Why are matrices useful for describing atomic systems with a limited number of states?

Matrices are particularly useful for describing atomic systems with a limited number of states because they facilitate numerical analysis calculations when performed on computers. While Schrödinger’s wave equation is often preferred for visualizing continuous electron distributions, matrix methods—originally developed by Werner Heisenberg—are highly efficient for systems defined by a finite number of states, such as the spin states of an electron.1

By representing atomic properties as arrays of numbers that follow specific algebraic rules, this approach provides a mathematically rigorous framework for analyzing quantum systems. Today, physicists utilize both wave functions and matrix methods depending on the specific application, leveraging the strengths of each to better understand quantum mechanics.1