Tiheydinmatriisi

Tiheydinmatriisi, often translated as density matrix, is a fundamental concept in quantum mechanics used to describe the state of a quantum system. It provides a more general way to represent a quantum state than the wavefunction alone, particularly when dealing with mixed states or subsystems of larger entangled systems. The density matrix is a mathematical object that encapsulates all possible information about the statistical ensemble of states the system might be in.

Mathematically, the density matrix is represented by a Hermitian, non-negative definite operator, denoted by the symbol $\rho$. For a pure quantum state described by a wavefunction $|\psi\rangle$, the density matrix is given by $\rho = |\psi\rangle\langle\psi|$. In this case, the density matrix has a trace of 1 and is idempotent ($\rho^2 = \rho$). However, for a mixed state, which is a probabilistic combination of multiple pure states, the density matrix is a weighted sum of individual pure state density matrices: $\rho = \sum_i p_i |\psi_i\rangle\langle\psi_i|$, where $p_i$ are the probabilities of being in state $|\psi_i\rangle$ and $\sum_i p_i = 1$. For mixed states, the density matrix is not idempotent.

The density matrix is essential for calculating expectation values of observables. The expectation value of an observable represented by an operator $A$ for a system in state $\rho$ is given by $\langle A \rangle = \text{Tr}(\rho A)$, where Tr denotes the trace of the operator. This formalism is particularly useful in quantum statistical mechanics and when dealing with open quantum systems that interact with their environment, leading to decoherence and the loss of quantum coherence. The time evolution of the density matrix is governed by the Liouville-von Neumann equation, analogous to the Schrödinger equation for pure states.