Numerical Mathematics

Numerical mathematics turns equations into finite calculations whose errors can be estimated. For quantum mechanics, the recurring tasks are building Hamiltonian matrices, finding eigenstates, integrating wavefunctions, and evolving states in time. A useful calculation answers both “what is the result?” and “how much could the approximation change it?”

Helpful background. Review matrix exponentials for time evolution and ordinary differential equations for stepping methods. Norms and Metrics explains how to measure state and matrix errors.

Enter this chapter

Begin with Floating-Point Arithmetic, Conditioning and Stability, and Discretization. They address three different limitations: finite arithmetic, sensitivity of the problem, and the replacement of a continuum by finite data.

Then choose one small problem, such as the infinite well or a two-level system, and carry it through a complete calculation. Benchmark Problems provides exact answers and explains what each example tests.

Two errors in one eigenvalue

Suppose a grid Hamiltonian $H_h$ gives an approximate eigenpair $(E_h,v_h)$. A small residual $\lVert H_hv_h-E_hv_h\rVert$ means that the matrix equation is nearly satisfied. It does not mean that the grid represents the continuum accurately.

In the infinite well, the three-point discretization gives low energies below the exact continuum energies, with error proportional to $h^2$. Solving the matrix to more digits leaves that error untouched. Halving $h$ reduces it by about four once the state is well resolved. Finite Difference Methods derives both spectra.

Choose a route

• A first stationary spectrum: Finite Differences → Matrix Diagonalization → Convergence Tests.
• A large Hamiltonian: Sparse Matrices → Sparse Eigensolvers. Compare a small instance with dense diagonalization first.
• A time-dependent wave packet: Time-Stepping Methods → Matrix Exponentials Numerically → Fast Fourier Transform. Check relative phases as well as norm.
• A continuum equation: Spectral Methods, ODE Solvers, and PDE Solvers explain complementary choices. Numerical Quadrature supplies normalization and matrix-element integrals.

For every route, use Error Estimates to distinguish spatial, temporal, solver, roundoff, and statistical errors. Refine one parameter at a time before checking their combined effect.

Check your understanding

A time integrator preserves $\lVert\psi\rVert$ exactly. Does that prove its transition probabilities are accurate?

Solution

No. It may rotate energy components by incorrect relative phases while preserving their total norm. Their interference, and hence transition probabilities, can then be wrong. Test a two-level system with known oscillation frequency and reduce the time step. Crank–Nicolson provides a concrete example in Time-Stepping Methods.

Page map

Page	Central question
Floating-Point Arithmetic	How does finite representation of real numbers affect quantum calculations?
Conditioning and Stability	Is sensitivity intrinsic to the problem or introduced by the algorithm?
Discretization	Which finite problem is replacing the continuum one?
Finite Difference Methods	How do local derivative stencils become operator matrices?
Spectral Methods	When do global basis expansions converge rapidly?
Numerical Quadrature	How are normalization and matrix-element integrals approximated reliably?
Matrix Diagonalization	How are dense Hermitian eigenpairs computed and checked?
Sparse Matrices	How should large local Hamiltonians be stored or applied?
Sparse Eigensolvers	How are selected eigenpairs extracted without dense diagonalization?
Time-Stepping Methods	Which time steps control stability, phase error, and unitarity?
Matrix Exponentials Numerically	When should one compute an exponential or only its action?
Fast Fourier Transform	How do discrete Fourier grids support derivatives and split evolution?
ODE Solvers	How are initial-value and shooting problems integrated and validated?
PDE Solvers	How are multidimensional quantum equations discretized and evolved?
Error Estimates	Which uncertainties limit the claimed observable?
Convergence Tests	Does the answer approach a stable limit under controlled refinement?
Benchmark Problems	Which exact or controlled cases expose implementation failures?

References

• G. H. Golub and C. F. Van Loan, Matrix Computations, 4th ed., Johns Hopkins University Press, 2013.
• E. Hairer, C. Lubich, and G. Wanner, Geometric Numerical Integration, 2nd ed., Springer, 2006.
• N. J. Higham, Accuracy and Stability of Numerical Algorithms, 2nd ed., SIAM, 2002.
• R. J. LeVeque, Finite Difference Methods for Ordinary and Partial Differential Equations, SIAM, 2007.
• Y. Saad, Numerical Methods for Large Eigenvalue Problems, revised ed., SIAM, 2011.
• D. J. Tannor, Introduction to Quantum Mechanics: A Time-Dependent Perspective, University Science Books, 2007.
• J. M. Thijssen, Computational Physics, 2nd ed., Cambridge University Press, 2007.
• L. N. Trefethen and D. Bau, Numerical Linear Algebra, SIAM, 1997.
• L. N. Trefethen, Spectral Methods in MATLAB, SIAM, 2000.