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, so that the eigenvectors are uniquely specified by the corresponding sets of eigenvalues.
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 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).
The compatibility theorem
Consider two observables, and , represented by the operators and . Then the following statements are equivalent:
and are compatible observables. and have a common eigenbasis. The operators and commute, meaning that .
Proofs
Discussion
We consider the two above observables and . Suppose there exists a complete set of kets whose every element is simultaneously an eigenket of and . Then we say that and are compatible. If we denote the eigenvalues of and corresponding to respectively by and , we can write
If the system happens to be in one of the eigenstates, say, , then both and can be simultaneously measured to any arbitrary level of precision, and we will get the results and respectively. This idea can be extended to more than two observables.
Examples of compatible observables
The Cartesian components of the position operator are , and . These components are all compatible. Similarly, the Cartesian components of the momentum operator , that is , and are also compatible.
Formal definition
A set of observables is called a CSCO if:
All the observables commute in pairs. If we specify the eigenvalues of all the operators in the CSCO, we identify a unique eigenvector (up to a phase) in the Hilbert space of the system.
If we are given a CSCO, we can choose a basis for the space of states made of common eigenvectors of the corresponding operators. We can uniquely identify each eigenvector (up to a phase) by the set of eigenvalues it corresponds to.
Discussion
Let us have an operator of an observable , which has all non-degenerate eigenvalues . As a result, there is one unique eigenstate corresponding to each eigenvalue, allowing us to label these by their respective eigenvalues. For example, the eigenstate of corresponding to the eigenvalue can be labelled as . Such an observable is itself a self-sufficient CSCO.
However, if some of the eigenvalues of are degenerate (such as having degenerate energy levels), then the above result no longer holds. In such a case, we need to distinguish between the eigenfunctions corresponding to the same eigenvalue. To do this, a second observable is introduced (let us call that ), which is compatible with . The compatibility theorem tells us that a common basis of eigenfunctions of and can be found. Now if each pair of the eigenvalues uniquely specifies a state vector of this basis, we claim to have formed a CSCO: the set . The degeneracy in is completely removed.
It may so happen, nonetheless, that the degeneracy is not completely lifted. That is, there exists at least one pair which does not uniquely identify one eigenvector. In this case, we repeat the above process by adding another observable , which is compatible with both and . If the basis of common eigenfunctions of , and is unique, that is, uniquely specified by the set of eigenvalues , then we have formed a CSCO: . If 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.
Suppose we are given a finite CSCO . Then we can expand any general state in the Hilbert space as
where are the eigenkets of the operators , and form a basis space. That is, , etc
If we measure in the state then the probability that we simultaneously measure is given by .