99. Appendix: notation, units and numerical methods

Notation

Symbol	Meaning
$\ket{\psi}$ , $\bra{\psi}$	A state vector and its dual
$\braket{\phi}{\psi}$	Inner product (conjugate-linear in the first slot)
$\hat A$	An operator (hatted)
$\vec r$	Bloch vector
$\rho$	Density matrix
$\psi_{n\ell m}$	Hydrogen wavefunction; principal, angular and magnetic quantum numbers
$a_0$	Bohr radius

To keep the numbers readable, every 3D scene uses natural units:

One-dimensional problems: $\hbar = m = 1$ . Energies and lengths are dimensionless; only ratios mean anything.
Hydrogen: $\hbar = m_e = e = 1$ (atomic units). Length is in Bohr radii, $a_0 = 0.529$ Å; energy is in hartree, $27.211$ eV.
Spin: in units of $\hbar/2$ , so the Pauli matrices have eigenvalues $\pm1$ .

Every number displayed in a scene is in these units. Multiply by the appropriate constants to return to SI.

Scene	Method	Main source of error
Bloch sphere	Rodrigues finite rotations	None (the norm is conserved exactly)
Hydrogen orbitals	Radial inverse CDF plus angular rejection sampling	Monte Carlo noise $\sim 1/\sqrt N$
One-dimensional wells	Three-point differences, QL for eigenvalues, inverse iteration for eigenvectors	Grid discretisation $O(\dd x^2)$ ; a finite window squeezes weakly bound states
Double slit	Direct Huygens summation (no far-field approximation)	A finite number of sub-sources (11 per slit)
Tunnelling	Split-operator, symmetric splitting	Time step $O(\dd t^3)$ ; the absorbing layer at the edges
Path integral	Monte Carlo sampling over paths	Sample size; discrete time steps

Full derivations of the numerical methods are in appendix C (in Chinese).

Linear algebra: Hermitian operators, eigendecomposition, tensor products → appendix A.1
Fourier analysis: position–momentum duality, and where the uncertainty relation comes from mathematically → appendix A.2
Special functions: Hermite (oscillator), Laguerre (hydrogen radial), Legendre and spherical harmonics (angular) → appendix A.3

The full table is in appendix B, or press f anywhere on the site for the quick-reference panel.