Boundary Conditions in Computation

Boundary conditions enter a numerical Hamiltonian through its endpoint rows, trial functions, or interface equations. They are part of the physical model. A large matrix with the wrong endpoint rule is a highly resolved calculation of a different problem. This page translates the physical boundary conditions into discrete operations and verification tests.

Required background. Boundary conditions supplies the physical domains and flux requirements; array representations supplies weighted inner products.

Dirichlet walls and eliminated unknowns

For $\psi(0)=\psi(L)=0$, keep only the $N$ interior samples with $h=L/(N+1)$. The centered kinetic matrix is

$$T=\frac{\hbar^2}{2mh^2} \begin{pmatrix} 2&-1&&\\-1&2&\ddots&\\&\ddots&\ddots&-1\\&&-1&2 \end{pmatrix}.$$

The missing endpoint values are zero, not extra unknowns. This produces an ordinary Hermitian eigenproblem. Keeping endpoint rows that impose $\psi=0$ while leaving incompatible endpoint columns in a matrix can destroy symmetry or introduce spurious eigenvalues. Eliminate constrained degrees of freedom consistently.

For nonzero prescribed boundary values in an inhomogeneous boundary-value solve, elimination contributes known terms to the right-hand side. A stationary eigenfunction of a homogeneous quantum domain does not normally have an arbitrary nonzero endpoint value prescribed independently of its amplitude.

Neumann conditions and the discrete metric

For a grid including both endpoints, reflecting ghost values enforces $\psi'(0)=\psi'(L)=0$. The left kinetic row is $t(2\psi_0-2\psi_1)$, with $t=\hbar^2/(2mh^2)$, while an interior row has coefficients $(-t,2t,-t)$.

This sample-space matrix is not symmetric in the Euclidean inner product. With trapezoidal weights

$$W=h\operatorname{diag}(1/2,1,\ldots,1,1/2),$$

it satisfies $T^\dagger W=WT$. Transforming by $W^{1/2}$ restores an ordinary Hermitian representation. Check the constant vector: the free Neumann kinetic operator must annihilate it. This catches a mistakenly retained Dirichlet endpoint term.

Other discretizations, such as cell-centered finite volumes or variational finite elements, implement Neumann data differently. Their endpoint formulas and metrics must be derived together; combining a row from one scheme with weights from another is not justified.

Periodic and twisted endpoints

For a ring of length $L$, use $N$ distinct samples $x_j=jh$, $h=L/N$, without storing the endpoint $L$ a second time. A twist obeys $\psi(x+L)=e^{i\theta}\psi(x)$. Consequently,

$$T_{0,N-1}=-t e^{-i\theta}, \qquad T_{N-1,0}=-t e^{i\theta}$$

for a nearest-neighbor kinetic stencil with $N>2$. These corner entries are conjugates. Its plane-wave modes have $k_n=(2\pi n+\theta)/L$ and discrete kinetic energies $4t\sin^2(k_nh/2)$. At $\theta=0$, the constant mode has zero energy.

The same phase on both corner entries generally makes the operator non-Hermitian. A twist can also be distributed through link phases by a gauge transformation, but the total phase around the ring must remain the same. Compare spectra under that redistribution as a gauge-consistency check.

Artificial boxes, interfaces, and outgoing waves

A bound state on the infinite line is often approximated by distant Dirichlet walls. Increase the box at fixed spatial resolution and monitor the target energy and tail weight. A small wavefunction value at one endpoint can be misleading if the boundary accidentally lies near a node. Inspect the tail over a region and compare several box sizes.

At a finite step in a constant-mass potential, impose continuity of $\psi$ and $\psi'$. At a delta interaction, the derivative jumps; matching methods implements that condition explicitly. Variable-mass kinetic operators require their own flux-continuity rule and must not inherit the constant-mass derivative rule without derivation.

A hard wall reflects a propagating packet. Enlarging the box only delays its return. Use absorbing boundaries when an open evolution is intended, and measure artificial reflection. Absorption deliberately creates a non-Hermitian effective problem; its spectrum and norm tests differ from those of a closed box.

For radial problems, regularity applies to the correct radial variable. In three dimensions, $u(r)=rR(r)$ has norm $\int|u|^2dr$ and usually obeys $u(0)=0$ for regular nonsingular states. That does not mean the original $s$-wave function $R(0)$ must vanish.

Exercises

A free twisted-ring matrix has corner entries both equal to $-t e^{i\theta}$. For which twists is that compatible with Hermiticity, and what is the general repair?

Solution

The two entries can be equal and conjugate only when $e^{i\theta}$ is real, so $\theta=0$ or $\pi$ modulo $2\pi$. In general use conjugate corner phases, with their orientation fixed by $\psi(x+L)=e^{i\theta}\psi(x)$.

References

• LeVeque, R. J., Finite Difference Methods for Ordinary and Partial Differential Equations, SIAM, 2007, author resources — endpoint stencils and discrete stability.
• Trefethen, L. N., Spectral Methods in MATLAB, SIAM, 2000, author resources — periodic grids and boundary-value discretization.