  We explore a new variational principle for studying one-dimensional quantum
systems in a trapping potential. We focus on the Fermi polaron problem, where a
single distinguishable impurity interacts through a contact potential with a
background of identical fermions. We can accurately describe this system at
arbitrary finite repulsion by constructing a truncated basis containing states
at both the limits of zero and infinite repulsion. We show how to construct
this basis and how to obtain energies, density matrices and correlation
functions, and provide results both for a harmonic well and a double well for
various particle numbers. The results are compared both with matrix product
states methods and with the analytical result for two particles in a harmonic
well.
