Wave-Mechanics Models and Lookup Tables

Use these tables when you know the kind of calculation you need but want to check a formula, boundary condition, normalization, or limit. Each entry points to the article that develops the result. For a first encounter with a model, read that explanation before relying on its table entry.

Choosing the formula that matches the problem

A familiar Hamiltonian expression can describe several different systems. Consider

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

On the full line it describes free motion with a continuous energy spectrum. On a hard interval it has sine eigenfunctions and discrete energies. On a periodic interval it has traveling-wave eigenfunctions, a zero-energy state, and pairs of opposite nonzero momenta. The coordinate range and boundary conditions determine which result applies.

Before using an entry, identify three things:

1. The model: its potential, geometry, and physical parameters.
2. The state convention: its normalization measure and the meaning of its labels.
3. The quantity sought: an energy, amplitude, probability, current, or density of states.

This short check prevents common errors. A scattering amplitude is not automatically the square root of a transmission probability when incoming and outgoing velocities differ. A radial function $R(r)$ is normalized with $r^2dr$, while $u(r)=rR(r)$ uses $dr$. A continuum plane wave uses a stated delta or flux normalization rather than unit probability over all space.

The tables are also useful after a calculation. Check dimensions, then take a limit with specified quantities held fixed. For example, a barrier becomes transparent as its width tends to zero at fixed height; shrinking its width while keeping its integrated strength fixed instead gives a delta interaction.

Compare models by what changes

The infinite well and finite well distinguish impenetrable walls from finite confinement: the latter permits evanescent tails and has only finitely many bound states on the line. A step has one interface; a finite barrier has two, allowing interference and tunneling. A harmonic oscillator has an infinite equally spaced ladder, whereas a two-state approximation to a double well retains a selected low-energy pair.

These models isolate mechanisms. A barrier calculation is not a complete theory of alpha decay, a tunneling microscope, or a Josephson junction: those applications add geometry, interactions, many-particle physics, or detector models. A double well is a family of potentials, not one universal closed-form solution. Use Common Hamiltonians to compare the model definitions and scales, then follow the linked derivation for their assumptions.

Reading sequence

Page	What you will learn
Notation and Conventions	Fourier signs, normalization measures, radial notation, charge, and current conventions
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

A compact worked check

Suppose a hard-well calculation gives $E_n=n^2\pi^2\hbar^2/(2mL^2)$. Its dependence on $m$ and $L$ has the right energy dimensions. It increases as confinement becomes tighter, and its allowed labels start at $n=1$ because the candidate $n=0$ sine function is identically zero. These checks explain the formula’s structure; the boundary-value derivation explains why the numerical factor is $\pi^2$.

For a numerical wavefunction sampled on a uniform grid, remember that $\Delta x\sum_i|\psi_i|^2\approx1$. A linear-algebra routine may instead return a vector with $\sum_i|v_i|^2=1$; then $\psi_i\approx v_i/\sqrt{\Delta x}$. Both conventions can be used, but probabilities must use the matching measure.

Exercises

For the same kinetic operator on an interval of length $L$, compare the lowest energy with hard-wall conditions and with periodic conditions.

Solution

Hard walls require the wavefunction to vanish at both ends, giving lowest energy $\pi^2\hbar^2/(2mL^2)$. Periodicity of the function and its derivative permits a constant normalized eigenfunction, with energy zero. The different answers reflect different physical domains, not a disagreement about the kinetic operator.

A scattering problem has equal masses on both sides, transmitted wavenumber $k_R=k_L/2$, and transmitted amplitude $t=1$. What is the transmitted flux fraction?

Solution

For open plane-wave channels with incident amplitude one, $T=(k_R/k_L)|t|^2=1/2$. The slower transmitted wave carries less probability per unit time at the same amplitude. This is a conversion from a specified amplitude to a flux; whether that amplitude actually solves a particular potential must be checked separately.

Which model comparison best distinguishes quantization caused by hard walls from bound states with evanescent tails?

Solution

Compare the infinite and finite square wells. Hard-wall eigenfunctions vanish at the walls, whereas finite-well eigenfunctions and their first derivatives match to exponential tails outside. The different matching conditions change both energies and normalization.

Why does the oscillator recur in molecular vibrations and field modes?

Solution

Near a nondegenerate stable minimum, the leading change in a smooth potential is quadratic. For coupled small oscillations, normal coordinates diagonalize that quadratic problem into oscillators. Free bosonic fields similarly decompose into oscillator modes. Anharmonic interactions couple the modes and limit the independent-oscillator approximation.

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.
• N. J. Higham, Accuracy and Stability of Numerical Algorithms, 2nd ed., SIAM, 2002.
• 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.
• 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.