  Most quantum tomographic methods can only be used for one-dimensional
problems. We show how to infer the quantum state of a non-relativistic
N-dimensional harmonic oscillator system by simple inverse Radon transforms.
The procedure is equally applicable to finding the joint quantum state of
several distinguishable particles in different harmonic oscillator potentials.
A requirement of the procedure is that the angular frequencies of the N
harmonic potentials are incommensurable. We discuss what kind of information
can be found if the requirement of incommensurability is not fulfilled and also
under what conditions the state can be reconstructed from finite time
measurements. As a further example of quantum state reconstruction in N
dimensions we consider the two related cases of an N-dimensional free particle
with periodic boundary conditions and a particle in an N-dimensional box, where
we find a similar condition of incommensurability and finite recurrence time
for the one-dimensional system.
