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  "body": "\n## Summary\n\nThe mathematical formulation of quantum mechanics describes a physical system by a complex [[Hilbert space|HM2HHE3J51]].\nStates are unit vectors or density operators, and measurable quantities are [[self-adjoint operators|0NEGN5CRSC]].\nMeasurement probabilities come from the spectral theorem through [[projection-valued measures|ATGT25E5FF]], and time evolution is a one-parameter unitary group generated by the Hamiltonian.\nDirac and von Neumann set out this framework around 1930, and it is also stated as the Dirac–von Neumann axioms.\nIt made quantum theory a part of functional analysis and gave rise to operator algebras.\n\n## Explanations\n\n### L1 human\n\nIn classical mechanics, the state of a system is a point, and a quantity such as energy is a function of that point.\nIn quantum mechanics, the state is a vector in a space of possibly infinite dimension, and a quantity is an operator that acts on these vectors.\nThe possible results of measuring the quantity are the points of the spectrum of the operator, and the chances of each result are computed from the state vector.\nFor example, a spin-$\\tfrac12$ particle lives in $\\C^2$, and its spin along one axis is the matrix $\\operatorname{diag}(1, -1)$, with possible results $+1$ and $-1$.\nBecause operators need not commute, some pairs of quantities cannot both have sharp values, which is the source of the uncertainty principle.\n",
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