Math Needed for Computational QM

A quantum calculation on a computer combines a physical model, a finite representation, and a numerical algorithm. Learning to separate their errors matters as much as learning to run the solver. This route selects the mathematics for Computational Quantum Mechanics; start with one small problem whose answer you can also calculate independently.

Matrices, grids, and the inner product

For a first matrix Hamiltonian, review bases and coordinates, Hermitian operators, and matrix diagonalization. You should be able to state what each array index means and calculate the norm of a state.

For a grid wavefunction, add discretization, boundary conditions, and quadrature. Raw samples $\psi_j=\psi(x_j)$ on a uniform grid are approximately normalized by $\Delta x\sum_j|\psi_j|^2=1$. If instead the stored amplitudes are $c_j=\sqrt{\Delta x}\,\psi_j$, the same discrete normalization becomes $\sum_j|c_j|^2=1$. These are two consistent conventions; mixing them produces a normalization error.

Select the method for the calculation

Calculation	Mathematical tools	A useful first check
A few energy levels of a large Hamiltonian	Sparse matrices and sparse eigensolvers	Eigenpair residuals and stability under basis enlargement
A bound-state wavefunction on a grid	Finite differences, boundary conditions, and quadrature	Grid refinement separately from box-size enlargement
Smooth periodic wave dynamics	Spectral methods and the FFT	Aliasing and boundary wraparound
Time evolution	Matrix exponentials, ODE solvers, and time stepping	Time-step refinement and phase accuracy as well as norm
Statistical averages	Monte Carlo and variance	Autocorrelation and the effective number of independent samples
Many-body states	Tensor products and the many-body route	Factor order, symmetry sectors, and fermionic signs
Density-matrix evolution	The open-systems route	Trace, Hermiticity, positivity, and convergence

Conditioning and stability explains why an accurate algorithm cannot remove a problem’s inherent sensitivity. Error estimates, convergence tests, and benchmark problems show how to judge the result.

Separate two kinds of error

For a normalized computed eigenvector $c$ of a Hermitian matrix, define

$$r=Hc-Ec.$$

The norm $\lVert r\rVert$ bounds the distance from $E$ to some eigenvalue of that matrix. A particular eigenvector requires a spectral gap to be well determined. Neither statement proves that a grid or truncated basis accurately represents a continuum system.

For example, a particle-in-a-box finite-difference matrix can be diagonalized nearly to machine precision while its energies still have appreciable grid error. Tightening the eigensolver tolerance will not remove that error; reducing the grid spacing will. Enlarging the computational box addresses a different error in problems whose physical domain is infinite.

For a time-independent closed-system Hamiltonian, norm and energy conservation are useful checks, but neither establishes phase accuracy. Compare a measured quantity or the state with a refined time step and an exactly solvable case.

Exercise: estimate a continuum limit

A second-order discretization gives energies $E(h)=5.10$, $E(h/2)=5.025$, and $E(h/4)=5.00625$ in the same units. Check the expected convergence ratio and estimate the limiting energy, assuming the leading error is $Ch^2$.

Solution

The successive differences are $0.075$ and $0.01875$, with ratio four as expected for second order. Canceling the leading error gives

$$E_0\approx\frac{4E(h/2)-E(h)}{3}=5.00.$$

The two finer results give the same extrapolation. This supports the assumed convergence regime for these data, but it does not test an incorrect boundary condition or a shared modeling error.

References

• R. J. LeVeque, Finite Difference Methods for Ordinary and Partial Differential Equations, SIAM, 2007.
• Y. Saad, Numerical Methods for Large Eigenvalue Problems, 2nd ed., SIAM, 2011.
• L. N. Trefethen and D. Bau III, Numerical Linear Algebra, SIAM, 1997.