Discretizing Schrödinger operators and boundaries

Use the nonrelativistic one-dimensional Hamiltonian with kinetic term minus hbar squared over twice the mass times the second derivative, plus a real potential. A centered uniform-grid stencil produces a tridiagonal matrix. Let a be the spatial spacing and define epsilon as the corresponding kinetic energy scale. For a uniform quadrature, use coefficients c-j equal to the sampled wavefunction multiplied by the square root of a, so their squared magnitudes sum to the discrete probability norm.

This coefficient convention makes ordinary matrix inner products match the chosen quadrature. Other bases, nonuniform grids or finite elements can introduce weights or an overlap matrix, which must then enter normalization and Hermiticity checks. A real symmetric stencil is Hermitian in its specified inner product; writing the same entries on an unrelated weighted grid does not preserve that property automatically. State the finite representation before using its eigenvectors as physical states.

Homogeneous Dirichlet conditions set the endpoint wavefunction to zero and produce a hard-wall problem. Periodic conditions instead connect the last grid point to the first and model a ring. These choices change the operator and its spectrum; they are not interchangeable implementation details. On a finite box meant to approximate a localized bound state on the full line, moving the walls farther away is a separate convergence control from decreasing the grid spacing.

The finite matrix also has its own kinetic dispersion. Applying the centered stencil to a grid plane wave gives an energy proportional to two minus twice the cosine of k times a. At small k times a it approximates the continuum quadratic energy, but near the grid scale the discrepancy is large. A stable wavepacket can consequently travel at the wrong group velocity. Norm conservation cannot reveal this phase and transport error by itself.

Multiplying a tiny Hamiltonian by a proposed eigenvector checks signs, indexing and endpoint handling directly. A zero matrix residual confirms an eigenpair of that matrix, not an exact continuum eigenpair. The same state can have an appreciable energy bias relative to a continuum box. Refining a while keeping the physical domain fixed tests the spatial approximation; simply adding more points while also moving the walls changes two controls at once.

The Hilbert-space dimension, represented momentum band and physical potential resolution must all be adequate for the observables of interest. A sharp barrier or rapidly varying state may need more resolution than a smooth low-energy state. Closed finite Hermitian evolution is unitary, whereas absorbing boundaries and open-system equations will deliberately use different norm or density-matrix rules later. Preserving those distinctions prevents every loss of norm from being called numerical failure and every preserved norm from being called physical success.