If you like you can think of Hermitian operators like \ hat { P } as an odd way of specifying an orthogonal basis with a real number attached to each basis vector .-- talk ) 15 : 50, 9 March 2008 ( UTC)
32.
From a geometric and algebraic point of view, the Stokes parameters stand in one-to-one correspondence with the closed, convex, 4-real-dimensional cone of nonnegative Hermitian operators on the Hilbert space "'C "'2.
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Thus if a bounded functional " f " of the trace-class Banach space and " f " is positive on the product pure states, then " f ", or its identification as a Hermitian operator, is an entanglement witness.
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Then, the same concept follows : any operator is the sum of a Hermitian operator and an anti-Hermitian one, so if time is some kind of disturbance in a Hermitian space, the CPT-symmetric laws will amplify, and the CPT-antisymmetric laws will cancel.
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This mathematical machinery gives a simple, direct way to compute a statistical property of the outcome of an experiment, once it is understood how to associate the initial state with a Hilbert space vector, and the measured quantity with an observable ( that is, a specific Hermitian operator ).
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All such nontrivial commutation relations for pairs of operators lead to corresponding Hermitian operators and, consider expectation values in a system in the state, the variances around the corresponding expectation values being ( " A " " " A " ) 2 } }, etc.
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Let be a Hamiltonian representing a weak physical disturbance, such as a potential energy produced by an external field . ( Thus, is formally a Hermitian operator . ) Let be a dimensionless parameter that can take on values ranging continuously from 0 ( no perturbation ) to 1 ( the full perturbation ).
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In quantum mechanics each dynamical variable ( e . g . position, translational momentum, orbital angular momentum, spin, total angular momentum, energy, etc . ) is associated with a Hermitian operator that acts on the state of the quantum system and whose eigenvalues correspond to the possible values of the dynamical variable.
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In linear algebra and functional analysis, the "'min-max theorem "', or "'variational theorem "', or "'Courant & ndash; Fischer & ndash; Weyl min-max principle "', is a result that gives a variational characterization of compact Hermitian operators on Hilbert spaces.
