Matrixmechanik

Matrixmechanik is a fundamental formulation of quantum mechanics developed by Werner Heisenberg, Max Born, and Pascual Jordan in 1925. It represents a radical departure from classical physics by describing physical quantities, such as position and momentum, not as continuous variables but as matrices. These matrices are infinite-dimensional operators that act on quantum states.

In matrix mechanics, the state of a quantum system is not represented by a wave function as in Schrödinger's wave mechanics, but by a vector in an abstract Hilbert space. Observables, like energy and angular momentum, are represented by Hermitian operators. The possible values an observable can take are the eigenvalues of its corresponding matrix, and the change in the quantum state over time is governed by the Heisenberg equation of motion, which involves the commutator of the observable with the Hamiltonian operator.

A key feature of matrix mechanics is its inherent non-commutativity. For instance, the position and momentum operators do not commute, meaning the order in which they are applied matters. This non-commutativity is directly related to the uncertainty principle, a cornerstone of quantum mechanics, stating that certain pairs of properties, like position and momentum, cannot be simultaneously known with arbitrary precision.

Matrix mechanics and Schrödinger's wave mechanics are mathematically equivalent, both providing a complete and consistent description of quantum phenomena. The matrix formulation offers a direct link to experimental measurements and is particularly useful in areas like quantum computing and the theory of angular momentum.