Physics & Astronomy International Journal Review Article Open Access Exact solutions of the Newton-Schrödinger equation, infinite derivative gravity and Schwarzschild atoms Abstract Volume 1 Issue 4 - 2017 Exact solutions to the stationary spherically symmetric Newton-Schrödinger equation are proposed in terms of integrals involving generalized Gaussians. The energy eigenvalues Carlos Castro Perelman are also obtained in terms of these integrals which agree with the numerical results in the Center for Theoretical Studies of Physical Systems, Clark Atlanta literature. A discussion of infinite derivative-gravity follows which allows generalizing the University, Georgia Newton-Schrödinger equation by replacing the ordinary Poisson equation with a modified Correspondence: Carlos Castro Perelman, Center for non-local Poisson equation associated with infinite-derivative gravity. We proceed to Theoretical Studies of Physical Systems, Clark Atlanta University, replace the nonlinear Newton-Schrödinger equation for a non-linear quantum-like Bohm- Atlanta, GA 30314, Georgia, Email Poisson equation involving Bohm’s quantum potential, and where the fundamental quantity is no longer the wave-function Ψ but the real-valued probability density ρ . Finally, we Received: October 07, 2017 | Published: November 10, 2017 discuss how the latter equations reflect a nonlinear feeding loop mechanism between matter and geometry which allows us to envisage a “Schwarzschild atom” as a spherically symmetric probability cloud of matter which curves the geometry, and in turn, the geometry back-reacts on this matter cloud perturbing its initial distribution over the space, which in turn will affect the geometry, and so forth until static equilibrium is reached. Keywords: quantum mechanics, Newton-Schrödinger equation, infinite derivative gravity, de Broglie-Bohm theory, Schwarzschild metric Introduction (such as with the proposed class of experiments put forward there), it would be sufficient to consider Newtonian gravity. This leads us The Newton-Schrödinger equation to consider what Penrose termed the Schrödinger-Newton equation. 7–17 Various arguments have been put forward from time to time to This equation has had a long history since the 1950’s. It is the name support the view that quantum state reduction is a phenomenon given to the system coupling the Schrödinger equation to the Poisson that occurs objectively, because of some gravitational influence.1–6 equation. In the case of a single particle, this coupling is effected as According to a particular argument put forward by Penrose,1 a follows: for the potential energy term in the Schrödinger equation take superposition of two quantum states, each of which would be stationary the gravitational potential energy determined by the Poisson equation on its own, but for which there is a significant mass displacement from a matter density proportional to the probability density obtained between the two states, ought to be unstable and reduce to one state from the wave-function. For a single particle of mass m the system consists of the following pair of partial differential equations: or the other within a certain characteristic average timescale TG . This argument is motivated by a conflict between the basic principles of The Newton-Schrödinger equation is nonlinear and nonlocal quantum mechanics and those of general relativity. It is accordingly modification of the Schrödinger equation given by proposed that TG can be calculated in situations for which velocities 2 and gravitational potentials are small in relativistic units, so that a ∂Ψ()rt, 2 i =− ∇Ψ()rt,++V ()() rt ,,Ψ rt mVrt ( ,,)(Ψ rt ) (1) Newtonian approximation is appropriate, and TG is the reciprocal, in ∂tm2 G Planckian units, of the gravitational self-energy EG of the difference Where V ()rt, is the external potential acting on the particle and between the mass distributions of the two states. mVG ( r, t) is the self- gravitational potential energy arising due a There is of course a substantial literature on the problem of mass density obtained from the wave function of the particle itself. 2 wave-function collapse and the related measurement problem. See, Given the Poisson equation sourced by a mass density ρ=m Ψ() rt, for example4–6 and references therein. For a different idea about 2 2 3 gravitationally-induced wave-function collapse see. It has been ∇ΨVrG (),t=4πGm () rt , (2) 1,2 pointed out by Penrose R et al. that one can regard the basic It leads to a self-gravitational potential stationary states, into which a superposition of such states is to decay 2 into (on a timescale of order /E ), as stationary solutions of the m Ψ()rt, 3 G V() rt,=−∫Gd r′ (3) Schrödinger equation where there is an additional term provided G rr− ′ by a certain gravitational potential. The appropriate gravitational Inserting equation (3) into equation (1) leads to the integro potential is the one which arises from the mass density given by the differential form of the nonlinear and expectation value of the mass distribution in the state determined by the wave-function. In the practical situations under consideration in1 Submit Manuscript | http://medcraveonline.com Phys Astron Int J. 2017;1(4):131‒137. 131 ©2017 Perelman. This is an open access article distributed under the terms of the Creative Commons Attribution License, which permits unrestricted use, distribution, and build upon your work non-commercially. Copyright: Exact solutions of the Newton-Schrödinger equation, infinite derivative gravity and Schwarzschild atoms ©2017 Perelman 132 nonlocal Newton-Schrödinger equation 2 2 2 ∂Ψ()rt, 2 23m Ψ()rt, Gm i =− ∇Ψ()rt,,,+VG ( rt)Ψ() rt) −∫m d r′ Ψ( r,t (4) ∂tm2 − ′ the Newtonian form − . Because the error function Erf( r) r r r admits a series expansion around r = 0 as Relevant work pertaining pseudo differential and hyper differential operators, nonlinear partial differential equations and their 2rr35 11 r 241r 7 Erfr( ) −+ − +... (9) applications can be found in.18–20 ππ20 π 840 π This equation is based on the assumption that the point-particle The potential Vr( ) is no longer singular at the origin r = 0 , but G is smeared over space such that its mass is distributed according to it behaves as its wave function. Namely, there is a mass cloud over space whose 2 24 net mass m=m∫ Ψ() rt,.d 3r′ coincides with the mass of the 2Gm r11 r Vr( ) − [1 −+ +...] (10) point-particle. The mass cloud is self-gravitating and experiences a G πσ 2σσ24 40 gravitational potential energy given by equation (3).1 The normalized Gaussian wave function 21 More recently, the authors showed that nonlinear Schrödinger 22 −r /2σ equations (NSEs) for individual particles do not follow from general 1 e Φ≡(r ) (11) relativity (GR) plus quantum field theory (QFT). Contrary to what is πσ3/4 3/2 commonly assumed, the NSEs are not the weak-field, non-relativistic Satisfies limit of the semi-classical Einstein equation. The wave-function in the ∞ NSEs makes sense only as that for a mean field describing a system 2 2 ∫ Φ=(r ) 4π r dr 1 (12) of N particles as N →∞ , not that of a single or finite many particles. They concluded that the origins and consequences of NSEs are very 0 different, and should be clearly demarcated from those of the semi- Let us minimize the energy functional 2 ∞∞ classical Einstein equation, the only legitimate representative of semi 222 = Φ ∇Φ2 ππ + Φ( ) Φ ( ) classical gravity, based on GR+QFT. Bearing this in mind, we proceed E ∫∫( (r) ( r))4r dr mVG [rr ]( ) 4r dr 13 2m 00 to find solutions to the NSE’s. Using this Gaussian as a trial function, and which generates the Solutions to the Newton-Schrödinger equation Gm r regular potential at the origin V []Φ=(r ) Erf given by G r σ Let us set the external potential to zero and look for stationary 2 -iEt/ equation (8) after solving Poisson’s equation (7). The integrals to be solutions Ψ(Φrr,t)=e ( ) such that the gravitational potential evaluated in equation (13) are of the form becomes time independent. Ö(r ) obeys the equation ∞ 22 2 42−x 3 1 − x 2 2 Ö ∫ x−=3 x e dx −π Erf ( x) − ex 2x − 3 (14) 2 2 (r ) 3 ( ) 8 4 ( ) EÖÖ(rr)=−−∫∇ ( ) Gm d r′ Ö(r ) (5) 0 2m r −r′ 2Which is also related to the incomplete gamma function The authors22–26 found numerical spherically-symmetric solutions 1 to equation (5). Variational forms of the stationary Newton- γπ;r= Erf (r) Schrödinger equation to find a lower bound for the ground state 2 27 energy have been studied by several authors, see references in, and ∞ 2 2 −x 1 − compared to numerical values in the literature. ∫ xErf( xx) e dx= 2 Erf( 2 x)− 2ex Erf ( ) (15) 4 2 0 1This may be in conflict with Born’s rule of interpreting Ψ(rt, ) After performing the definite integrals we find that as the probability density of finding a particle at the point r if one 3 π 222 Gm has abandoned the notion of point-particles. At the moment we shall E (σ ) = − (16) not be concerned with this. 4 mσ 2 π σ If one replaces a delta function point-mass source distribution 2 3π mδ32( rmr)= δπ( )/4 r for a normalized Gaussian mass distribution σ = (17) min 22 3 of width Gm 22 And inserting this value of σ into E (σ ) gives me−r /σ ∞ ρ(r)= ⇒=m∫ ρπ( r)4 r2 dr (6) 2 Gm25 Gm25 3/2 3 E = = −0.119 (18) πσ 0 min 3/2 22 3π The solution to Poisson’s equation Which is a satisfactory value since it is above the lower energy bound 2 25 25 ∇=VrG ( ) 4πρ G( r) (7) 32 Gm Gm = = − ( ) 2 E 0.360 19 Is given in terms of the error function Erf( r) as follows bound 9π 222 27 25 2 r/σ 2 Derived by.