The behaviour of solitons under the effect of a harmonic potential is investigated in a Galilean-invariant field theory. On a zero background, these solitons behave as classical particles and are expected to reproduce the dynamics of a classical harmonic oscillator. However, on a non-zero chaotic background, as an ensemble, these solitons reproduce quantum behaviour described by the Schrödinger equation. Numerical simulations are used to investigate the extent to which the ensemble dynamics of the solitons reproduce states of the quantum harmonic oscillator, and study their dynamics.