Matrix-Mechanics Examples

Matrix mechanics is the natural language for finite-dimensional systems, truncated bases, spin, and numerical Hamiltonians. A good example should identify the basis before writing the matrix.

Example Routes

Need	Learn	Main Check
Linear-map meaning of a matrix	Matrices as Linear Maps	Columns encode action on basis vectors
Eigenvalue calculation	Eigenvalues and Eigenvectors	Eigenvectors are not unique until normalized and phased
Diagonalization	Diagonalization	Matrix must have enough eigenvectors
Spectral projectors	Spectral Decomposition	Degenerate eigenspaces require projectors
Two-state and Pauli Hamiltonians	Two-State Hamiltonians	Basis and coupling conventions; effective-field direction sets the eigenbasis
Numerical diagonalization	Matrix Diagonalization Numerically	Convergence and conditioning

Minimal Worked Check

For

$$H= \begin{pmatrix} \epsilon & \Delta \\ \Delta & -\epsilon \end{pmatrix},$$

the characteristic equation is

$$\det(H-EI)=E^2-(\epsilon^2+\Delta^2)=0.$$

Thus

$$E_\pm=\pm\sqrt{\epsilon^2+\Delta^2}.$$

This is the algebraic core behind many two-level examples. The physical interpretation depends on what the two basis states represent: spin projections, localized wells, two atomic levels, or a truncated subspace.

What to Record

• ordered basis;
• matrix elements and their units;
• whether the matrix is Hermitian;
• eigenvalues and normalized eigenvectors;
• phase convention for eigenvectors;
• projectors if degeneracy is present;
• limiting cases, such as $\Delta\to0$ or $\epsilon\to0$.

Common Mistakes

• Writing a matrix without specifying the basis.
• Treating an eigenvector’s overall phase as physical.
• Forgetting that degenerate eigenvectors can be rotated within the degenerate subspace.
• Diagonalizing $H$ but expanding the initial state in the old basis when computing measurement probabilities.
• Trusting numerical eigenvectors without checking residuals $\lVert H\psi-E\psi\rVert$.

References

• R. Shankar, Principles of Quantum Mechanics, 2nd ed., Springer, 1994.
• J. J. Sakurai and J. Napolitano, Modern Quantum Mechanics, 3rd ed., Cambridge University Press, 2020.
• G. H. Golub and C. F. Van Loan, Matrix Computations, 4th ed., Johns Hopkins University Press, 2013.

Exercises

1. In the Hamiltonian above, what happens to the spectrum when $\Delta=0$?

Solution

The eigenvalues become $E_\pm=\pm\lvert\epsilon\rvert$. If $\epsilon>0$, the original basis already diagonalizes the Hamiltonian with diagonal entries $\epsilon$ and $-\epsilon$.