  Coulomb interactions can have a decisive effect on the ground state of
electronic systems. The simplest system in which interactions can play an
interesting role is that of two electrons on a string. In the presence of
strong interactions the two electrons are predicted to form a Wigner molecule,
separating to the ends of the string due to their mutual repulsion. This
spatial structure is believed to be clearly imprinted on the energy spectrum,
yet to date a direct measurement of such a spectrum in a controllable
one-dimensional setting is still missing. Here we use an ultra-clean suspended
carbon nanotube to realize this system in a tunable potential. Using tunneling
spectroscopy we measure the excitation spectra of two interacting carriers,
electrons or holes, and identify seven low-energy states characterized by their
spin and isospin quantum numbers. These states fall into two multiplets
according to their exchange symmetries. The formation of a strongly-interacting
Wigner molecule is evident from the small energy splitting measured between the
two multiplets, that is quenched by an order of magnitude compared to the
non-interacting value. Our ability to tune the two-electron state in space and
to study it for both electrons and holes provides an unambiguous demonstration
of the fundamental Wigner molecule state.
