The Central Spin Echo Problem

My publication record as a theoretical chemist is diverse, to put it mildly. I’ve written papers on everything from high-dimensional functional representation to quantum search algorithms to proton-coupled electron transfer to superadiabatic dynamics. However, my favorite projects involves a problem I’ve been mulling over for almost 18 years. It emerges naturally from spin physics, but is basically a mathematical problem that raises basic questions that could be applicable to many other problems.

I’ll sketch the problem below in the hopes that brighter minds than mine can solve it. The outline of this article will be: 1) A Big Question 2) The Physical Observation 3) The Required Proof and 4) A Few Hints.

The Big Question

\textrm{Suppose we want to know whether the two quantum mechanical operators} \hat{A} and \hat{B} share exactly the same set of eigenvectors. The brute for solution to this problem is to obtain a complete set of eigenvectors and eigenvalues for \hat{A} and 14 \hat{B} such that \vert\psi_n \rangle and \vert\phi_n \rangle are eigenvectors of \hat{A} and \hat{B} corresponding to eigenvalues a_n and b_n respectively.