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  "body": "\n## Definition\n\nIn the mathematical formulation of quantum mechanics, a physical system is described by the following data and rules.\n\n1. States: the system has a complex separable [[Hilbert space|HM2HHE3J51]] $H$. A pure state is a unit vector $\\psi \\in H$. The vectors $\\psi$ and $e^{i\\alpha}\\psi$ with $\\alpha \\in \\R$ give the same state.\n2. Observables: each measurable quantity is a [[self-adjoint operator|0NEGN5CRSC]] $A$ on $H$, possibly [[unbounded|KRJX5RA0PW]], with [[projection-valued measure|ATGT25E5FF]] $P^A$ from the spectral theorem, so that $A = \\int_\\R \\lambda \\, dP^A(\\lambda)$.\n3. Measurement (Born rule): measuring $A$ in the state $\\psi$ gives a result in a Borel set $E \\subseteq \\R$ with probability $\\langle P^A(E)\\psi, \\psi \\rangle$. So the possible results lie in the spectrum of $A$. For $\\psi$ in the domain of $A$, the expected value is $\\langle A\\psi, \\psi \\rangle$.\n4. Dynamics: there is a self-adjoint operator $\\hat H$, the Hamiltonian. A state evolves by $\\psi(t) = e^{-it\\hat H/\\hbar}\\psi(0)$, where $\\hbar$ is the reduced Planck constant. The operators $e^{-it\\hat H/\\hbar}$ form a strongly continuous one-parameter group of [[unitary operators|MM0EJ2GDE4]].\n5. Composite systems: the Hilbert space of a system made of two parts with spaces $H_1$ and $H_2$ is the [[tensor product|EY0X2NKA3S]] $H_1 \\otimes H_2$.\n\nMixed states are density operators: positive [[trace class operators|Z2CP530WDE]] $\\rho$ with $\\operatorname{tr}\\rho = 1$; the probability in clause 3 is then $\\operatorname{tr}(\\rho P^A(E))$.\n\n## Remarks\n\nExample: a spin-$\\tfrac12$ system has $H = \\C^2$ and spin observable $\\sigma_z = \\operatorname{diag}(1, -1)$, whose spectral projections are $P(\\set{1}) = \\operatorname{diag}(1, 0)$ and $P(\\set{-1}) = \\operatorname{diag}(0, 1)$.\nFor $\\psi = (1, 1)/\\sqrt 2$, the two results each have probability $\\tfrac12$, and the expected value is $\\langle \\sigma_z \\psi, \\psi \\rangle = 0$.\n\nNon-example: the matrix $N = \\begin{pmatrix} 0 \u0026 1 \\\\ 0 \u0026 0 \\end{pmatrix}$ is not self-adjoint, so clause 2 does not make it an observable.\nIt has no spectral decomposition into orthogonal projections: its only eigenvalue is $0$, but $N \\ne 0$.\n\nWhen $\\psi$ is a unit vector, the pure state $\\psi$ corresponds to the density operator $\\rho = \\langle \\cdot, \\psi \\rangle \\psi$, the projection onto $\\C\\psi$.\nBy Stone's theorem, every strongly continuous one-parameter unitary group has the form of clause 4 for a unique self-adjoint generator.\n",
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