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The '''Schrödinger-Newton equations''' are modifications of the [[Schrödinger equation]] and derived from [[Gauss' law for gravity]]. These equations were first considered by Runi and Bonazzola<ref>See Runi and Bonazzola "Systems of Self-Gravitating Particles in General Relativity and the Concept of an Equation of State".</ref> in the theory of self-gravitating [[boson star]]s.
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With self-gravitating boson stars, the Schrodinger-Newton equations are the non-relativistic limit of the governing [[Klein-Gordon equation]]s. Runi and Bonazzola considered the problem of finding stationary boson-stars in the non-relativistic spherically symmetric case, which corresponds to finding stationary spherically symmetric solutions of the Schrodinger-Newton equations.
 
But the Schrödinger-Newton equation has importance for the nonrelativistic limit of [[quantum gravity]],<ref>See Domenico Giulini, André Großardt, "The Schrödinger-Newton equation as non-relativistic limit of self-gravitating Klein-Gordon and Dirac fields"</ref> and as a way we can explain gravity causing quantum wave function collapse.
 
== Overview ==
We consider the usual [[Schrödinger equation]] with the gravitational potential included
: <math>i\hbar \frac{\partial\Psi}{\partial t} = -\frac{\hbar^2}{2m} \nabla ^2 \Psi + V \Psi  + m \Phi \Psi </math>
where the [[gravitational potential]] <math>\Phi</math> satisfies the field equation
:<math>\nabla^2 \Phi = 4 \pi G \rho |\Psi|^2 = 4\pi G\langle\rho\rangle</math>
where <math>\rho</math> is the classical mass density.
 
After some calculation, we find the gravitational potential can be written down as
:<math>\Phi (\mathbf{x},t) = -G m \int_{}^{} \frac{  | \Psi(\mathbf{y},t) |^2}{|\mathbf{x} - \mathbf{y}|} \, d^3 \mathbf{y}</math>
It can be shown that these equations conserve probability, momentum etc., as the Schrödinger equation does.
 
Their [[Lie point symmetry|Lie point symmetries]] are [[rotation]]s, [[translation]]s, [[scaling (geometry)|scaling]]s, a phase change in time, and a [[Galilean transformation]] of sorts that looks like the [[equivalence principle]] at work.<ref>See Robertshaw and Tod, specifically section 2.2</ref>
 
== Quantum wave function collapse ==
 
The idea gravity causes (or somehow influences) [[wavefunction collapse]] using the Schrodinger-Newton equation dates back to the 1980s.<ref>See Diósi.</ref>  
 
[[Roger Penrose]] proposed that the Schrodinger-Newton equations mathematically describe the basis states involved in a gravitationally-induced [[wavefunction collapse]] scheme.<ref>See Penrose 1996 and 1998.</ref> Basically Penrose suggests that a superposition of two or more quantum states, which have a significant amount of mass displacement between the states, ought to be unstable and reduce to one of the states within a finite time. He hypothesises there exists a "preferred" set of states which could collapse no further, specifically the [[Stationary state]]s.
 
[[K.R.W. Jones]] considered an extension of this proposal to gravitational collapse of an environmentally decohered quantum state. The interpretation differs in that he advocated this as a completion of the Schrödinger interpretation of a material wavefunction. The role of gravity in the scheme of Jones is to suppress macroscopic dispersion.
 
However, each of these proposals involves the same Schrödinger–Newton system of equations.
 
== Notes ==
{{Reflist}}
 
==References==
* L. Diósi, "Gravitation and quantum-mechanical localization of macro-objects", ''Physics Letters A'' '''105''' 4–5 (1984) 199–202 [http://dx.doi.org/10.1016/0375-9601(84)90397-9 doi:10.1016/0375-9601(84)90397-9]
* Roger Penrose, "On Gravity's Role in Quantum State Reduction", ''General Relativity and Gravitation'' '''28''' 5 (1996) 581-600. DOI:[http://dx.doi.org/10.1007/BF02105068 10.1007/BF02105068]
* Roger Penrose, "Quantum computation, entanglement and state reduction",  ''Phil. Trans. R. Soc. Lond.'' A '''356''' no. 1743 (1998) 1927-1939. DOI:[http://dx.doi.org/10.1098/rsta.1998.0256 10.1098/rsta.1998.0256]
* P.J. Salzman, [[Steve Carlip|S. Carlip]], "A possible experimental test of quantized gravity" [http://arxiv.org/abs/gr-qc/0606120 arXiv:gr-qc/0606120]
* Domenico Giulini, André Großardt, "The Schrödinger-Newton equation as non-relativistic limit of self-gravitating Klein-Gordon and Dirac fields" [http://arxiv.org/abs/1206.4250 arXiv:1206.4250] [gr-qc]
* Domenico Giulini, André Großardt, "Gravitationally induced inhibitions of dispersion according to the Schrödinger-Newton Equation" ''Class. Quantum Grav.'' '''28''' (2011) 195026. Eprint [http://arxiv.org/abs/1105.1921 arXiv:1105.1921] [gr-qc]
* R. Harrison "A numerical study of the Schrodinger-Newton equations", Ph.D Thesis [http://eprints.maths.ox.ac.uk/41/1/harrison.pdf online].
*  R. Harrison, I. Moroz, K.P. Tod, "A numerical study of the Schrodinger-Newton equation 1: Perturbing the spherically-symmetric stationary states" [http://arxiv.org/abs/math-ph/0208045/ arXiv:math-ph/0208045]
* R. Harrison, I. Moroz, K.P. Tod, "A numerical study of the Schrodinger-Newton equation 2: the time-dependent problem" [http://arxiv.org/abs/math-ph/0208046 arXiv:math-ph/0208046]
*  Oliver Robertshaw, Paul Tod, "Lie point symmetries and the geodesic approximation for the Schrödinger-Newton equations" [http://arxiv.org/abs/math-ph/0509066 arXiv:math-ph/0509066]
* R.Runi, S.Bonazzola, "Systems of Self-Gravitating Particles in General Relativity and the concept of an Equation of State", ''Phys.Rev.'' '''187''' 5 (1969) 1767-1783. doi:[http://dx.doi.org/10.1103/PhysRev.187.1767 10.1103/PhysRev.187.1767]
* J. R. van Meter, "Schrödinger-Newton 'collapse' of the wave function", ''Class.Quant.Grav.'' '''28''' 21 (2011) 215013. Eprint [http://arxiv.org/abs/1105.1579 arXiv:1105.1579] [quant-ph]
* K. R. W. Jones, "Newtonian Quantum Gravity", Aust. J. of Phys. '''48''' 6 (1995) 1055-1082. Eprint [http://arxiv.org/abs/quant-ph/9507001 arxiv:9507001] [quant-ph]
 
{{DEFAULTSORT:Schrodinger-Newton Equations}}
[[Category:Equations]]
[[Category:Quantum gravity]]

Latest revision as of 21:23, 24 March 2014

If a man experiences anxiety during sex it can alter the balance between the parasympathetic and the sympathetic nervous system and lead to the loss of an erection. Symptoms of infertility in individuals whose age is more than 35 years old are very common compared to those who are still in their prime years of life. Disorders that narrow arteries and decrease blood inflow on the blood vessels can cause erectile dysfunction. When you want an erection, just reach for the pump. A better and simple way to isolate between physiological and psychological impotence is to check if the patient ever had an erection.

Not only that, the process of cell repair can also be done well ( the regenerate ) , that old cells replaced with better ones. This environment thus promotes the growth of the anaerobic bacteria. The same high fat diet that narrows arteries and blocks flow to the heart also narrows the arteries that carry blood to the penis. Read effective Herbal Treatment for Male Infertility. Alcohol or drug abuse: Drugs such as antidepressants, sedatives, and antihypertensives have been known to contribute to impotence.

It is the concentrations of UVB and UVA rays that are reported to cause the problems, but there are ways to protect your skin while getting that glowing tan at the same time. However, recent research indicates that as many as 75 percent of all erectile disorder cases result from medical problems, usually related to the vascular (blood supply) system in the penis. Cutting down on your alcohol intake or giving up smoking could make all the difference. After trying it once, you may find your partner requesting you keep it on. Even though you can create an exercise routine to do each one of the separately, you are going to realize that the best result will be using some kind of circuit training.

PROPECIA®, also marketed under the Proscar brand name, is a drug that comes in a tablet containing 1 mg finasteride. Nowadays lots of pharmacies are providing the generic medications only because all these medicines are manufactured by the branded companies like Zydus cadila pvt ltd. Finally, another great tip would be to exercise at least 30 minutes a day. 'This herb is a noted aphrodisiac (arouse sexual desire), demulcent (soothing), and diaphoretic. If you are looking for a legitimate resource for Cialis and Levitra you can visit UK Med.

It can ruin relationships between people, making it extremely difficult for them to manage sexual relationships. Ingesting medium chain fatty acids is rather valuable for people, as they're quickly burnt by the human body filling it with the needed degree of stamina. Many persons, therefore, suffer from sexual dysfunctions. An ability to maintain a penile erection requires a healthy & sound nervous system, healthy arteries in and near corpora cavernosa, smooth muscles & fibrous tissues in corpora cavernosa and abundant levels of nitric oxide. This medication can benefit a number of mood disorders.

If you have any concerns concerning the place and how to use natural cures for erectile dysfunction, you can contact us at our own web page.

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