Schrödinger-Newton equation

The Schrödinger-Newton equation (also Newton-Schrödinger or Schrödinger-Poisson equation ) is a non-linear modification of the Schrödinger equation , taking Newton's law of gravitation into account . This results in a self-interaction, since the wave function is assumed to have a mass. The equation can be written either as an integro-differential equation or as a system of equations consisting of Schrödinger's and Poisson's equations .

The Schrödinger – Newton equation was first considered by Remo Ruffini and Silvano Bonazzola in connection with the self-gravity of boson stars.

Lajos Diósi and Roger Penrose later discussed that the Schrödinger-Newton equation could explain the collapse of the wave function . Matter has quantum properties, whereas gravity remains a classic theory.

In addition, the Schrödinger-Newton equation is used as a Hartree approximation for the mutual gravitational attraction in a system with a large number of particles.

Overview

Written as a system of equations, the Schrödinger-Newton equation results from the linear Schrödinger equation, expanded by a gravitational potential ${\displaystyle \Phi }$

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

here is the non-gravitational potential; the gravitational potential satisfies the Poisson equation ${\displaystyle V}$ ${\displaystyle \Phi }$

${\displaystyle \nabla ^{2}\Phi =4\pi Gm|\Psi |^{2}\,.}$

Because of the coupling of the wave function and the gravitational potential and because of the term , the system of equations is non-linear. ${\displaystyle |\Psi |^{2}}$

The integro-differential form of the equation is

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

This equation results from the system of equations given above, assuming that the gravitational potential vanishes at infinity.

Mathematically, the Schrödinger-Newton equation is a Hartree equation for the case . The equation has many of the properties of the linear Schrödinger equation. In particular, the total probability and the energy are retained; furthermore, the equation is invariant with respect to a Galilei transformation . Solutions to the Schrödinger-Newton equation have already been examined analytically and numerically; the stationary equation, which results from the separation of the variables, has an infinite set of solutions, of which only the stationary ground state is stable. ${\displaystyle n=2}$

Relationship to semi-classical and quantum gravity

The Schrödinger-Newton equation results from the assumption that gravity also behaves classically on a fundamental level and that the wave function is mass-related. Effects of general relativity are neglected. In the event that the assumption is correct, the Schrödinger-Newton equation is a fundamental equation for a single particle; a generalization to multi-particle systems is described below. In the event that the assumption is incorrect, the Schrödinger-Newton equation is only an approximation for the gravitational attraction in a system with a large number of particles.

Schrödinger-Newton equation for multi-particle systems

In the event that the Schrödinger-Newton equation is a fundamental equation, there is a corresponding equation for multi-particle systems, which Diósi derived analogously to the one-particle equation under the assumption of semi-classical gravity:

${\displaystyle {\begin{aligned}\mathrm {i} \hbar {\frac {\partial \Psi (t,\mathbf {x_{1}} ,\dots ,\mathbf {x_{N}} )}{\partial t}}={\Bigg (}&-\sum _{i=1}^{N}{\frac {\hbar ^{2}}{2m_{i}}}\nabla _{i}^{2}+\sum _{i\not =j}V_{ij}(|\mathbf {x_{i}} -\mathbf {x_{j}} |)\\&-G\sum _{i,j=1}^{N}m_{i}m_{j}\int \mathrm {d} ^{3}\mathbf {y_{1}} \cdots \mathrm {d} ^{3}\mathbf {y_{N}} \,{\frac {|\Psi (t,\mathbf {y_{1}} ,\dots ,\mathbf {y_{N}} )|^{2}}{|\mathbf {x_{i}} -\mathbf {y_{j}} |}}{\Bigg )}\Psi (t,\mathbf {x_{1}} ,\dots ,\mathbf {x_{N}} )\,.\end{aligned}}}$

The potential contains all mutual linear interactions, e.g. B. the Coulomb potential , whereas the gravitational potential results from the mass distribution of all particles. ${\displaystyle V_{ij}}$

With a Born-Oppenheimer approximation , the -particle equation can be separated. One equation describes the relative movement, the other describes the dynamics of the center of gravity of the wave function. The gravitational interaction plays only a minor role for the relative movement, since it is usually weak compared to the other interactions. But it has a significant influence on the movement of the center of gravity. ${\displaystyle N}$

Influence of gravity

A rough determination of the quantities that result in differences between the Schrödinger equation and the Schrödinger-Newton equation is possible by using a Gaussian distribution .

For a radially symmetric Gaussian distribution

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

the linear Schrödinger equation has the solution

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

The maximum of the probability density is at ${\displaystyle 4\pi r^{2}|\Psi |^{2}}$

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

For the acceleration, i.e. the second derivative with respect to time , one obtains from this after a short calculation ${\displaystyle t}$

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

This is done with the acceleration due to gravity

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

compared. There currently is and equating the amounts of the accelerations result in this distance and thus ${\displaystyle t=0}$ ${\displaystyle r_{p}=\sigma }$ ${\displaystyle \textstyle {\frac {\hbar ^{2}}{m^{2}\sigma ^{3}}}={\frac {Gm}{\sigma ^{2}}}}$

${\displaystyle m^{3}\sigma ={\frac {\hbar ^{2}}{G}}\approx 1{,}7\times 10^{-58}\,\mathrm {m\,kg^{3}} \,.}$

This equation allows us to determine a critical dimension for a given mass, and vice versa. ${\displaystyle \sigma }$

Numerical calculations show that this equation gives a good estimate of the parameter range in which gravitational influences become significant.

For a hydrogen atom ( = atomic mass unit ) the critical size is approximately 10 ²² meters; for a particle with a mass of one microgram one gets 10 ⁻³¹ meters. In the range of 10 ¹⁰ atomic mass units, the critical size is in the micrometer range, so that an experimental test of the Schrödinger-Newton equation may be possible in the future. ${\displaystyle m}$

Collapse of the wave function

The idea that gravity causes (or at least influences) the collapse of the wave function was proposed by Károlyházy as early as the 1960s.

The Schrödinger – Newton equation was proposed by Diósi as a mathematical description in this context.

Roger Penrose discussed that a superposition of two or more quantum states, which differ significantly in the mass distribution, is unstable and therefore changes into one of the states. His hypothesis is that there is a preferred set of states (the stationary states of the Schrödinger-Newton equation) that do not collapse any further, but are stable. A macroscopic, massive system can therefore never be in a superposition of states, since the non-linear gravitational self-interaction immediately leads to a collapse into a stationary state of the Schrödinger-Newton equation. According to Penrose, a measurement of a quantum system leads on the one hand to entanglement with the macroscopic environment and thus to decoherence ; at the same time, the entanglement with the massive measurement system leads to a reduction to a certain (the measured) state due to the gravitational self-interaction.

Problems and open questions

There are three fundamental problems in interpreting the Schrödinger-Newton equation as the cause of the collapse of the wave function.

Numerical simulations show that when the wave function “collapses” to a stationary solution, a small part of the wave function tends to reach infinity. This would mean that even in the case of a completely reduced state, a particle can be measured at a distant location with a low probability. The Schrödinger-Newton equation can only partially be used as an explanation and the effect of the environment through decoherence must be taken into account.

A second problem is that Born's probability interpretation is not explained. To solve the measurement problem, it is not sufficient that a collapse of the wave function occurs. It must also be explained that the probability density with which a particle is measured at a certain location can be calculated using the square of the magnitude of the wave function. It is unclear whether, on closer analysis, it can be shown that this probability density occurs.

A final problem arises from the interpretation of the wave function as a real physical object. The wave function can thus be a quantity that can at least in principle be measured. Due to the non-local nature of the wave function, it could therefore be possible to transmit information with faster than light speed, which is in contradiction to the theory of relativity. It is unclear whether this problem is actually relevant on closer inspection.

Individual evidence

1. ^ Remo Ruffini, Silvano Bonazzola: Systems of Self-Gravitating Particles in General Relativity and the Concept of an Equation of State . In: Physical Reviews . 187, No. 5, 1969, pp. 1767-1783. bibcode : 1969PhRv..187.1767R . doi : 10.1103 / PhysRev.187.1767 .
2. ↑ ^{a b c} L. Diósi: Gravitation and quantum-mechanical localization of macro-objects . In: Physics Letters A . 105, 1984, pp. 199-202. arxiv : 1412.0201 . bibcode : 1984PhLA..105..199D . doi : 10.1016 / 0375-9601 (84) 90397-9 .
3. ^ ^{A b} Roger Penrose: On Gravity's Role in Quantum State Reduction . In: General Relativity and Gravitation . 28, No. 5, 1996, pp. 581-600. bibcode : 1996GReGr..28..581P . doi : 10.1007 / BF02105068 .
4. ↑ ^{a b} Roger Penrose: Quantum computation, entanglement and state reduction . In: Phil. Trans. R. Soc. Lond. A . 356, No. 1743, 1998, pp. 1927-1939. bibcode : 1998RSPTA.356.1927P . doi : 10.1098 / rsta.1998.0256 .
5. ^ ^{A b} Roger Penrose: On the Gravitization of Quantum Mechanics 1: Quantum State Reduction . In: Foundations of Physics . 44, 2014, pp. 557-575. bibcode : 2014FoPh ... 44..557P . doi : 10.1007 / s10701-013-9770-0 .
6. ↑ ^{a b} p Carlip: Is quantum gravity necessary? . In: Classical and Quantum Gravity . 25, 2008, p. 154010. arxiv : 0803.3456 . bibcode : 2008CQGra..25o4010C . doi : 10.1088 / 0264-9381 / 25/15/154010 .
7. ↑ Elliott H. Lieb: Existence and uniqueness of the Minimizing Solution of Choquard's Nonlinear Equation . In: Studies of Applied Mathematics . 57, 1977, pp. 93-105.
8. ↑ Oliver Robertshaw, Paul Tod: Lie point symmetries and an approximate solution for the Schrödinger – Newton equations . In: Nonlinearity . 19, 2006, pp. 1507-1514. arxiv : math-ph / 0509066 . bibcode : 2006Nonli..19.1507R . doi : 10.1088 / 0951-7715 / 19/7/002 .
9. ↑ ^{a b c} Domenico Giulini, André Großardt: Gravitationally induced inhibitions of dispersion according to the Schrödinger – Newton Equation . In: Classical and Quantum Gravity . 28, 2011, p. 195026. arxiv : 1105.1921 . bibcode : 2011CQGra..28s5026G . doi : 10.1088 / 0264-9381 / 28/19/195026 .
10. ↑ Irene M. Moroz, Roger Penrose, Paul Tod: Spherically-symmetric solutions of the Schrödinger-Newton equations . In: Classical and Quantum Gravity . 15, 1998, pp. 2733-2742. bibcode : 1998CQGra..15.2733M . doi : 10.1088 / 0264-9381 / 15/9/019 .
11. ^ Paul Tod, Irene M. Moroz: An analytical approach to the Schrödinger – Newton equations . In: Nonlinearity . 12, 1999, pp. 201-216. bibcode : 1999Nonli..12..201T . doi : 10.1088 / 0951-7715 / 12/2/002 .
12. ^ ^{A b} R. Harrison, I. Moroz, KP Tod: A numerical study of the Schrödinger – Newton equations . In: Nonlinearity . 16, 2003, pp. 101-122, arxiv : math-ph / 0208045 (part 1) and arxiv : math-ph / 0208046 (part 2); based on R. Harrison: A numerical study of the Schrödinger – Newton equations , bibcode : 2003Nonli..16..101H , doi: 10.1088 / 0951-7715 / 16/1/307 .
13. ↑ Mohammad Bahrami, André Großardt, Sandro Donadi, Angelo Bassi: The Schrödinger-Newton equation and its foundations . In: New J. Phys. . 2014. arxiv : 1407.4370 .
14. ↑ ^{a b} J. R. van Meter: Schrödinger-Newton 'collapse' of the wave function . In: Classical and Quantum Gravity . 28, No. 21, 2011, p. 215013. arxiv : 1105.1579 . bibcode : 2011CQGra..28u5013V . doi : 10.1088 / 0264-9381 / 28/21/215013 .
15. ↑ F. Károlyházy: Gravity and Quantum Mechanics of Macroscopic Objects . In: Nuovo Cimento A . 42, 1966, pp. 390-402. bibcode : 1966NCimA..42..390K . doi : 10.1007 / BF02717926 .