  We quantize the 1-dimensional 3-body problem with harmonic and inverse square
pair potential by separating the Schr\"odinger equation following the classic
work of Calogero, but allowing all possible self-adjoint boundary conditions
for the angular and radial Hamiltonians. The inverse square coupling constant
is taken to be $g=2\nu (\nu-1)$ with ${1/2} <\nu< {3/2}$ and then the angular
Hamiltonian is shown to admit a 2-parameter family of inequivalent
quantizations compatible with the dihedral $D_6$ symmetry of its potential term
$9 \nu (\nu -1)/\sin^2 3\phi$. These are parametrized by a matrix $U\in U(2)$
satisfying $\sigma_1 U \sigma_1 = U$, and in all cases we describe the
qualitative features of the angular eigenvalues and classify the eigenstates
under the $D_6$ symmetry and its $S_3$ subgroup generated by the particle
exchanges. The angular eigenvalue $\lambda$ enters the radial Hamiltonian
through the potential $(\lambda -{1/4})/r^2$ allowing a 1-parameter family of
self-adjoint boundary conditions at $r=0$ if $\lambda <1$. For $0<\lambda<1$
our analysis of the radial Schr\"odinger equation is consistent with previous
results on the possible energy spectra, while for $\lambda <0$ it shows that
the energy is not bounded from below rejecting those $U$'s admitting such
eigenvalues as physically impermissible. The permissible self-adjoint angular
Hamiltonians include, for example, the cases $U=\pm {\bf 1}_2, \pm \sigma_1$,
which are explicitly solvable and are presented in detail. The choice $U=-{\bf
1}_2$ reproduces Calogero's quantization, while for the choice $U=\sigma_1$ the
system is smoothly connected to the harmonic oscillator in the limit $\nu \to
1$.
