Continuum States in Finite Boxes

A finite box turns a continuum into discrete normalizable eigenvectors. Those vectors can approximate local continuum physics, but their normalization and level spacing depend on the box. Increasing the number of grid points at fixed box size improves the discrete box problem; it does not by itself recover infinite-volume scattering.

Required background. Scattering states and boundary conditions defines continuum states, while boundary conditions in computation distinguishes physical and numerical boundaries.

Radial box quantization

Consider one short-range central potential and a reduced radial function $u_\ell(r)$ regular at the origin. Impose a hard wall at radius $R$ outside the potential. At sufficiently large $kR$ for the asymptotic free form,

$$u_\ell(r)\propto \sin(kr-\ell\pi/2+\delta_\ell(k)),$$

so the box levels approximately satisfy

$$k_nR-\ell\pi/2+\delta_\ell(k_n)=n\pi.$$

For $\ell=0$ and a strictly finite-range potential this free sine form is exact outside the range. For higher partial waves at finite $R$, use the exact Riccati–Bessel boundary equation rather than treating the asymptotic sine as exact.

The levels depend on the wall through $R$, while the scattering phase is a property of the potential with its chosen exterior convention. Changing $R$ helps distinguish both contributions.

Converting discrete and continuum normalization

Let $\langle n|m\rangle=\delta_{nm}$ and $\langle E|E'\rangle=\delta(E-E')$. A local quadrature replacement of the continuum resolution of identity gives

$$|E_n\rangle\approx\frac{|n\rangle}{\sqrt{\Delta E_n}}.$$

Consequently a transition strength per unit energy is approximately $|\langle n|O|i\rangle|^2/\Delta E_n$, not the unscaled discrete strength. The local spacing must be resolved smoothly; isolated box levels near avoided crossings require care.

For the free $s$ wave, $u_n=\sqrt{2/R}\sin(n\pi r/R)$ and $\Delta E_n\approx\pi\hbar^2k_n/(\mu R)$. Hence

$$u_E(r)\approx \sqrt{\frac{2\mu}{\pi\hbar^2k}}\sin(kr),$$

which has the required energy normalization. This provides a direct unit and normalization test.

Density shifts and spectral smoothing

Differentiating the asymptotic quantization relation gives

$$\frac{dn}{dE} \approx \frac{\mu R}{\pi\hbar^2k} +\frac1\pi\frac{d\delta_\ell}{dE}.$$

Subtracting a free-box density with the same boundary convention isolates a phase-dependent density shift. Numerically, a finite list of delta functions must be smoothed or integrated over an energy interval before comparing a density. Choose a smoothing width larger than the local level spacing but smaller than the physical feature of interest, and vary it.

A narrow resonance can produce rapid phase variation and a redistribution of nearby box levels. A level that moves slowly under changing $R$ is a stabilization hint, not by itself a measured resonance width.

Separate the numerical limits

At fixed $R$, refine the spatial discretization until individual levels and local matrix elements stabilize. Then increase $R$ and repeat. For an outgoing wave packet, estimate the time at which reflected amplitude can return from the wall; observables measured afterward are contaminated unless reflection is physically intended.

A finite Hermitian matrix has only real eigenvalues. Its positive-energy levels are not complex resonance poles. Extract scattering information through boundary matching, level-density analysis, or a separately justified continuation.

Long-range Coulomb potentials change the exterior phase and density relation. A free-wave subtraction with no Coulomb correction does not converge to the short-range formula.

Exercises

If a free radial box doubles in radius while a fixed energy window is retained, how do level spacing and normalized box amplitudes scale?

Solution

The local spacing scales as $R^{-1}$ and the box amplitude as $R^{-1/2}$. Dividing the box eigenfunction by $\sqrt{\Delta E}$ removes this radius dependence. A raw discrete transition strength therefore changes with $R$ even when the physical continuum strength density is already converged.

References

• Descouvemont, Pierre, and Daniel Baye, “The R-matrix theory,” Reports on Progress in Physics 73, 036301 (2010), arXiv:1001.0678 — internal-region bases and channel matching.
• Taylor, John R., Scattering Theory: The Quantum Theory of Nonrelativistic Collisions, Dover Publications (2006; original edition 1972) — scattering normalization, amplitudes, and analytic structure.