Schrödinger-Newton equations
Encyclopedia
The Schrödinger–Newton equations are modifications of the Schrödinger equation
and derived from Gauss' law for gravity
, proposed by Roger Penrose
in The Road to Reality, that mathematically describe the basis states involved in a gravitationally-induced wavefunction collapse
scheme:
where
is a quasi-Newtonian potential
given by
and
is the classical mass density.
It can be shown that these equations conserve probability, momentum etc., as the Schrödinger equation does.
Their Lie point symmetries
are rotation
s, translation
s, scaling
s, a phase change in time, and a Galilean transformation
of sorts that looks like the equivalence principle
at work.
Schrödinger equation
The Schrödinger equation was formulated in 1926 by Austrian physicist Erwin Schrödinger. Used in physics , it is an equation that describes how the quantum state of a physical system changes in time....
and derived from Gauss' law for gravity
Gauss' law for gravity
In physics, Gauss's law for gravity, also known as Gauss's flux theorem for gravity, is a law of physics which is essentially equivalent to Newton's law of universal gravitation...
, proposed by Roger Penrose
Roger Penrose
Sir Roger Penrose OM FRS is an English mathematical physicist and Emeritus Rouse Ball Professor of Mathematics at the Mathematical Institute, University of Oxford and Emeritus Fellow of Wadham College...
in The Road to Reality, that mathematically describe the basis states involved in a gravitationally-induced wavefunction collapse
Wavefunction collapse
In quantum mechanics, wave function collapse is the phenomenon in which a wave function—initially in a superposition of several different possible eigenstates—appears to reduce to a single one of those states after interaction with an observer...
scheme:
where
is a quasi-Newtonian potentialPotential
*In linguistics, the potential mood*The mathematical study of potentials is known as potential theory; it is the study of harmonic functions on manifolds...
given by
and
is the classical mass density.It can be shown that these equations conserve probability, momentum etc., as the Schrödinger equation does.
Their Lie point symmetries
Lie point symmetry
Towards the end of the nineteenth century, Sophus Lie introduced the notion of Lie group in order to study the solutions of ordinary differential equations . He showed the following main property: the order of an ordinary differential equation can be reduced by one if it is invariant under...
are rotation
Rotation
A rotation is a circular movement of an object around a center of rotation. A three-dimensional object rotates always around an imaginary line called a rotation axis. If the axis is within the body, and passes through its center of mass the body is said to rotate upon itself, or spin. A rotation...
s, translation
Translation
Translation is the communication of the meaning of a source-language text by means of an equivalent target-language text. Whereas interpreting undoubtedly antedates writing, translation began only after the appearance of written literature; there exist partial translations of the Sumerian Epic of...
s, scaling
Scaling
Scaling may refer to:* Scaling , a linear transformation that enlarges or diminishes objects* Reduced scales of semiconductor device fabrication processes...
s, a phase change in time, and a Galilean transformation
Galilean transformation
The Galilean transformation is used to transform between the coordinates of two reference frames which differ only by constant relative motion within the constructs of Newtonian physics. This is the passive transformation point of view...
of sorts that looks like the equivalence principle
Equivalence principle
In the physics of general relativity, the equivalence principle is any of several related concepts dealing with the equivalence of gravitational and inertial mass, and to Albert Einstein's assertion that the gravitational "force" as experienced locally while standing on a massive body is actually...
at work.