In this note we present a notion of harmonic oscillator on the Heisenberg group H n which forms the natural analogue of the harmonic oscillator on ℝ n under a few reasonable assumptions: the harmonic oscillator on H n should be a negative sum of squares of operators related to the sub-Laplacian on H n , essentially self-adjoint with purely discrete spectrum, and its eigenvectors should be smooth functions and form an orthonormal basis of L 2 ( H n ) . This approach leads to a differential operator on H n which is determined by the (stratified) Dynin–Folland Lie algebra. We provide an explicit expression for the operator as well as an asymptotic estimate for its eigenvalues.