# Can there be an infinite set of commuting quantum mechanical operators, and how are their common eigenfunctions to be determined?

The quantum mechanical atomic kinetic energy and momentum operators commute, and essentially are powers of the gradient operator: it seems that any power of the gradient commutes with any other, and thus would comprise an infinite set of commuting operators that would have a common eigenfunction.

My question more succinctly is to determine what other operators commute with these "gradient" operators, i.e., what are the other commuting operators (besides the identity operator) that with these "gradient" operators form a complete set of commuting operators?

The operators being used are the atomic operators using hydrogenic atomic quantum numbers (to start).

This is involved in research that is seeking a common, closed-form eigenvector solution to a set of differential eigenvalue equations (each equation corresponding to an operator) that explicitly includes electron-electron interactions, including a fundamental wave-mechanical spin-dependent model for electron correlation.

I cannot find that the kinetic and momentum operators alone comprise a complete set, moreover, when I try to solve the corresponding set of differential eigenvalue equations, I get a hodgepodge of functions with coefficients, which I cannot otherwise but arbitrarily evaluate. This problem might be resolved if there was at least another operator that would add another equation to the set that would either reduce the array of possible functions that are solutions, or resolve these coefficients.

A simple approach could be to use the matrix form of the operator of interest. Suppose we want to have all the possible operators that commute with $$\hat{O}$$. It means that they are diagonal in the representation where $$\hat{O}$$ is diagonal. In this representation we can write $$O_{ij}= \delta_{i,j}O_i$$ Any operator diagonal in the same representation will commute with $$\hat{O}$$, i.e., any commuting operator will have the matrix form $$X_{ij} = \delta_{i,j}X_i,$$ where $$X_i=f(O_i)$$ is an arbitrary function of the eigenvalues of $$\hat{O}$$. Note that it does not have to be an analytical function - any vector of the appropriate dimensionality as would suffice.

One could then add to these operators all the operators obtained by taking a direct product between the operators determined above and operators acting in different spaces.

• As was indicated, the operators of interest are the atomic KE and momentum operators with eigenvalues that are functions of hydrogenic quantum numbers with analytic eigenfunctions. Can you provide or suggest an answer that is explicitly analytic and not a solution in a general matrix and vector representation?
– Doc
Dec 2 '20 at 12:29
• Firstly, it is not clear from your question what specific atomic operators you mean - how are they different from the standard kinetic energy and momentum operators? Do you simply mean that they should be considered in polar coordinates with atomic wave functions as the basis? - but then you are already talking about their matrix form. Dec 2 '20 at 13:03
• Secondly, you can take the recipe proposed in my answer: any function of a momentum operator, $f(\hat{p})$, will commute with this operator. Note, however, that the functions are understood either as Taylor expansions or in terms of the spectral decomposition, as in my answer. But requiring analyticity is an additional constraint, which limits the number of possible operators. Dec 2 '20 at 13:03
• The atomic wavefunctions with which I am most familiar are the hydrogenic wavefunctions in spherical polar coordinates that are the eigenfunctions of the atomic Total Energy (-Z^2/2n^2), Total Angular Momentum Squared l(l+1), and Azimuthal Angular Momentum (m) operators (their respective eigenvalues).
– Doc
Dec 2 '20 at 15:33
• The above hydrogenic wavefunctions are not eigenfunctions of the Kinetic Energy (Z^2/2n^2) and Momentum (Z/n) operators (their respective eigenvalues), and these operators do not commute with the above operators. Analogously, what are the atomic operators that with the Kinetic Energy and Momentum operators would form a complete set of commuting operators?
– Doc
Dec 2 '20 at 15:33