Tiheydematriisi

Tiheydematriisi, sometimes translated as density matrix, is a fundamental concept in quantum mechanics used to describe the state of a quantum system. Unlike a pure quantum state, which can be represented by a single wavefunction, a density matrix can describe both pure states and mixed states. A mixed state is a statistical ensemble of multiple pure states, where each state has a certain probability of being realized. The density matrix provides a unified mathematical framework to handle both situations.

Mathematically, the density matrix $\rho$ is a linear operator that is Hermitian, positive semi-definite, and has a trace equal to one ($\text{Tr}(\rho) = 1$). For a pure state $|\psi\rangle$, the density matrix is given by $\rho = |\psi\rangle\langle\psi|$. For a mixed state, which is a statistical mixture of pure states $|\psi_i\rangle$ with probabilities $p_i$, the density matrix is $\rho = \sum_i p_i |\psi_i\rangle\langle\psi_i|$.

The density matrix is crucial for calculating expectation values of observables. The expectation value of an observable represented by an operator $A$ is given by $\langle A \rangle = \text{Tr}(\rho A)$. This formula applies to both pure and mixed states, simplifying calculations in many quantum mechanical problems, especially those involving statistical mixtures or interactions with an environment. The time evolution of the density matrix is governed by the von Neumann equation, $\frac{d\rho}{dt} = -\frac{i}{\hbar}[H, \rho]$, where $H$ is the Hamiltonian of the system.