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In quantum mechanics, a complete set of commuting observables (CSCO) is a set of commuting operators whose common eigenvectors can be used as a basis to express any quantum state. In the case of operators with discrete spectra, a CSCO is a set of commuting observables whose simultaneous eigenspaces span the Hilbert space and are linearly ...
Specifically, the position and momentum operators in quantum mechanics, usually denoted and , satisfy the canonical commutation relation: [,] = where is the identity operator. It follows that X {\displaystyle X} and P {\displaystyle P} commute with their commutator.
In the phase space formulation of quantum mechanics, eigenstates of the quantum harmonic oscillator in several different representations of the quasiprobability distribution can be written in closed form. The most widely used of these is for the Wigner quasiprobability distribution.
The commutator of two operators acting on a Hilbert space is a central concept in quantum mechanics, since it quantifies how well the two observables described by these operators can be measured simultaneously. The uncertainty principle is ultimately a theorem about such commutators, by virtue of the Robertson–Schrödinger relation. [7]
According to the standard mathematical formulation of quantum mechanics, quantum observables such as ^ and ^ should be represented as self-adjoint operators on some Hilbert space. It is relatively easy to see that two operators satisfying the above canonical commutation relations cannot both be bounded .
Since translation operators all commute with each other (see above), and since each component of the momentum operator is a sum of two scaled translation operators (e.g. ^ = (^ ((,,)) ^ ((,,)))), it follows that translation operators all commute with the momentum operator, i.e. ^ ^ = ^ ^ This commutation with the momentum operator holds true ...
Canonical commutation rule for position q and momentum p variables of a particle, 1927.pq − qp = h/(2πi).Uncertainty principle of Heisenberg, 1927. The uncertainty principle, also known as Heisenberg's indeterminacy principle, is a fundamental concept in quantum mechanics.
Dirac's rule of thumb suggests that statements in quantum mechanics which contain a commutator correspond to statements in classical mechanics where the commutator is supplanted by a Poisson bracket multiplied by iħ. This makes the operator expectation values obey corresponding classical equations of motion, provided the Hamiltonian is at most ...