac{\lvert q\rvert B}{h},$$

which tends to zero with $B$. The free density of states is recovered from the combined contribution of many levels as their spacing collapses. Tracking one fixed level discards both the changing state count and the levels that enter any fixed energy window.