Schrödinger–Newton equation

The Schrödinger–Newton equation, sometimes referred to as the Newton–Schrödinger or Schrödinger–Poisson equation, is a nonlinear modification of the Schrödinger equation with a Newtonian gravitational potential, where the gravitational potential emerges from the treatment of the wave function as a mass density, including a term that represents interaction of a particle with its own gravitational field. The inclusion of a self-interaction term represents a fundamental alteration of quantum mechanics. in connection with self-gravitating boson stars. In this context of classical general relativity it appears as the non-relativistic limit of either the Klein–Gordon equation or the Dirac equation in a curved space-time together with the Einstein field equations.

The equation also describes fuzzy dark matter and approximates classical cold dark matter described by the Vlasov–Poisson equation in the limit that the particle mass is large.

Later on it was proposed as a model to explain the quantum wave function collapse by Lajos Diósi and Roger Penrose, from whom the name "Schrödinger–Newton equation" originates. In this context, matter has quantum properties, while gravity remains classical even at the fundamental level. The Schrödinger–Newton equation was therefore also suggested as a way to test the necessity of quantum gravity.

In a third context, the Schrödinger–Newton equation appears as a Hartree approximation for the mutual gravitational interaction in a system of a large number of particles. In this context, a corresponding equation for the electromagnetic Coulomb interaction was suggested by Philippe Choquard at the 1976 Symposium on Coulomb Systems in Lausanne to describe one-component plasmas. Elliott H. Lieb provided the proof for the existence and uniqueness of a stationary ground state and referred to the equation as the Choquard equation.

Overview

As a coupled system, the Schrödinger–Newton equations are the usual Schrödinger equation with a self-interaction gravitational potential

<math display="block"> \mathrm i \hbar\ \frac{\partial\Psi}{\ \partial t\ } = -\frac{\ \hbar^2 }{\ 2\ M\ }\ \nabla ^2 \Psi\; +\; V\ \Psi\; +\; M\ \Phi\ \Psi\ ,</math>

where is an ordinary potential, and the gravitational potential <math>\ \Phi\ ,</math> representing the interaction of the particle with its own gravitational field, satisfies the Poisson equation

<math display="block">\ \nabla^2 \Phi = 4 \pi\ G\ M\ |\Psi|^2 ~.</math>

Because of the back coupling of the wave-function into the potential, it is a nonlinear system.

Replacing <math>\ \Phi \ </math> with the solution to the Poisson equation produces the integro-differential form of the Schrödinger–Newton equation for a particle with mass M:

<math display="block">\mathrm i \hbar\ \frac{\ \partial \Psi\ }{ \partial t } = \left[\ -\frac{\ \hbar^2 }{\ 2\ M\ }\ \nabla ^2 \; + \; V \; - \; G\ m^2 \int \frac{\ | \Psi(t,\mathbf{y}) |^2}{\ |\mathbf{x} - \mathbf{y}|\ } \; \mathrm{d}^3 y\ \right] \Psi ~,</math>

whose Laplacian could also be reduced into a spherial symmerical one set up by

<math display="block">\mathrm i \hbar\ \frac{\ \partial \Psi\ }{ \partial t } = \left[\ -\frac{\ \hbar^2 }{\ 2\ M\ }\ \left(\frac 2 r \frac{\partial }{\partial r}+\frac{\partial^2}{\partial r^2}\right) \; + \; V \; - \; G\ M^2 \int \frac{\ | \Psi(t,\mathbf{y}) |^2}{\ |\mathbf{x} - \mathbf{y}|\ } \; \mathrm{d}^3 y\ \right] \Psi ~.</math>

It is obtained from the above system of equations by integration of the Poisson equation under the assumption that the potential must vanish at infinity.

As it is known, the wave function in the three dimensional space satisfies

<math display="block"> \rho_{\text{mass(\mathbf x) = M|\mathbf \Psi(\mathbf x)|^2, ~.</math>

thus, the Poisson equation derived by Newtonian mechanics finally takes the form:

<math display="block"> \nabla^2\Phi = 4\pi G \rho(\mathbf x) = 4\pi G M |\Psi(\mathbf x)|^2 ~,</math>

which is the shown equation above.

Mathematically, the Schrödinger–Newton equation is a special case of the Hartree equation for . The equation retains most of the properties of the linear Schrödinger equation. In particular, it is invariant under constant phase shifts, leading to conservation of probability and exhibits full Galilei invariance. In addition to these symmetries, a simultaneous transformation

<math display="block"> m \mapsto \mu\ m \ ,\qquad t \mapsto \mu^{-5} t \ ,\qquad \mathbf{x} \mapsto \mu^{-3} \mathbf{x} \ ,\qquad \psi(t, \mathbf{x}) \mapsto \mu^{9/2} \psi(\mu^5 t, \mu^3 \mathbf{x}) </math>

maps solutions of the Schrödinger–Newton equation to solutions.

The stationary equation, which can be obtained in the usual manner via a separation of variables, possesses an infinite family of normalisable solutions of which only the stationary ground state is stable.

Relation to semi-classical and quantum gravity

The Schrödinger–Newton equation can be derived under the assumption that gravity remains classical, even at the fundamental level, and that the right way to couple quantum matter to gravity is by means of the semiclassical Einstein equations. In this case, a Newtonian gravitational potential term is added to the Schrödinger equation, where the source of this gravitational potential is the expectation value of the mass density operator or mass flux-current.

In this regard, if gravity is fundamentally classical, the Schrödinger–Newton equation is a fundamental one-particle equation, which can be generalised to the case of many particles (see below).

If, on the other hand, the gravitational field is quantised, the fundamental Schrödinger equation remains linear. The Schrödinger–Newton equation is then only valid as an approximation for the gravitational interaction in systems of a large number of particles and has no effect on the centre of mass.

Many-body equation and centre-of-mass motion

If the Schrödinger–Newton equation is considered as a fundamental equation, there is a corresponding N-body equation that was already given by Diósi

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. show that this equation gives a rather good estimate of the mass regime above which effects of the Schrödinger–Newton equation become significant.

For an atom the critical width is around 10<sup>22</sup> metres, while it is already down to 10<sup>−31</sup> metres for a mass of one microgram. The regime where the mass is around while the width is of the order of micrometres is expected to allow an experimental test of the Schrödinger–Newton equation in the future. A possible candidate are interferometry experiments with heavy molecules, which currently reach masses up to .

Quantum wave function collapse

The idea that gravity causes (or somehow influences) the wavefunction collapse dates back to the 1960s and was originally proposed by Károlyházy.

The Schrödinger–Newton equation was proposed in this context by Diósi.

about a similar thought experiment proposed by Eppley & Hannah).