Dimensionless Parameters Table

Dimensionless parameters are the knobs that actually control a canonical wave-mechanics problem. Mass, length, field strength, barrier width, and energy may look independent in dimensional form, but the physics usually depends on combinations such as $kL$, $\kappa a$, $V_0/E_L$, $x/\ell$, $r/a_0$, or $A/(2\pi\ell_B^2)$.

This page is a lookup table. The derivation of scaling conventions is Dimensionless Variables and Scaling; the table below records the combinations most often needed when reading, solving, or numerically checking canonical model pages.

General Template

For a one-dimensional problem with a geometric length $L$, define

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

If a potential has characteristic height $V_0$, a standard strength parameter is

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

Different authors may choose $L$ to be a full width, half-width, oscillator length, Bohr radius, or magnetic length. The exact numerical factors depend on that choice, but the physical question is stable: how large is the potential, phase, action, or flux compared with the natural quantum scale?

Lookup Table

System	Natural variables	Main dimensionless parameters	What they control	Check in limits
Free particle	$kx$, $Et/\hbar$	$kL$, $pL/\hbar$, $Et/\hbar$	phase accumulation over a length or time	$kL\ll1$ means little phase variation over the region; $kL\gg1$ means rapid oscillation
Infinite square well	$\xi=x/L$, $\varepsilon=E/E_L$	quantum number $n$, eigenvalue $\varepsilon_n=n^2\pi^2$	node count and energy scale	all dimensional energies scale as $1/L^2$
Finite square well	$x/L$, $E/E_L$	depth $\lambda=V_0/E_L$, decay length ratio $L/\ell_{\mathrm{tail}}$	number of bound states and leakage outside the well	$\lambda\to\infty$ approaches the infinite well; shallow wells lose excited states
Delta potential	$x/\ell_\delta$	$\ell_\delta=\hbar^2/(m\alpha)$ for $V=-\alpha\delta(x)$; also $k\ell_\delta$ in scattering	bound-state size and scattering strength	small $k\ell_\delta$ gives strong low-energy scattering from the point interaction
Potential step	$k_1x$, $k_2x$	$E/V_0$, $k_2/k_1$	impedance mismatch between asymptotic regions	$V_0\to0$ gives no reflection; current ratios still matter
Rectangular barrier	$ka$, $\kappa a$, $E/V_0$	opacity $\kappa a=a\sqrt{2m(V_0-E)}/\hbar$ for $E\lt V_0$	tunneling suppression and resonance widths	$\kappa a\gg1$ gives exponential suppression; $a\to0$ at fixed height gives transparency
Transfer-matrix chains	region phases $k_ja_j$	interface ratios $k_{j+1}/k_j$, total phase, determinant checks	interference, resonances, and current conservation	unit determinant and current conservation expose algebra mistakes
Gaussian wave packet	$(x-x_0)/\sigma_x$, $t/t_0$	$\sigma_x\sigma_p/\hbar$, $t/t_0$ with $t_0=2m\sigma_x^2/\hbar$ for the common minimum packet convention	spreading and near-plane-wave behavior	broad packets have small momentum spread; long times require boundary checks
Harmonic oscillator	$\xi=x/\ell$, $\tau=\omega t$	$\ell=\sqrt{\hbar/(m\omega)}$, $\varepsilon=E/(\hbar\omega)$, coherent-state amplitude $\alpha$	universal oscillator shape, spectrum, and classicality	$\lvert\alpha\rvert^2\gg1$ gives large action compared with $\hbar$
Coupled oscillators	normal coordinates $Q_a/\ell_a$	frequency ratios $\omega_a/\omega_b$, coupling-to-detuning ratios	normal-mode splitting and degeneracy	equal frequencies create enhanced degeneracies and strong mixing
Two-level system	dimensionless time $\Omega t$	detuning ratio $\delta/\Omega$, coupling ratio $\Delta/\delta$, gap time $\Delta E\,t/\hbar$	mixing angle, transition probability, and adiabaticity estimates	large detuning suppresses transitions; near degeneracy invalidates nondegenerate perturbation
Three-dimensional box	$x_i/L_i$	aspect ratios $L_x/L_y$, $L_y/L_z$, scaled energy $E/E_{L_x}$	degeneracy patterns and density of states	taking one length to infinity makes only that direction continuous
Central potentials	$r/L$, reduced radial coordinate $u/L^{1/2}$	angular parameter $\ell(\ell+1)$, strength $2mL^2V_0/\hbar^2$	centrifugal barrier and radial localization	large $\ell$ pushes probability away from the origin
Hydrogenic Coulomb problem	$\rho=Zr/a_0$, $E/E_{\mathrm H}$	$Zr/a_0$, principal quantum number $n$, Coulomb strength encoded in $a_0/Z$	orbital size, binding energy, and Rydberg scaling	radii grow like $n^2a_0/Z$ and energies scale as $-Z^2/n^2$
Rigid rotor	angles on $S^2$	$E/B_{\mathrm{rot}}$, $J$, with $B_{\mathrm{rot}}=\hbar^2/(2I)$	rotational ladder and degeneracy	$J\gg1$ approaches classical angular momentum magnitude
Minimal coupling	kinetic momentum scale, gauge phase	flux phase $q\Phi_B/\hbar$, magnetic length if $B$ is uniform	gauge-covariant phase and magnetic localization	only gauge-invariant quantities should survive physical comparisons
Landau levels	$x/\ell_B$, $\omega_ct$	$\ell_B=\sqrt{\hbar/(\lvert q\rvert B)}$, $A/(2\pi\ell_B^2)$, $E/(\hbar\omega_c)$	cyclotron radius scale, level spacing, and degeneracy	$B\to0$ requires tracking both vanishing spacing and vanishing degeneracy density
Semiclassical checks	action divided by $\hbar$	$S_0/\hbar$, de Broglie wavelength over variation length	WKB validity and classical correspondence	large action alone is not enough near turning points

Common Parameter Families

Several different-looking parameters express the same comparison:

• phase parameters such as $kL$ compare a wavelength with a length scale;
• opacity parameters such as $\kappa a$ compare an evanescent decay length with a barrier width;
• strength parameters such as $V_0/E_L$ compare potential energy with localization kinetic energy;
• flux parameters such as $A/(2\pi\ell_B^2)$ compare area with the flux quantum scale;
• action parameters such as $S_0/\hbar$ compare classical action with the quantum of action;
• detuning parameters such as $\delta/\Omega$ compare an energy mismatch with a coupling.

The notation can vary. What matters is that each parameter is dimensionless and tied to a physical question.

Scaling Choices That Often Differ

For a finite square well, some authors use the full width $L$ while others use the half-width $a=L/2$. A depth parameter written with $L$ differs by a factor of four from one written with $a$. Both are acceptable if the convention is stated.

For the harmonic oscillator, some derivations write the stationary equation with energy $E/(\hbar\omega)$, while others multiply through by $2$ and use $2E/(\hbar\omega)$. The physics is unchanged; the oscillator length $\ell=\sqrt{\hbar/(m\omega)}$ is the key scale.

For hydrogen, some pages use the Bohr radius $a_0$ and keep $Z$ explicit, while others use the hydrogenic length $a_0/Z$. Do not compare radial plots or expectation values until the convention is clear.

For Landau levels, the magnetic length sets the wavefunction scale, while the flux count sets degeneracy. A calculation can get $\hbar\omega_c$ right and still get the state count wrong if it ignores boundary conditions.

Common Mistakes

• Calling a quantity dimensionless because it has been written in natural units without naming the scale.
• Mixing full-width and half-width conventions in square-well formulas.
• Treating $\kappa a$ and $ka$ as interchangeable.
• Reporting a numerical grid spacing without comparing it to the smallest physical length scale.
• Comparing two oscillator plots without checking whether $x$ is measured in $\ell$.
• Forgetting that $A/(2\pi\ell_B^2)$ is a degeneracy estimate, not a single-particle energy.
• Hiding the control parameter by setting too many constants to one at the start.

Where This Is Used

• Dimensionless Variables and Scaling derives the general scaling rules summarized here.
• Limiting Cases Table uses these parameters to state small, large, and singular limits.
• Common Hamiltonians lists the model families whose controls appear in the table.
• Benchmark Problems turns several of these dimensionless checks into numerical validation tests.
• Numerical Notebooks Index explains how computational notebooks should record parameter choices.

References

• D. J. Griffiths and D. F. Schroeter, Introduction to Quantum Mechanics, 3rd ed., Cambridge University Press, 2018.
• R. Shankar, Principles of Quantum Mechanics, 2nd ed., Springer, 1994.
• C. Cohen-Tannoudji, B. Diu, and F. Laloë, Quantum Mechanics, Wiley, 1977.
• N. J. Higham, Accuracy and Stability of Numerical Algorithms, 2nd ed., SIAM, 2002.

Exercises

1. A finite square well is written with full width $L$ and depth $V_0$. Define a natural kinetic energy and a dimensionless depth.

Solution

A convenient kinetic scale is

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

The dimensionless depth is

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

If a half-width $a=L/2$ is used instead, the numerical value of the strength parameter changes by a factor of four, so the convention must be stated.

2. For a rectangular barrier with $E\lt V_0$, explain the meaning of $\kappa a$.

Solution

Inside the barrier, the evanescent decay constant is

$$\kappa=\frac{\sqrt{2m(V_0-E)}}{\hbar}.$$

Thus $\kappa a$ is the barrier width measured in decay lengths. When $\kappa a\gg1$, transmission is exponentially suppressed; when $\kappa a\ll1$, the barrier is thin on the evanescent scale.

3. A Landau-level calculation reports the correct spacing $\hbar\omega_c$ but no degeneracy estimate. Which dimensionless parameter is missing?

Solution

The missing parameter is the number of flux-scale area units,

$$N_\phi \simeq \frac{A}{2\pi\ell_B^2}.$$

It controls the ideal bulk degeneracy of each Landau level in area $A$. The energy spacing alone does not determine the number of available states.