Propagator of a charged particle with a spin in uniform magnetic and perpendicular electric fields

Letters in Mathematical Physics

Volumn 84, Issue 2-3, 2008, Pages 159-178

  Cordero Soto, Ricardo ^a   Lopez, Raquel M ^a   Suazo, Erwin ^a   Suslov, Sergei K ^a  

^a ARIZONA STATE UNIVERSITY   (United States)

Author keywords

Forced harmonic oscillator; Fourier transform; Green function; Landau levels; Nonlinear Schr dinger equation; Propagator; Riccati differential equation; Schr dinger equation in electromagnetic field; The Cauchy initial value problem

Indexed keywords

EID: 46649105468     PISSN: 03779017     EISSN: 15730530     Source Type: Journal    
DOI: 10.1007/s11005-008-0239-6     Document Type: Article

References (48)

1. Andrews G.E., Askey R.A., Roy R.: Special Functions. Cambridge University Press, Cambridge (1999)
2. Arrighini G.P., Durante N.L.: More on the quantum propagator of a particle in a linear potential. Am. J. Phys. 64(8), 1036-1041 (1996)
3. Banica, V., Vega, L.: On the Dirac delta as initial condition for nonlinear Schrödinger equations. arXiv:0706.0819v1 [math.AP] 6 June (2007)
4. Bogoliubov, N.N., Shirkov, D.V.: Introduction to the Theory of Quantized Fields. 3rd edn. Wiley (1980)
5. Beauregard L.A.: Propagators in nonrelativistic quantum mechanics. Am. J. Phys. 34, 324-332 (1966)
6. Bettencourt L.M.A., Cintrón-Arias A., Kaiser D.I., Castillo-Chávez C.: The power of a good idea: quantitative modeling of the spread of ideas from epidemiological models. Phisica A 364, 513-536 (2006)
7. Brown L.S., Zhang Y.: Path integral for the motion of a particle in a linear potential. Am. J. Phys. 62(9), 806-808 (1994)
8. Feynman R.P. The Principle of Least Action in Quantum Mechanics. Ph.D. thesis, Princeton University, 1942
9. reprinted In: Brown, L.M. (ed.) Feynman's Thesis-A New Approach to Quantum Theory, pp. 1-69 World Scientific Publishers, Singapore (2005)
10. Feynman, R.P.: Space-time approach to non-relativistic quantum mechanics. Rev. Mod. Phys. 20(2), 367-387 (1948)
11. reprinted In: Brown, L.M. (ed.) Feynman's Thesis-A New Approach to Quantum Theory, pp. 71-112. World Scientific Publishers, Singapore (2005)
12. Feynman R.P.: The theory of positrons. Phys. Rev. 76(6), 749-759 (1949)
13. Feynman R.P.: Space-time approach to quantum electrodynamics. Phys. Rev. 76(6), 769-789 (1949)
14. Feynman R.P., Hibbs A.R.: Quantum Mechanics and Path Integrals. McGraw-Hill, New York (1965)
15. Flügge S.: Practical Quantum Mechanics. Springer, Berlin (1999)
16. Gottfried K., Yan T.-M.: Quantum Mechanics: Fundamentals, 2nd edn. Springer, Berlin (2003)
17. Haaheim D.R., Stein F.M.: Methods of solution of the Riccati differential equation. Math. Magazine 42(2), 233-240 (1969)
18. Holstein B.R.: The linear potential propagator. Am. J. Phys. 65(5), 414-418 (1997)
19. Holstein B.R.: The harmonic oscillator propagator. Am. J. Phys. 67(7), 583-589 (1998)
20. Howland J.: Scattering theory for Hamiltonians periodic in time. Indiana Univ. Math. J. 28(3), 471-494 (1979)
21. Jackson J.D.: Classical Electrodynamics. 2nd edn. John Wiley, New York (1975)
22. Jafaev, D.R.: Wave operators for the Schrödinger equation. Teoret. Mat. Fiz. 45(2): 224-234 (1980, in Russian)
23. Kenig C.E., Ponce G., Vega L.: On the ill-posedness of some canonical dispersive equations. Duke Math. J. 106(3), 617-633 (2001)
24. Landau L.D., Lifshitz E.M.: Quantum Mechanics: Nonrelativistic Theory. Pergamon Press, Oxford (1977)
25. Lopez, R.M., Suslov, S.K.: The Cauchy problem for a forced harmonic oscillator. arXiv:0707.1902v8 [math-ph], 27 December 2007
26. Maslov V.P., Fedoriuk M.V.: Semiclassical Approximation in Quantum Mechanics. Reidel, Dordrecht (1981)
27. Meiler, M., Cordero-Soto, R., Suslov, S.K.: Solution of the Cauchy problem for a time-dependent Schrödinger equation. arXiv: 0711.0559v4 [math-ph], 5 December 2007
28. Merzbacher E.: Quantum Mechanics, 3rd edn. Wiley, New York (1998)
29. Messia A.: Quantum Mechanics. Dover Publications, New York (1999)
30. Naibo V., Stefanov A.: On some Schrödinger and wave equations with time dependent potentials. Math. Ann. 334(2), 325-338 (2006)
31. Nardone P.: Heisenberg picture in quantum mechanics and linear evolutionary systems. Am. J. Phys. 61(3), 232-237 (1993)
32. Nikiforov A.F., Suslov S.K., Uvarov V.B.: Classical Orthogonal Polynomials of a Discrete Variable. Springer, Berlin (1991)
33. Nikiforov A.F., Uvarov V.B.: Special Functions of Mathematical Physics. Birkhäuser, Basel, Boston (1988)
34. Paliouras J.D., Meadows D.S.: Complex Variables for Scientists and Engineers, 2nd edn. Macmillan Publishing Company, New York (1990)
35. Rainville E.D.: Special Functions. The Macmillan Company, New York (1960)
36. Rainville E.D.: Intermediate Differential Equations. Wiley, New York (1964)
37. Pérez-Garcia V.M., Torres P., Montesinos G.D.: The method of moments for nonlinear Schrödinger equations: theory and applications. SIAM J. Appl. Math. 67(4), 990-1015 (2007)
38. Robinett R.W.: Quantum mechanical time-development operator for the uniformly accelerated particle. Am. J. Phys. 64(6), 803-808 (1996)
39. Rodnianski I., Schlag W.: Time decay for solutions of Schrödinger equations with rough and time-dependent potential. Invent. Math. 155(3), 451-513 (2004)
40. Rozanova O.S.: Hydrodynamic approach to constructing solutions of nonlinear Schrödinger equations in the critical case. Proc. Am. Math. Soc. 133, 2347-2358 (2005)
41. Schlag, W.: Dispersive estimates for Schrödinger operators: a survay. arXiv: math/0501037v3 [math.AP], 10 January 2005
42. Staffilani G., Tataru D.: Strichartz estimates for a Schrödinger operator with nonsmooth coefficients. Commun. Partial Differ. Eqn. 27(5,6), 1337-1372 (2002)
43. Stein E.M.: Harmonic Analysis: Real-Variable Methods, Orthogonality, and Oscillatory Integrals. Princeton University Press, Princeton (1993)
44. Styer, D.: Additions and Corrections to Feynman and Hibbs. See an updated version on the author's web site: http://www.oberlin.edu/physics/dstyer/ FeynmanHibbs/errors.pdf
45. Suslov, S.K., Trey, B.: The Hahn polynomials in the nonrelativistic and relativistic Coulomb problems. J. Math. Phys. 49(1), published on line 22 January 2008. http://link.aip.org/link/?JMP/49/012104 (2008)
46. Thomber N.S., Taylor E.F.: Propagator for the simple harmonic oscillator. Am. J. Phys. 66(11), 1022-1024 (1998)
47. Watson G.N.: A Treatise on the Theory of Bessel Functions, 2nd edn. Cambridge University Press, Cambridge (1944)
48. Yajima K.: Scattering theory for Schrödinger equations with potentials periodic in time. J. Math. Soc. Japan 29(4), 729-743 (1977)