We consider the quantum Calogero model, which describes N non-distinguishable
quantum particles on the real line confined by a harmonic oscillator potential
and interacting via two-body interactions proportional to the inverse square of
the inter-particle distance. We elaborate a novel solution algorithm which
allows us to obtain fully explicit formulas for its eigenfunctions, for
arbitrary coupling parameter and particle number. We also show that our method
applies, with minor changes, to all Calogero models associated with classical
root systems.