Intro to Numerical Quantum Mechanics

Basic numerical solution of quantum states, energies, and discretized wavefunctions.

Conceptual context

Numerical quantum mechanics transforms a continuous differential equation into a discrete object that a computer can handle. $x$ becomes a grid, $\psi$ becomes a collection of amplitudes, and the $E$ operator becomes a matrix $H$. The result is not a closed formula but a set of approximate $E$ values and $\psi _vec$ objects. Obtaining them is only the first step. A credible solution must also satisfy normalization, boundary conditions, and convergence as $\Delta x$ is reduced.

🟢 Essential level

The basic idea is to replace continuous space with a line of grid points. At each point, one stores a value of $\psi$, and curvature is approximated from $\psi ᵢ₋₁$, the central value, and $\psi ᵢ₊₁$. These ingredients build a matrix $H$ acting on $\psi _vec$. Its eigenvalues are allowed $E$ values, and its eigenvectors describe the states. A bound state is localized and usually decays near the domain edges. Excited states have more nodes. The numerical state must be normalized so that $P_tot$ is close to one. Stability should be observed both in the computed eigenvalue and in the spatial shape. The calculation is then repeated on a finer grid. Convergence means that $A$, $B$, and the $\psi$ shape become mutually consistent. A good solution preserves its physical interpretation under refinement.

The grid is therefore a physical approximation, not merely a storage format. Its spacing controls what spatial structure can be represented, while its boundaries decide which tails are retained or discarded.

🔵 Formal level

Basic formulas

The physical starting point relates curvature, $V$, $m$, $ℏ$, $E$, and $\psi$.

One-dimensional time-independent Schrödinger equation

Curvature is approximated with a centered relation involving $\psi ᵢ₋₁$, $\psi ᵢ₊₁$, and $\Delta x$.

Second derivative by finite differences

This approximation generates a tridiagonal $K$ matrix. $U$ is usually placed on the diagonal, and both contributions form $H$.

Finite-difference discretized Hamiltonian

The differential problem becomes an algebraic eigenvalue problem.

Quantum matrix eigenvalue problem

Real student questions

Why are energies discrete? Because boundary conditions turn the problem into an eigenvalue problem.

Does a negative $\psi$ mean negative probability? No. Probability uses the squared modulus.

Is a finer grid always better? It lowers truncation error but increases cost and may expose roundoff.

Does normalization prove correctness? No. Energy, shape, boundaries, and convergence must also be checked.

How do I match the same state? Use nodes, parity, shape, and overlap.

Final synthesis

Numerical quantum mechanics turns Schrödinger’s equation into a matrix problem. A trustworthy result combines converged $E$, stable $\psi$, correct normalization, sufficient domain size, and explicit control of $\Delta x$.