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

2. 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

3. 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

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

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

6. 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

7. 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

8. 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

9. 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

10. 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

11. 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

12. 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

13. 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

14. 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

15.  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/ 15
Marco A. Reyes: QM lessons for standard cosmology

16. 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/ 16
Marco A. Reyes: QM lessons for standard cosmology

17. 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/ 17
Marco A. Reyes: QM lessons for standard cosmology

18. 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/
Marco A. Reyes: QM lessons for standard cosmology

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

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

21. 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)
Marco A. Reyes: QM lessons for standard cosmology

22. 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

23. 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

24. 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