  The nature of the proton wave function can be elucidated by studying the
matrix elements of spin-dependent quark density operators SDDs, defined as
matrix elements of density operators in proton states of definite
spin-polarization. These have an infinite variety of non-spherical shapes. The
matrix elements of the SDDs are closely related to specific transverse momentum
dependent TMD parton distributions accessible in the angular dependence of
certain semi-inclusive processes. New measurements would allow the direct
exhibition of the non-spherical nature of the proton. The TMDs can be computed
using lattice QCD so that the non-spherical shapes could eventually be measured
experimentally and computed using fundamental theory.
