Limiting Cases Table

Limiting cases are physical error checks. A formula for a canonical system should reduce to a simpler model when a wall becomes infinite, a barrier disappears, a packet becomes broad, a magnetic field turns off, or a quantum number becomes large. If it does not, either the calculation has a mistake or the limit has been taken while holding the wrong quantity fixed.

This table is a diagnostic layer. The canonical derivations live on the linked model pages; the entries here state what behavior should be recovered and what subtlety usually causes wrong answers.

How to Use the Table

Before trusting a result, identify the dimensionless control parameter and ask what happens in the small, large, or degenerate limit. The most useful checks are often qualitative:

• a finite wall should not become a hard wall until the penetration length becomes negligible;
• a tunneling barrier should become transparent when its width or height disappears;
• a discrete spectrum should become dense when the confining length goes to infinity;
• a quantum probability distribution should approach a classical time-spent distribution only after suitable averaging;
• a singular limit, such as a delta potential, depends on which product of parameters is held fixed.

The last point is important. The limit $a\to0$ of a rectangular barrier is not unique: if the height stays fixed the barrier disappears, while if $V_0a$ stays fixed it becomes a delta interaction.

Lookup Table

Model or result	Limit	Expected behavior	Good diagnostic	Common trap
Infinite square well from a finite well	well depth $V_0\to\infty$ at fixed width $L$	exterior penetration length goes to zero and bound energies approach $E_n=n^2\pi^2\hbar^2/(2mL^2)$	compare the lowest finite-well energies with the hard-wall spectrum	imposing hard-wall boundary conditions before the depth is actually infinite
Shallow finite well	$V_0\to0^+$ for an attractive one-dimensional well	only the lowest bound state survives; excited states disappear through threshold	track bound-state count as depth increases	expecting every infinite-well level to have a shallow-well descendant
Delta potential from a narrow well	width $a\to0$, depth $V_0\to\infty$, area $V_0a$ fixed	the wavefunction remains continuous but its derivative has a finite jump	recover the matching rule for $V(x)=\lambda\delta(x)$	letting both width and area vanish, which gives the free particle instead
Potential step	$V_0\to0$ or $E\gg V_0$	reflection tends to zero and transmission tends to one	check probability current, not just amplitude	using $\lvert A\rvert^2$ ratios when incoming and transmitted wave numbers differ
Rectangular barrier, transparent limit	barrier width $a\to0$ at fixed height, or height $V_0\to0$	$T\to1$ and $R\to0$	the transfer matrix should approach the identity	confusing this with the fixed-area delta-barrier limit
Rectangular barrier, opaque limit	$\kappa a\gg1$ for $E\lt V_0$	transmission is exponentially small, with leading scale $T\propto e^{-2\kappa a}$	plot $\log T$ against $a$	treating an evanescent wave as an ordinary probability density moving through the barrier
Resonant barrier or well	phase accumulated in the interior equals an integer multiple of $\pi$	transmission can approach one even with nonzero interfaces	look for zeros of the reflection amplitude	assuming every barrier suppresses transmission monotonically
Free particle from a box	$L\to\infty$	level spacing tends to zero and sums over $n$ become integrals over $k$	verify the density of states before replacing sums by integrals	keeping box-normalized amplitudes while using continuum normalization
Gaussian wave packet	initial width $\sigma_x\to\infty$ at fixed mean momentum	momentum spread tends to zero and the packet approaches a plane wave locally	check $\sigma_x\sigma_p=\hbar/2$ for the minimum packet	forgetting that an exact plane wave is not square-normalizable
Harmonic oscillator, large quantum number	$n\gg1$	turning points approach $A_n\simeq \sqrt{2E_n/(m\omega^2)}$ and the coarse-grained density approaches the classical time-spent distribution	compare support near the classical turning points	expecting pointwise convergence of the oscillatory wavefunction
Harmonic oscillator, weak confinement	$\omega\to0$	spacing $\hbar\omega$ goes to zero and oscillator length $\ell=\sqrt{\hbar/(m\omega)}$ diverges	state which energy, coordinate scale, or quantum number is held fixed	calling the limit a free particle without tracking the widening states
Coherent state	large amplitude $\lvert\alpha\rvert\gg1$	center follows the classical oscillator orbit while the packet shape stays Gaussian	compare $\langle x(t)\rangle$ and $\langle p(t)\rangle$ with the classical solution	mistaking small relative uncertainty for zero absolute uncertainty
Two-level avoided crossing	coupling $\Delta\to0$	eigenvalues cross and eigenvectors become diabatic basis states away from degeneracy	diagonalize $a_0I+\mathbf b\cdot\boldsymbol\sigma$ as $\Delta$ changes	using nondegenerate perturbation theory at the crossing
Rabi oscillations	detuning $\lvert\delta\rvert\gg\Omega$	transition amplitude is suppressed by the small mixing angle	check the maximum transition probability, not only the oscillation frequency	assuming large detuning merely changes the period
Three-dimensional box	one side length $L_z\to\infty$	the $z$ spectrum becomes continuous while transverse levels remain discrete	replace $\sum_{n_z}$ by an integral only in that direction	treating the whole spectrum as continuous too early
Central radial problem	angular momentum $\ell\gg1$	centrifugal barrier pushes probability away from the origin	inspect the effective potential $V_{\mathrm{eff}}(r)$	forgetting the radial measure and judging only $R(r)$
Hydrogen atom, Rydberg limit	$n\gg1$ at fixed $Z$	radius scale grows like $n^2a_0/Z$ and binding energy approaches zero as $-1/n^2$	check both the energy and radial scale	treating large-$n$ states as small corrections to the ground state
Hydrogen atom, continuum threshold	$n\to\infty$	bound levels accumulate at $E=0^-$ and connect to Coulomb scattering states	verify the sign convention for the zero of energy	expecting equally spaced levels near ionization
Rigid rotor	$J\gg1$	energy grows as $J^2$ and angular momentum becomes approximately classical in magnitude	compare $E_J=\hbar^2J(J+1)/(2I)$ with $L^2/(2I)$	replacing $J(J+1)$ by $J^2$ at small $J$
Landau levels	$B\to0$	cyclotron spacing $\hbar\omega_c$ and degeneracy density $1/(2\pi\ell_B^2)$ go to zero; the free-particle continuum is recovered after summing many levels	track both spacing and degeneracy	looking at one fixed Landau level and expecting the full free continuum
Strong magnetic field	$B\to\infty$	magnetic length $\ell_B=\sqrt{\hbar/(\lvert q\rvert B)}$ shrinks and Landau-level spacing grows	compare the physical length scale with $\ell_B$	ignoring boundary effects when $\ell_B$ is not small compared with the sample

Order-of-Limits Warnings

Several canonical limits do not commute. A long-time wave-packet limit taken before a large-box limit can show boundary recurrence, while the same packet on the line disperses indefinitely. A weak-barrier limit at fixed area gives a delta interaction, while a weak-barrier limit at fixed height gives no interaction. A classical limit of the oscillator can mean $n\to\infty$ at fixed $\hbar\omega$, $\hbar\to0$ at fixed classical energy, or a coherent state with large action compared with $\hbar$.

When writing a derivation or notebook, state the limiting variables explicitly. Good phrases are “at fixed width,” “at fixed area,” “after coarse graining,” “with energy measured relative to the continuum threshold,” and “with the box-normalized density of states included.”

Quick Consistency Checks

Use these checks when debugging formulas:

• Does the answer have the correct dimension after the limiting parameter is removed?
• Does a reflection coefficient vanish when the interface disappears?
• Does a bound-state wavefunction become less extended when the binding energy becomes more negative?
• Does a continuum result include a density-of-states factor after a box limit?
• Does a large-quantum-number result require averaging before comparison with classical mechanics?
• Does a gauge-dependent wavefunction claim reduce to a gauge-invariant observable in a physical limit?

For numerical work, pair each limit with convergence data. A finite-difference square well should approach the analytic hard-wall spectrum as both the grid spacing and the finite-wall penetration error are reduced. A tunneling notebook should show norm conservation separately from asymptotic reflection and transmission accounting.

Common Mistakes

• Taking a limit without saying what is held fixed.
• Using the right limiting formula with the wrong normalization convention.
• Assuming a finite barrier becomes an infinite wall merely because $E\lt V_0$.
• Forgetting that classical correspondence can require coarse graining.
• Comparing one member of a degenerating spectrum with an entire continuum.
• Treating a singular limit as if it were a smooth perturbation.
• Calling a numerical trend a limit without checking more than one parameter value.

Where This Is Used

• Common Hamiltonians identifies the model parameters whose limits are tested here.
• Spectra and Eigenfunctions Table supplies the formulas that should reduce correctly.
• Boundary Conditions Table explains why hard-wall, finite-wall, and singular limits impose different matching rules.
• Dimensionless Parameters Table identifies the small, large, and fixed quantities used in the limits.
• Canonical Plots Gallery gives visual signatures for many of the same limits.
• Benchmark Problems turns several of these checks into concrete validation tasks.
• Numerical Notebooks Index records how notebook artifacts should state and test their limiting behavior.

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.
• L. D. Landau and E. M. Lifshitz, Quantum Mechanics: Non-Relativistic Theory, 3rd ed., Pergamon, 1977.

Exercises

1. A rectangular barrier has height $V_0$ and width $a$. Explain why the limits $a\to0$ at fixed $V_0$ and $a\to0$, $V_0\to\infty$ at fixed $V_0a$ give different physics.

Solution

At fixed height, the area of the barrier goes to zero, so the transfer matrix approaches the free-particle transfer matrix and the barrier becomes transparent. At fixed area, the barrier becomes singular rather than disappearing. The limiting interaction is a delta potential, which leaves the wavefunction continuous but produces a finite derivative jump.

2. Why is the statement “the large-$n$ harmonic oscillator becomes the classical oscillator” incomplete?

Solution

The wavefunction remains oscillatory and does not converge pointwise to a classical trajectory. The useful statement is that the probability density, after suitable coarse graining, approaches the classical time-spent distribution between the turning points, while wave packets such as coherent states can have expectation values that follow classical motion.

3. In the limit $B\to0$, why is it misleading to follow only the lowest Landau level?

Solution

The Landau-level spacing goes to zero, but so does the degeneracy density of each level. The free-particle continuum is recovered by considering the combined contribution of many levels, not by tracking one fixed level in isolation.