Schrödinger–Newton equation

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

$${\displaystyle i\hbar \ {\frac {\partial \Psi }{\ \partial t\ }}=-{\frac {\ \hbar ^{2}}{\ 2\ m\ }}\ \nabla ^{2}\Psi \;+\;V\ \Psi \;+\;m\ \Phi \ \Psi \ ,}$$

where  V  is an ordinary potential, and the gravitational potential ${\displaystyle \ \Phi \ ,}$ representing the interaction of the particle with its own gravitational field, satisfies the Poisson equation

$${\displaystyle \ \nabla ^{2}\Phi =4\pi \ G\ m\ |\Psi |^{2}~.}$$

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

Replacing ${\displaystyle \ \Phi \ }$ with the solution to the Poisson equation produces the integro-differential form of the Schrödinger–Newton equation:

$${\displaystyle i\hbar \ {\frac {\ \partial \Psi \ }{\partial t}}=\left[\ -{\frac {\ \hbar ^{2}}{\ 2\ m\ }}\ \nabla ^{2}\;+\;V\;-\;G\ m^{2}\int {\frac {\ |\Psi (t,\mathbf {y} )|^{2}}{\ |\mathbf {x} -\mathbf {y} |\ }}\;\mathrm {d} ^{3}\mathbf {y} \ \right]\Psi ~.}$$

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 n = 2 . 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

$${\displaystyle m\to \mu \ m\ ,\qquad t\to \mu ^{-5}t\ ,\qquad \mathbf {x} \to \mu ^{-3}\mathbf {x} \ ,\qquad \psi (t,\mathbf {x} )\to \mu ^{9/2}\psi (\mu ^{5}t,\mu ^{3}\mathbf {x} )}$$

maps solutions of the Schrödinger–Newton equation to solutions.[11][12] 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.[13][14][15]

If the Schrödinger–Newton equation is considered as a fundamental equation, there is a corresponding N-body equation that was already given by Diósi[5] and can be derived from semiclassical gravity in the same way as the one-particle equation:

$${\displaystyle {\begin{aligned}i\hbar {\frac {\partial }{\partial t}}\Psi (t,\mathbf {x} _{1},\dots ,\mathbf {x} _{N})={\bigg (}&-\sum _{j=1}^{N}{\frac {\hbar ^{2}}{2m_{j}}}\nabla _{j}^{2}+\sum _{j\neq k}V_{jk}{\big (}|\mathbf {x} _{j}-\mathbf {x} _{k}|{\big )}\\&-G\sum _{j,k=1}^{N}m_{j}m_{k}\int \mathrm {d} ^{3}\mathbf {y} _{1}\cdots \mathrm {d} ^{3}\mathbf {y} _{N}\,{\frac {|\Psi (t,\mathbf {y} _{1},\dots ,\mathbf {y} _{N})|^{2}}{|\mathbf {x} _{j}-\mathbf {y} _{k}|}}{\bigg )}\Psi (t,\mathbf {x} _{1},\dots ,\mathbf {x} _{N}).\end{aligned}}}$$

The potential ${\displaystyle V_{jk}}$ 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 distributions 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 ${\displaystyle V_{jk}}$. But it has a significant influence on the centre-of-mass motion. While ${\displaystyle V_{jk}}$ 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:

$${\displaystyle i\hbar {\frac {\partial \psi _{c}(t,\mathbf {R} )}{\partial t}}=\left({\frac {\hbar ^{2}}{2M}}\nabla ^{2}-G\int \mathrm {d} ^{3}\mathbf {R'} \,\int \mathrm {d} ^{3}\mathbf {y} \,\int \mathrm {d} ^{3}\mathbf {z} \,{\frac {|\psi _{c}(t,\mathbf {R'} )|^{2}\,\rho _{c}(\mathbf {y} )\rho _{c}(\mathbf {z} )}{\left|\mathbf {R} -\mathbf {R'} -\mathbf {y} +\mathbf {z} \right|}}\right)\psi _{c}(t,\mathbf {R} ),}$$

where M is the total mass, R is the relative coordinate, ${\displaystyle \psi _{c}}$ the centre-of-mass wave-function, and ${\displaystyle \rho _{c}}$ is the mass density of the many-body system (e.g. a molecule or a rock) relative to its centre of mass.[18]

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.[19]

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.

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.[9] For a spherically symmetric Gaussian,

$${\displaystyle \Psi (t=0,r)=(\pi \sigma ^{2})^{-3/4}\exp \left(-{\frac {r^{2}}{2\sigma ^{2}}}\right),}$$

the free linear Schrödinger equation has the solution

$${\displaystyle \Psi (t,r)=(\pi \sigma ^{2})^{-3/4}\left(1+{\frac {i\hbar t}{m\sigma ^{2}}}\right)^{-3/2}\exp \left(-{\frac {r^{2}}{2\sigma ^{2}\left(1+{\frac {i\hbar t}{m\sigma ^{2}}}\right)}}\right).}$$

The peak of the radial probability density ${\displaystyle 4\pi r^{2}|\Psi |^{2}}$ can be found at

$${\displaystyle r_{p}=\sigma {\sqrt {1+{\frac {\hbar ^{2}t^{2}}{m^{2}\sigma ^{4}}}}}.}$$

Now we set the acceleration

$${\displaystyle {\ddot {r}}_{p}={\frac {\hbar ^{2}}{m^{2}r_{p}^{3}}}}$$

of this peak probability equal to the acceleration due to Newtonian gravity:

$${\displaystyle {\ddot {r}}=-{\frac {Gm}{r^{2}}},}$$

using that ${\displaystyle r_{p}=\sigma }$ at time ${\displaystyle t=0}$. This yields the relation

$${\displaystyle m^{3}\sigma ={\frac {\hbar ^{2}}{G}}\approx 1.7\times 10^{-58}~{\text{m}}\,{\text{kg}}^{3},}$$

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[12][1] 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 equation in the future. A possible candidate are interferometry experiments with heavy molecules, which currently reach masses up to 10000 atomic mass units.

The idea that gravity causes (or somehow influences) the wavefunction collapse dates back to the 1960s and was originally proposed by Károlyházy.[20] The Schrödinger–Newton equation was proposed in this context by Diósi.[5] 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

$${\displaystyle a_{0}\approx {\frac {\hbar ^{2}}{Gm^{3}}}.}$$

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

$${\displaystyle a_{0}^{(R)}\approx a_{0}^{1/4}R^{3/4}.}$$

Assuming a usual density around 1000 kg/m³, a critical radius can be calculated for which ${\displaystyle a_{0}^{(R)}\approx R}$. This critical radius is around a tenth of a micrometer.

Roger Penrose proposed that the Schrödinger–Newton equation mathematically describes the basis states involved in a gravitationally induced wavefunction collapse scheme.[6][7][8] 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.

• Nonlinear Schrödinger equation
• Semiclassical gravity
• Penrose interpretation
• Poisson's equation