QM lessons for standard cosmology Marco A. Reyes S. Departamento de Física DCI-Ugto XIII Mexican Workshop on Particles and Fields

Motivation I’ll refer to previous works on 1D QM In particular, I’ll refer to the appearance of the Riccati eq in different aspects of QM problems I shall speak about two of our works on cosmology, and some work in progress Marco A. Reyes: QM lessons for standard cosmology

Quantum Mechanics Erwin Schödinger proposed his model in 1926 Solved the hydrogen atom problem: the static 3D eq is separable in 3 ODE Marco A. Reyes: QM lessons for standard cosmology

Numerical calculations Hydrogen atom The radial eq is solved with the use of associated Laguerre polynomials EL 1834-1886 Numerical calculations The polar eq is solved with the use of the asociated Legendre polynomials A-ML 1752-1833 Laplace’s Eq Marco A. Reyes: QM lessons for standard cosmology

Simple Harmonic Oscillator The 1D differential eq for the SHO is solved with the use of the Hermite polynomials Ch.H 1822-1901 Prob. Sequence Very few QM problems (or 2O) have exact solutions numerical approx, variations, …   sequence Marco A. Reyes: QM lessons for standard cosmology

Bound States In general, each 1D QM problem has one bound state solution (for the involved parameters) that satisfies the BC of the problem B. Mielnik y M.A.Reyes, J.Phys. A 29 (1996) 6009. Even if there is no exact solution, it is possible to find the form of the exp-decaying tails that act as the asymptotes that determine that solution Marco A. Reyes: QM lessons for standard cosmology

Classical Schrödinger eq Schrödinger dynamics in phase space whose solutions define angular asymptotes are driven by the evolution matrix a solution is obtained when  brings e+ to e- Marco A. Reyes: QM lessons for standard cosmology

Angular Riccati Equation In terms of the angular Riccati eq, the asymptotes determinen the exact ocurrence of a bound state: squeezing B. Mielnik y M.A.Reyes, J.Phys. A 29 (1996) 6009. C.F.Delgado, B.Mielnik y M.A.Reyes, J.Phys. A 237 (1998) 359. Marco A. Reyes: QM lessons for standard cosmology

SUSY QM SUSY appears in particle physics: new particles, dark matter, cosmology… In 1984 Mielnik proposed to redefine creation anihilation ops of the SHO Marco A. Reyes: QM lessons for standard cosmology

SUSY QM for SHO Mielnik solves the Riccati eq for (x) to find new hamiltonians which are isoespectral to the original one Marco A. Reyes: QM lessons for standard cosmology

New potentials with the same spectrum Same for Dirac and KG eqs SUSY QM for SHO New potentials with the same spectrum Same for Dirac and KG eqs Marco A. Reyes: QM lessons for standard cosmology

SUSY type applications Orthogonal Polynomials – Special Functions M.A.Reyes, D.Jiménez y H.C.Rosu, Rev. Mex. Fís. 49 (2003) 358 H.C.Rosu y M.A.Reyes, Phys. Rev. E 51 (1995) 5112 And also Difference Equations Supersymmetric time-continuous discrete random walks H.C.Rosu, M.A.Reyes y O.Obregón, Rev. Mex. Fís. 43 (1997) 224 The three step master equation: class of parametric stationary solutions H.C.Rosu, M.A.Reyes y F.Valencia, Int. Jour. Theor. Phys. 44 (2005) 1565 Marco A. Reyes: QM lessons for standard cosmology

Riccati eq in GR We have also applied the SUSY approach to relativity problems … J.Socorro, M.A.Reyes y F.A.Gelbert, Phys. Lett. A 313 (2003) 338 …therefore, whenever I see an equation like which may be transformed into I think I could try to find solutions Marco A. Reyes: QM lessons for standard cosmology

Standard Cosmology: FRW, (t) What I didn’t know was that there was a set of coupled eqs and that this is an area where very few exact* solutions are known Decided to reproduce typical commutative problems:  2 and  4 Marco A. Reyes: QM lessons for standard cosmology

 FRW + (t) I found that one part of the eqs is easily solved solve for Φ(t), and then find V(Φ) and H(t) I’ve been told that these are actually the dynamics found when solving for slow roll M.A.Reyes, arXiv: gr-qc/0806.2292 15 Marco A. Reyes: QM lessons for standard cosmology

FRW + (t) The exact solutions found in this work are to be compared with typical potentials used in the literature M.A.Reyes, arXiv: gr-qc/0806.2292 16 Marco A. Reyes: QM lessons for standard cosmology

Solutions .vs. Non-linearity Why is it that this solution does not behave as found numerically? Overdamping? Non-linearity? The logistic eq posses only one solution M.A.Reyes, arXiv: gr-qc/0806.2292 17 Marco A. Reyes: QM lessons for standard cosmology

QM .vs. FRW + (t) If there appears a Riccati eq, what about a Scrhodinger eq?  non singular  Many analogies are found: =E, a3=ψ, and Slow Roll becomes the WKB approx N.Barbosa-Cendejas y M.A.Reyes, arXiv: gr-qc/1001.0084 Marco A. Reyes: QM lessons for standard cosmology

Schrödinger picture of SC N.Barbosa-Cendejas y M.A.Reyes, arXiv: gr-qc/1001.0084 Marco A. Reyes: QM lessons for standard cosmology

Big crunch .vs. Forever Inflation  , and a(t), need to be non singular N.Barbosa-Cendejas y M.A.Reyes, arXiv: gr-qc/1001.0084 Marco A. Reyes: QM lessons for standard cosmology

5D Membranes Similar analogies allow us to find solutions for the graviton normal modes in a 5D membrane model of Weyl geometry the Schrödinger type eq in this case has two bound states (normal modes) N.Barbosa-Cendejas, A.Herrera-Aguilar , M.A.Reyes y C.Schubert Phys. Rev. D 77 (2008) 126013 Marco A. Reyes: QM lessons for standard cosmology

New SHO Factorization We have worked on redefining Mielnik’s factorization for the SHO The SL equation reduces to the QM-SHO problem when =0 and to the Hermite eq when  M.A.Reyes, H.C.Rosu and M.Ranferi, Phys. Lett. A 375 (2011) 2145 Marco A. Reyes: QM lessons for standard cosmology

Generalized Hermite functions Here we show how the SL eigenfunctions run from the Hermite polinomials to the SHO eigenfunctions when the parameter  is varied M.A.Reyes, H.C.Rosu and M.Ranferi, Phys. Lett. A 375 (2011) 2145 Marco A. Reyes: QM lessons for standard cosmology

Ongoing & Future We are working on the generalization of the SHO factorization to include Mielnik’s work: preSUSY We are also working on making the exact solutions method useful for cosmologists We are working on other QM problems Marco A. Reyes: QM lessons for standard cosmology