Tæthedermatricer

Tæthedermatricer, sometimes translated as density matrices, are a fundamental concept in quantum mechanics used to describe the state of a quantum system, particularly when that system is part of a larger ensemble or when its exact quantum state is unknown. Unlike a pure state, which is described by a wave function, a density matrix can represent both pure and mixed states. A mixed state arises from a statistical mixture of different pure states, where the probabilities of occupying each pure state are known.

Mathematically, a density matrix is a Hermitian, positive-semidefinite operator with a trace of one. For a pure state $|\psi\rangle$, the density matrix is given by $\rho = |\psi\rangle\langle\psi|$. For a mixed state, it is a sum of density matrices for individual pure states, weighted by their probabilities: $\rho = \sum_i p_i |\psi_i\rangle\langle\psi_i|$. The expectation value of an observable $A$ for a system described by a density matrix $\rho$ is given by $\langle A \rangle = \text{Tr}(\rho A)$.

Density matrices are crucial for understanding concepts such as entanglement, decoherence, and open quantum systems. They allow for a consistent description of quantum systems that interact with their environment or are prepared in statistical ensembles. The evolution of a density matrix in time is governed by the von Neumann equation, which is the quantum mechanical analogue of Liouville's equation for classical statistical mechanics.