  We evaluate the chemical potential of a one-dimensional quantum dot, coupled
to an atomic force microscope tip. The dot is described within the Luttinger
liquid framework and the conductance peaks positions as a function of the tip
location are calculated in the linear and non-linear transport regimes for an
arbitrary number of particles. The differences between the chemical potential
oscillations induced by Friedel and Wigner terms are carefully analyzed in the
whole range of interaction strength. It is shown that Friedel oscillations,
differently from the Wigner ones, are sensitive probes to detect excited spin
states and collective spin density waves involved in the transport.
