Properties that probe electron kinetic energy

This post is inspired by [a question](https://chemistry.stackexchange.com/questions/121356/is-it-possible-for-the-kinetic-energy-integral-to-be-negative) regarding the meaning of off-diagonal elements of the KE matrix (in some AO basis). One answer suggests that a diagonalized KE matrix might not be very useful. I take the liberty of quoting: 



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> I am not sure it really makes sense to give much of an interpretation to this, however, because what one is really interested in is diagonalizing the Fock matrix, of which the kinetic energy is only one part.
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Indeed, we usually are interested in the wavefunctions (eigenfunctions of the Hamiltonian) whose eigenvalues are the energies of electrons inhabiting the orbitals. We like to think of electrons as existing primarily as standing waves with constant energy. Such states are central to computing electronic spectra and ionization energies, for instance. 


This leads me to ask two related questions: 


1) Is a diagonalized KE matrix, that is, an orbital representation in terms of kinetic energy (rather than full Hamiltonian) ever useful to chemists? 


2) What if any spectroscopies probe the instantaneous value of the kinetic energy of electrons (or some portion of the KE spectrum)?


