Canonical Plots Gallery

Canonical wave-mechanics systems are often recognized by their plots before they are recognized by their formulas. Hard-wall nodes, evanescent tails, tunneling suppression, Gaussian spreading, oscillator ladders, angular lobes, and Landau-level degeneracy each have a characteristic visual signature.

The gallery below is a visual index, not a replacement for the linked derivations. The three panels use dimensionless coordinates. Amplitudes, probability densities, and energies have different vertical scales; compare the behavior within a panel rather than heights across panels.

A compact visual index for standard wave-mechanics models. The panels show exact sine shapes in a hard box, Gaussian densities with widths in a ratio of two, and equally spaced oscillator levels above a parabolic potential. The first panel uses amplitude offsets, not energy offsets; the Gaussian curves have equal area when integrated over the full line.

Recognizing Wavefunction Shapes

Use this page as a recognition aid. When a calculation produces a wavefunction, density, spectrum, or numerical plot, compare it with the relevant panel and then follow the linked derivation for the actual assumptions and normalization conventions.

A good plot should make three things visible:

• the physical setup, such as walls, barriers, wells, angular constraints, or magnetic field;
• the boundary or asymptotic behavior, such as nodes, continuity, decay, or oscillatory scattering tails;
• the scale or quantum number being varied, such as $n$, $L$, $\omega$, $\ell_B$, or time.

Related Plots and Models

Model or plot	What to Look For	Main Lesson	Full explanation
Infinite well eigenfunctions	Nodes at hard walls and increasing oscillation count with $n$	Boundary conditions quantize energy	Infinite Square Well
Finite-well tails	Oscillatory interior with exponential decay outside the well	Finite barriers do not force $\psi$ to vanish	Finite Square Well
Barrier tunneling	Evanescent behavior inside the barrier and smaller transmitted wave	Transmission is a current ratio, not just an amplitude sketch	Rectangular Barrier Tunneling
Gaussian spreading	An initially unchirped free Gaussian broadens as time increases	Free-particle dispersion changes width while preserving norm	Wave Packet Spreading
Oscillator eigenstates	Parabolic potential, equally spaced levels, alternating parity	Quadratic potentials produce a universal ladder	Quantum Harmonic Oscillator
Coherent-state motion	Phase-space orbit of the packet center	Coherent states preserve their Gaussian shape in a harmonic potential	Coherent States
Hydrogen orbitals	Radial localization and angular lobes	Central potentials separate into radial and angular structure	Hydrogen Atom
Spherical harmonics	Angular lobes and nodal surfaces	Angular wavefunctions are geometry, not radial dynamics	Spherical Harmonics
Landau-level wavefunctions	Oscillator-like transverse states at different guiding centers	Uniform magnetic fields quantize cyclotron motion and create degeneracy	Landau Levels

Reading Axes and Scales

Check whether the vertical axis shows an amplitude, a density, or an energy. A negative amplitude is allowed; a negative probability density is not. Curves drawn at different energy offsets are not meant to have their heights compared directly. For radial plots, distinguish $\lvert R(r)\rvert^2$ from the shell probability density $r^2\lvert R(r)\rvert^2$.

Common Mistakes

• Treating schematic vertical offsets as physical energy values.
• Comparing unnormalized wavefunction heights between different systems.
• Plotting $\psi$ when the physical question is about $\lvert\psi\rvert^2$.
• Hiding boundary behavior by cropping the plot too tightly.
• Drawing tunneling as a literal classical path through a wall rather than an evanescent wave.
• Showing hydrogen orbitals without saying whether the plot is an amplitude, probability density, or surface of constant probability density.
• Plotting gauge-dependent Landau wavefunctions without stating the gauge.

Exercises

1. For a finite-well bound state, why do the tails outside the well matter physically even though the particle is classically forbidden there?

Solution

The finite outside region is not excluded from the Hilbert space. For a bound state below the outside potential, the solution outside the well is evanescent rather than zero. These tails affect normalization, matching conditions, bound-state energies, tunneling intuition, and overlap with nearby wells.

2. A numerical plot of a Landau-gauge wavefunction shows a Gaussian-like oscillator state centered at a value $x_0$. What should be checked before calling it a physical density plot?

Solution

One should state the gauge, the magnetic length $\ell_B$, the relation between $x_0$ and the conserved momentum, and the normalization convention. Plot $\lvert\psi\rvert^2$, not just its real part, and check its normalization. A gauge transformation of the same state changes its local phase but leaves its density unchanged. Different basis states in a degenerate Landau level can have different densities; that is a change of state, not a gauge effect.

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.
• L. D. Landau and E. M. Lifshitz, Quantum Mechanics: Non-Relativistic Theory, 3rd ed., Pergamon, 1977.
• R. Shankar, Principles of Quantum Mechanics, 2nd ed., Springer, 1994.