  Guided by ordinary quantum mechanics we introduce new fuzzy spheres of
dimensions d=1,2: we consider an ordinary quantum particle in D=d+1 dimensions
subject to a rotation invariant potential well V(r) with a very sharp minimum
on a sphere of unit radius. Imposing a sufficiently low energy cutoff to
`freeze' the radial excitations makes only a finite-dimensional Hilbert
subspace accessible and on it the coordinates noncommutative \`a la Snyder; in
fact, on it they generate the whole algebra of observables. The construction is
equivariant not only under rotations - as Madore's fuzzy sphere -, but under
the full orthogonal group O(D). Making the cutoff and the depth of the well
dependent on (and diverging with) a natural number L, and keeping the leading
terms in 1/L, we obtain a sequence S^d_L of fuzzy spheres converging (in a
suitable sense) to the sphere S^d as L diverges (whereby we recover ordinary
quantum mechanics on S^d). These models may be useful in condensed matter
problems where particles are confined on a sphere by an (at least
approximately) rotation-invariant potential, beside being suggestive of
analogous mechanisms in quantum field theory or quantum gravity.
