Complete set of commuting observables

In <a href="/facts/Quantum_mechanics/BRc3Mzgr">quantum mechanics</a>, a complete set of commuting observables (CSCO) is a set of <a href="/facts/Commutative/WaYxJLxd">commuting</a> <a href="/facts/Operator_(physics)/rCJw8UFz">operators</a> whose common eigenvectors can be used as a basis to express any quantum <a href="/facts/Quantum_state/7jbxd7Dx">state</a>. 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 independent, so that the eigenvectors are uniquely specified by the corresponding sets of eigenvalues.
In some simple cases, like <a href="/facts/Bound_state/NndW5JQl">bound state</a> problems in one dimension, the energy spectrum is nondegenerate, and energy can be used to uniquely label the eigenstates. In more complicated problems, the energy spectrum is <a href="/facts/Degenerate_energy_levels/ulrnMXTR">degenerate</a>, and additional observables are needed to distinguish between the eigenstates.
Since each pair of observables in the set commutes, the observables are all compatible so that the measurement of one observable has no effect on the result of measuring another observable in the set. It is therefore not necessary to specify the order in which the different observables are measured. Measurement of the complete set of observables constitutes a complete measurement, in the sense that it projects the <a href="/facts/Quantum_state/7jbxd7Dx">quantum state</a> of the system onto a unique and known vector in the basis defined by the set of operators. That is, to prepare the completely specified state, we have to take any state arbitrarily, and then perform a succession of measurements corresponding to all the observables in the set, until it becomes a uniquely specified vector in the <a href="/facts/Hilbert_space/aDFMh4lV">Hilbert space</a> (up to a phase).

Complete set of commuting observables open-in-new

Complete set of commuting observables