ISSUE 2007 VOLUME 2 - The World of Mathematical Equations

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April, 2007 PROGRESS IN PHYSICS Volume 2 Observation of orthopositronium anomalies gives one physical grounds to broaden the present-day SM. It now appears appropriate to analyze “anti-Compton scattering” in connection with the detection of notoph in the proposed program of additional measurements, which aim at proving the existence of a connection between gravity and electromagnetism [3]. We may add that the concept of the supersymmetric version of a spin- 1/2 quasi-particle and a hole as supersymmetric partners has been discussed in the literature [9]. To sum up: one should carry out additional measurements because the result, inconceivable in the frame of the SM, becomes an expected result in the program of experimentum crucis (Fig. 3). A positive result of this crucial experiment would mean the birth of new physics that would be complementary to the Standard Model. References Submitted on January 24, 2007 Accepted on January 30, 2007 1. Vallery R. S., Gidley D. W. and Zitzewitz P. W. Resolution of orthopositronium-lifetime puzzle. Phys. Rev. Lett., 2003, v. 90(20), 203402 (1–4). 2. Levin B. M., Kochenda L. M., Markov A. A. and Shantarovich V. P. Time spectra of annihilation of positron ( 22 Na) in gaseous neon of various isotopic compositions. Sov. J. Nucl. Phys., 1987, v. 45(6), 1119–1120. 3. Kotov B. A., Levin B. M. and Sokolov V. I. Orthopositronium: on the possible relation of gravity to electricity. Preprint-1784, A. F. Ioffe Physical Technical Institute of Russian Academy of Sciences. St.-Petersburg, 2005; arXiv: quant-ph/0604171. 4. Andreev A. F. Spontaneously broken complete relativity. JETF Lett., 1982, v. 36(3), 100. 5. Levin B. M. Orthopositronium: annihilation of positron in gaseous neon. arXiv: quant-ph/0303166. 6. Ogievetskii V. I. and Polubarinov I. V. Notoph and its possible interactions. Sov. J. Nucl. Res., 1966, v. 4(1), 156. 7. Levin B. M. On the kinematics of one-photon annihilation of orthopositronium. Phys. At. Nucl., 1995, v. 58 (2). 332. 8. Synge J. L. Anti-Compton scattering. Proc. Roy. Ir. Acad., 1974, v.A74 (9), 67. 9. Lee C. J. Spin- 1/2 particle and hole as supersymmetry partners. Phys. Rev. A, 1994, v.50, R4 (4–6). B. M. Levin. A Proposed Experimentum Crucis for the Orthopositronium Lifetime Anomalies 55
Volume 2 PROGRESS IN PHYSICS April, 2007 Numerical Solution of Time-Dependent Gravitational Schrödinger Equation 1 Introduction Vic Christianto ∗ , Diego L. Rapoport † and Florentin Smarandache ‡ ∗ Sciprint.org — a Free Scientific Electronic Preprint Server, http://www.sciprint.org E-mail: admin@sciprint.org † Dept. of Sciences and Technology, Universidad Nacional de Quilmes, Bernal, Argentina E-mail: diego.rapoport@gmail.com ‡ Department of Mathematics, University of New Mexico, Gallup, NM 87301, USA E-mail: smarand@unm.edu In recent years, there are attempts to describe quantization of planetary distance based on time-independent gravitational Schrödinger equation, including Rubcic & Rubcic’s method and also Nottale’s Scale Relativity method. Nonetheless, there is no solution yet for time-dependent gravitational Schrödinger equation (TDGSE). In the present paper, a numerical solution of time-dependent gravitational Schrödinger equation is presented, apparently for the first time. These numerical solutions lead to gravitational Bohr-radius, as expected. In the subsequent section, we also discuss plausible extension of this gravitational Schrödinger equation to include the effect of phion condensate via Gross-Pitaevskii equation, as described recently by Moffat. Alternatively one can consider this condensate from the viewpoint of Bogoliubov-deGennes theory, which can be approximated with coupled timeindependent gravitational Schrödinger equation. Further observation is of course recommended in order to refute or verify this proposition. In the past few years, there have been some hypotheses suggesting that quantization of planetary distance can be derived from a gravitational Schrödinger equation, such as Rubcic & Rubcic and also Nottale’s scale relativity method [1, 3]. Interestingly, the gravitational Bohr radius derived from this gravitational Schrödinger equation yields prediction of new type of astronomical observation in recent years, i.e. extrasolar planets, with unprecedented precision [2]. Furthermore, as we discuss in preceding paper [4], using similar assumption based on gravitational Bohr radius, one could predict new planetoids in the outer orbits of Pluto which are apparently in good agreement with recent observational finding.. Therefore one could induce from this observation that the gravitational Schrödinger equation (and gravitational Bohr radius) deserves further consideration. In the meantime, it is known that all present theories discussing gravitational Schrödinger equation only take its time-independent limit. Therefore it seems worth to find out the solution and implication of time-dependent gravitational Schrödinger equation (TDGSE). This is what we will discuss in the present paper. First we will find out numerical solution of time-independent gravitational Schrödinger equation which shall yield gravitational Bohr radius as expected [1, 2, 3]. Then we extend our discussion to the problem of time-dependent gravitational Schrödinger equation. In the subsequent section, we also discuss plausible extension of this gravitational Schrödinger equation to include the effect of phion condensate via Gross-Pitaevskii equation, as described recently by Moffat [5]. Alternatively one can consider this phion condensate model from the viewpoint of Bogoliubov-deGennes theory, which can be approximated with coupled time-independent gravitational Schrödinger equation. To our knowledge this proposition of coupled timeindependent gravitational Schrödinger equation has never been considered before elsewhere. Further observation is of course recommended in order to verify or refute the propositions outlined herein. All numerical computation was performed using Maple. Please note that in all conditions considered here, we use only gravitational Schrödinger equation as described in Rubcic & Rubcic [3], therefore we neglect the scale relativistic effect for clarity. 2 Numerical solution of time-independent gravitational Schrödinger equation and time-dependent gravitational Schrödinger equation First we write down the time-independent gravitational Schrödinger radial wave equation in accordance with Rubcic & Rubcic [3]: d2R 2 dR + dr2 r dr + 8πm2E ′ H2 R + + 2 4π r 2GMm2 H2 R − ℓ (ℓ + 1) r2 R = 0 . When H, V , E ′ represents gravitational Planck constant, Newtonian potential, and the energy per unit mass of the 56 V. Christianto, D. L. Rapoport and F. Smarandache. Numerical Solution of Time-Dependent Gravitational Schrödinger Equation (1)
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