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The Schrödinger–Newton equation, sometimes referred to as the Newton–Schrödinger or Schrödinger–Poisson equation, is a nonlinear modification of the
Schrödinger equation The Schrödinger equation is a partial differential equation that governs the wave function of a non-relativistic quantum-mechanical system. Its discovery was a significant landmark in the development of quantum mechanics. It is named after E ...
with a Newtonian gravitational potential, where the gravitational potential emerges from the treatment of the
wave function In quantum physics, a wave function (or wavefunction) is a mathematical description of the quantum state of an isolated quantum system. The most common symbols for a wave function are the Greek letters and (lower-case and capital psi (letter) ...
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. It can be written either as a single integro-differential equation or as a coupled system of a Schrödinger and a
Poisson equation Poisson's equation is an elliptic partial differential equation of broad utility in theoretical physics. For example, the solution to Poisson's equation is the potential field caused by a given electric charge or mass density distribution; with th ...
. In the latter case it is also referred to in the plural form. The Schrödinger–Newton equation was first considered by Ruffini and Bonazzola in connection with self-gravitating boson stars. In this context of classical
general relativity General relativity, also known as the general theory of relativity, and as Einstein's theory of gravity, is the differential geometry, geometric theory of gravitation published by Albert Einstein in 1915 and is the current description of grav ...
it appears as the non-relativistic limit of either the
Klein–Gordon equation The Klein–Gordon equation (Klein–Fock–Gordon equation or sometimes Klein–Gordon–Fock equation) is a relativistic wave equation, related to the Schrödinger equation. It is named after Oskar Klein and Walter Gordon. It is second-order i ...
or the
Dirac equation In particle physics, the Dirac equation is a relativistic wave equation derived by British physicist Paul Dirac in 1928. In its free form, or including electromagnetic interactions, it describes all spin-1/2 massive particles, called "Dirac ...
in a curved space-time together with the
Einstein field equations In the General relativity, general theory of relativity, the Einstein field equations (EFE; also known as Einstein's equations) relate the geometry of spacetime to the distribution of Matter#In general relativity and cosmology, matter within it. ...
. The equation also describes fuzzy dark matter and approximates classical
cold dark matter In cosmology and physics, cold dark matter (CDM) is a hypothetical type of dark matter. According to the current standard model of cosmology, Lambda-CDM model, approximately 27% of the universe is dark matter and 68% is dark energy, with only a sm ...
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 Sir Roger Penrose (born 8 August 1931) is an English mathematician, mathematical physicist, Philosophy of science, philosopher of science and Nobel Prize in Physics, Nobel Laureate in Physics. He is Emeritus Rouse Ball Professor of Mathematics i ...
, 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 Quantum gravity (QG) is a field of theoretical physics that seeks to describe gravity according to the principles of quantum mechanics. It deals with environments in which neither gravitational nor quantum effects can be ignored, such as in the v ...
. 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 The coulomb (symbol: C) is the unit of electric charge in the International System of Units (SI). It is defined to be equal to the electric charge delivered by a 1 ampere current in 1 second, with the elementary charge ''e'' as a defining c ...
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 In classical mechanics, the gravitational potential is a scalar potential associating with each point in space the work (energy transferred) per unit mass that would be needed to move an object to that point from a fixed reference point in the ...
i \hbar\ \frac = -\frac\ \nabla ^2 \Psi\; +\; V\ \Psi\; +\; m\ \Phi\ \Psi\ , where is an ordinary potential, and the gravitational potential \ \Phi\ , representing the interaction of the particle with its own gravitational field, satisfies the Poisson equation \ \nabla^2 \Phi = 4 \pi\ G\ m\ , \Psi, ^2 ~. Because of the back coupling of the wave-function into the potential, it is a
nonlinear system In mathematics and science, a nonlinear system (or a non-linear system) is a system in which the change of the output is not proportional to the change of the input. Nonlinear problems are of interest to engineers, biologists, physicists, mathem ...
. Replacing \ \Phi \ with the solution to the Poisson equation produces the integro-differential form of the Schrödinger–Newton equation: i \hbar\ \frac = \left -\frac\ \nabla ^2 \; + \; V \; - \; G\ m^2 \int \frac \; \mathrm^3 \mathbf\ \right\Psi ~. 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. 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 m \to \mu\ m \ ,\qquad t \to \mu^ t \ ,\qquad \mathbf \to \mu^ \mathbf \ ,\qquad \psi(t, \mathbf) \to \mu^ \psi(\mu^5 t, \mu^3 \mathbf) 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 and can be derived from semiclassical gravity in the same way as the one-particle equation: \begin i \hbar \frac\Psi(t,\mathbf_1,\dots, \mathbf_N) = \bigg(&-\sum_^N \frac \nabla_j^2 + \sum_ V_\big(, \mathbf_j - \mathbf_k, \big) \\ &- G\sum_^N m_j m_k \int \mathrm^3 \mathbf_1 \cdots \mathrm^3 \mathbf_N \, \frac \bigg) \Psi(t,\mathbf_1,\dots,\mathbf_N). \end The potential V_ contains all the mutual linear interactions, e.g. electrodynamical Coulomb interactions, while the gravitational-potential term is based on the assumption that all particles perceive the same gravitational potential generated by all the
marginal distribution In probability theory and statistics, the marginal distribution of a subset of a collection of random variables is the probability distribution of the variables contained in the subset. It gives the probabilities of various values of the variable ...
s for all the particles together. In a Born–Oppenheimer-like approximation, this ''N''-particle equation can be separated into two equations, one describing the relative motion, the other providing the dynamics of the centre-of-mass wave-function. For the relative motion, the gravitational interaction does not play a role, since it is usually weak compared to the other interactions represented by V_. But it has a significant influence on the centre-of-mass motion. While V_ 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: i \hbar \frac = \left(\frac\nabla^2 -G \int \mathrm^3 \mathbf \, \int \mathrm^3 \mathbf \, \int \mathrm^3 \mathbf \, \frac \right) \psi_c(t,\mathbf), where is the total mass, is the relative coordinate, \psi_c the centre-of-mass wave-function, and \rho_c 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, \Psi(t=0,r) = (\pi \sigma^2)^ \exp\left(-\frac\right), the free linear Schrödinger equation has the solution \Psi(t,r) = (\pi \sigma^2)^ \left(1+\frac\right)^ \exp\left(-\frac\right). The peak of the radial probability density 4 \pi r^2 , \Psi, ^2 can be found at r_p = \sigma \sqrt. Now we set the acceleration \ddot_p = \frac of this peak probability equal to the acceleration due to Newtonian gravity: \ddot = -\frac, using that r_p = \sigma at time t = 0. This yields the relation m^3 \sigma = \frac \approx 1.7 \times 10^~\text\,\text^3, 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 mass regime above which 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 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 Interferometry is a technique which uses the ''interference (wave propagation), interference'' of Superposition principle, superimposed waves to extract information. Interferometry typically uses electromagnetic waves and is an important inves ...
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 In various interpretations of quantum mechanics, wave function collapse, also called reduction of the state vector, occurs when a wave function—initially in a superposition of several eigenstates—reduces to a single eigenstate due to i ...
dates back to the 1960s and was originally proposed by Károlyházy. The Schrödinger–Newton equation was proposed in this context by Diósi. There the equation provides an estimation for the "line of demarcation" between microscopic (quantum) and macroscopic (classical) objects. The stationary ground state has a width of a_0 \approx \frac. For a well-localised homogeneous sphere, i.e. a sphere with a centre-of-mass wave-function that is narrow compared to the radius of the sphere, Diósi finds as an estimate for the width of the ground-state centre-of-mass wave-function a_0^ \approx a_0^ R^. Assuming a usual density around , a critical radius can be calculated for which a_0^ \approx R. This critical radius is around a tenth of a micrometre.
Roger Penrose Sir Roger Penrose (born 8 August 1931) is an English mathematician, mathematical physicist, Philosophy of science, philosopher of science and Nobel Prize in Physics, Nobel Laureate in Physics. He is Emeritus Rouse Ball Professor of Mathematics i ...
proposed that the Schrödinger–Newton equation mathematically describes the basis states involved in a gravitationally induced
wavefunction collapse In various interpretations of quantum mechanics, wave function collapse, also called reduction of the state vector, occurs when a wave function—initially in a superposition of several eigenstates—reduces to a single eigenstate due to i ...
scheme. Penrose suggests that a superposition of two or more quantum states having a significant amount of mass displacement ought to be unstable and reduce to one of the states within a finite time. He hypothesises that there exists a "preferred" set of states that could collapse no further, specifically, the stationary states of the Schrödinger–Newton equation. A macroscopic system can therefore never be in a spatial superposition, since the nonlinear gravitational self-interaction immediately leads to a collapse to a stationary state of the Schrödinger–Newton equation. According to Penrose's idea, when a quantum particle is measured, there is an interplay of this nonlinear collapse and environmental decoherence. The gravitational interaction leads to the reduction of the environment to one distinct state, and decoherence leads to the localisation of the particle, e.g. as a dot on a screen.


Problems and open matters

Three major problems occur with this interpretation of the Schrödinger–Newton equation as the cause of the wave-function collapse: # Excessive residual probability far from the collapse point # Lack of any apparent reason for the
Born rule The Born rule is a postulate of quantum mechanics that gives the probability that a measurement of a quantum system will yield a given result. In one commonly used application, it states that the probability density for finding a particle at a ...
# Promotion of the previously strictly hypothetical
wave function In quantum physics, a wave function (or wavefunction) is a mathematical description of the quantum state of an isolated quantum system. The most common symbols for a wave function are the Greek letters and (lower-case and capital psi (letter) ...
to an observable (real) quantity. First, numerical studies agreeingly find that when a wave packet "collapses" to a stationary solution, a small portion of it seems to run away to infinity. This would mean that even a completely collapsed quantum system still can be found at a distant location. Since the solutions of the linear Schrödinger equation tend towards infinity even faster, this only indicates that the Schrödinger–Newton equation alone is not sufficient to explain the wave-function collapse. If the environment is taken into account, this effect might disappear and therefore not be present in the scenario described by Penrose. A second problem, also arising in Penrose's proposal, is the origin of the
Born rule The Born rule is a postulate of quantum mechanics that gives the probability that a measurement of a quantum system will yield a given result. In one commonly used application, it states that the probability density for finding a particle at a ...
: To solve the
measurement problem In quantum mechanics, the measurement problem is the ''problem of definite outcomes:'' quantum systems have superpositions but quantum measurements only give one definite result. The wave function in quantum mechanics evolves deterministically ...
, a mere explanation of why a wave-function collapses – e.g., to a dot on a screen – is not enough. A good model for the collapse process ''also'' has to explain why the dot appears on different positions of the screen, with probabilities that are determined by the squared absolute-value of the wave-function. It might be possible that a model based on Penrose's idea could provide such an explanation, but there is as yet no known reason that Born's rule would naturally arise from it. Thirdly, since the gravitational potential is linked to the wave-function in the picture of the Schrödinger–Newton equation, the wave-function must be interpreted as a real object. Therefore, at least in principle, it becomes a measurable quantity. Making use of the nonlocal nature of entangled quantum systems, this could be used to send signals faster than light, which is generally thought to be in contradiction with causality. It is, however, not clear whether this problem can be resolved by applying the right collapse prescription, yet to be found, consistently to the full quantum system. Also, since gravity is such a weak interaction, it is not clear that such an experiment can be actually performed within the parameters given in our universe (see the referenced discussion about a similar thought experiment proposed by Eppley & Hannah ).


See also

*
Nonlinear Schrödinger equation In theoretical physics, the (one-dimensional) nonlinear Schrödinger equation (NLSE) is a nonlinear variation of the Schrödinger equation. It is a classical field equation whose principal applications are to the propagation of light in nonli ...
*
Semiclassical gravity Semiclassical gravity is an approximation to the theory of quantum gravity in which one treats matter and energy fields as being quantum and the gravitational field as being classical. In semiclassical gravity, matter is represented by quantum ...
* Penrose interpretation *
Poisson's equation Poisson's equation is an elliptic partial differential equation of broad utility in theoretical physics. For example, the solution to Poisson's equation is the potential field caused by a given electric charge or mass density distribution; with t ...


References

{{DEFAULTSORT:Schrodinger-Newton equation Gravity Quantum gravity Equations Nonlinear partial differential equations Schrödinger equation