is usually weak compared to the other interactions represented by . But it has a significant influence on the centre-of-mass motion. While only depends on relative coordinates and therefore does not contribute to the centre-of-mass dynamics at all, the nonlinear Schrödinger–Newton interaction does contribute. In the aforementioned approximation, the centre-of-mass wave-function satisfies the following nonlinear Schrödinger equation: where is the total mass, is the relative coordinate, the centre-of-mass wave-function, and is the mass density of the many-body system (e.g. a molecule or a rock) relative to its centre of mass. In the limiting case of a wide wave-function, i.e. where the width of the centre-of-mass distribution is large compared to the size of the considered object, the centre-of-mass motion is approximated well by the Schrödinger–Newton equation for a single particle. The opposite case of a narrow wave-function can be approximated by a harmonic-oscillator potential, where the Schrödinger–Newton dynamics leads to a rotation in phase space. In the context where the Schrödinger–Newton equation appears as a Hartree approximation, the situation is different. In this case the full N-particle wave-function is considered a product state of N single-particle wave-functions, where each of those factors obeys the Schrödinger–Newton equation. The dynamics of the centre-of-mass, however, remain strictly linear in this picture. This is true in general: nonlinear Hartree equations never have an influence on the centre of mass. Significance of effects A rough order-of-magnitude estimate of the regime where effects of the Schrödinger–Newton equation become relevant can be obtained by a rather simple reasoning. For a spherically symmetric Gaussian, the free linear Schrödinger equation has the solution The peak of the radial probability density can be found at Now we set the acceleration of this peak probability equal to the acceleration due to Newtonian gravity: using that at time . This yields the relation which allows us to determine a critical width for a given mass value and conversely. We also recognise the scaling law mentioned above. Numerical simulations show that this equation gives a rather good estimate of the regime where effects of the Schrödinger–Newton equation become significant. For an atom the critical width is around 1022 metres, while it is already down to 10−31 metres for a mass of one microgram. The regime where the mass is around 1010 atomic mass units while the width is of the order of micrometers is expected to allow an experimental test of the Schrödinger–Newton