Complete set of commuting observables
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 independent, so that the eigenvectors are uniquely specified by the corresponding sets of eigenvalues.In some simple cases, like bound state 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 degenerate, and additional observables are needed to distinguish between the eigenstates.Measurement of the complete set of observables constitutes a complete measurement, in the sense that it projects the quantum state 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 Hilbert space (up to a phase).) that form a complete eigenbasis for each of the two compatible observablesbe a complete set of orthonormal eigenkets of the self-adjoint operatorA non-trivial solution exists if This is an equation of ordercan be simultaneously measured to any arbitrary level of precision, and we will get the resultsSimilarly, the Cartesian components of the momentum operatorWe can uniquely identify each eigenvector (up to a phase) by the set of eigenvalues it corresponds to.In such a case, we need to distinguish between the eigenfunctions corresponding to the same eigenvalue.The compatibility theorem tells us that a common basis of eigenfunctions ofuniquely specifies a state vector of this basis, we claim to have formed a CSCO: the setIt may so happen, nonetheless, that the degeneracy is not completely lifted.In this case, we repeat the above process by adding another observableIf not, we add one more compatible observable and continue the process till a CSCO is obtained.The same vector space may have distinct complete sets of commuting operators.Then we can expand any general state in the Hilbert space as whereFor a complete set of commuting operators, we can find a unitary transformation which will simultaneously diagonalize all of them.It can be shown that the square of the angular momentum operator,are generators of rotation, it can be shown that Therefore, a commuting set consists ofThe solution of the problem tells us that disregarding spin of the electrons, the setbe any basis state in the Hilbert space of the hydrogenic atom.completely specifies a unique eigenstate of the Hydrogenic atom.However, if we express the Hamiltonian in the basis of the translation operator, we will find thatWe consider the case of two systems, 1 and 2, with respective angular momentum operatorsEquivalently, there exists another set of basis states for the system, in terms of the total angular momentum operatorThus we may also specify a unique basis state in the Hilbert space of the complete system by the set of eigenvalues