Exact Analytical and Numerical Solutions to the Time-Dependent Schrödinger Equation for a One-Dimensional Potential Exhibiting Non-Exponential Decay at All Times
ABSTRACT
The departure at large times from exponential decay in the case of resonance wavefunctions is mathematically demonstrated. Then, exact, analytical solutions to the time-dependent Schrödinger equation in one dimension are developed for a time-independent potential consisting of an infinite wall and a repulsive delta function. The exact solutions are obtained by means of a superposition of time-independent solutions spanning the given Hilbert space with appropriately chosen spectral functions for which the resulting integrals can be evaluated exactly. Square-integrability and the boundary conditions are satisfied. The simplest of the obtained solutions is presented and the probability for the particle to be found inside the potential well as a function of time is calculated. The system exhibits non-exponential decay for all times; the probability decreases at large times as . Other exact solutions found exhibit power law behavior at large times. The results are generalized to all normalizable solutions to this problem. Additionally, numerical solutions are obtained using the staggered leap-frog algorithm for select potentials exhibiting the prevalence of non-exponential decay at short times.
The departure at large times from exponential decay in the case of resonance wavefunctions is mathematically demonstrated. Then, exact, analytical solutions to the time-dependent Schrödinger equation in one dimension are developed for a time-independent potential consisting of an infinite wall and a repulsive delta function. The exact solutions are obtained by means of a superposition of time-independent solutions spanning the given Hilbert space with appropriately chosen spectral functions for which the resulting integrals can be evaluated exactly. Square-integrability and the boundary conditions are satisfied. The simplest of the obtained solutions is presented and the probability for the particle to be found inside the potential well as a function of time is calculated. The system exhibits non-exponential decay for all times; the probability decreases at large times as . Other exact solutions found exhibit power law behavior at large times. The results are generalized to all normalizable solutions to this problem. Additionally, numerical solutions are obtained using the staggered leap-frog algorithm for select potentials exhibiting the prevalence of non-exponential decay at short times.
Cite this paper
nullA. Petridis, L. Staunton, J. Vermedahl and M. Luban, "Exact Analytical and Numerical Solutions to the Time-Dependent Schrödinger Equation for a One-Dimensional Potential Exhibiting Non-Exponential Decay at All Times," Journal of Modern Physics, Vol. 1 No. 2, 2010, pp. 124-136. doi: 10.4236/jmp.2010.12018.
nullA. Petridis, L. Staunton, J. Vermedahl and M. Luban, "Exact Analytical and Numerical Solutions to the Time-Dependent Schrödinger Equation for a One-Dimensional Potential Exhibiting Non-Exponential Decay at All Times," Journal of Modern Physics, Vol. 1 No. 2, 2010, pp. 124-136. doi: 10.4236/jmp.2010.12018.
References
[1] S. A. Khalfin, “Contribution to the Decay Theory of a Quasi-Stationary State,” Soviet Journal of Experimental and Theoretical Physics, Vol. 6, 1958, pp. 1053-1063. [2] R. Winter, “Evolution of a Quasi-Stationary State,” Phy- sical Review, Vol. 123, No. 4, 1961, pp. 1503-1507. [3] V. Weisskopf and E. Wigner, “Berechnung der natür- lichen Linienbreite auf Grund der Diracschen Lichtt- heorie,” Zeitschrift für Physik, Vol. 63, No. 1-2, 1930, pp. 54-73. [4] J. J. Sakurai, “Modern Quantum Mechanics,” The Benja- min-Cummings Publishing Company, 1985. [5] C. Dullemond, “Fermi’s ‘Golden Rule’ and Non-Expo- nential Decay,” arXiv:quant-ph/0202105, 2003. [6] V. P. Oleinik and J. D. Arepjev, “On the Tunneling of Electrons out of the Potential Well in an Electric Field,” Journal of Physics A, Vol. 17, No. 9, 1984, pp. 1817-1827. [7] M. I. Shirokov, “Exponential Character of Decay Laws,” Soviet Journal of Nuclear Physics, Vol. 21, 1975, pp. 347-353. [8] V. V. Flambaum and F. M. Izrailev, Unconventional decay law for excited states in closed many-body systems. Physical Review E, Vol. 64, No. 2, 2001, pp. 026124- 026130. [9] P. G. Silvestrov, “Stretched Exponential Decay of a Quasiparticle in a Quantum Dot,” Physical Review B, Vol. 64, No. 11, 2001, pp. 113309-113313. [10] L. Accardi, S. V. Kozyrev and I. V. Volovich, “Non- Exponential Decay for Polaron Model,” Physics Letters A, Vol. 260, No. 1-2, 1999, pp. 31-38. [11] G. Wilk and Z. Wlodarczyk, “Nonexponential Decays and Nonextensivity,” Physics Letters A, Vol. 290, No. 1-2, 2001, pp. 55-58. [12] A. N. Petridis, L. P. Staunton, M. Luban and J. Vermedahl, Talk Given at the Fall Meeting of the Division of Nuclear Physics of the American Physical Society, Tucson, Arizona, Unpublished, 2003. [13] S. R.Wilkinson, et al., “Experimental Evidence for Non- Exponential Decay in Quantum Tunnelling,” Nature, Vol. 387, 1997, pp. 575-577. [14] N. G. Kelkar, M. Nowakowski and K. P. Khemchandani, “Hidden Evidence of Nonexponential Nuclear Decay,” Physical Review C, Vol. 70, No. 2, 2004, pp. 24601- 24605. [15] C. Rothe, S. I Hintschich and A. P. Monkman, “Violation of the Exponential-Decay Law at Long Times,” Physical Review Letters, Vol. 96, No. 16, 2006, pp. 163601- 163604. [16] B. L. Burrows and M. Cohen, “Exact Time-Dependent Solutions for a Double-Well Model,” Journal of Physics A, Vol. 36, No. 46, 2003, pp. 11643-11653. [17] R. M. Cavalcanti, P. Giacconi and R. Soldati, “Decay in a Uniform Field: An Exactly Solvable Model,” Journal of Physics A, Vol. 36, No. 48, 2003, pp. 12065-12080. [18] A. Messiah, “Quantum Mechanics,” Dover Publishers, Mineola, 1999. [19] The expressions for this wavefunction and the probability density are very long and complicated. They are available from the authors upon request.
[1] S. A. Khalfin, “Contribution to the Decay Theory of a Quasi-Stationary State,” Soviet Journal of Experimental and Theoretical Physics, Vol. 6, 1958, pp. 1053-1063. [2] R. Winter, “Evolution of a Quasi-Stationary State,” Phy- sical Review, Vol. 123, No. 4, 1961, pp. 1503-1507. [3] V. Weisskopf and E. Wigner, “Berechnung der natür- lichen Linienbreite auf Grund der Diracschen Lichtt- heorie,” Zeitschrift für Physik, Vol. 63, No. 1-2, 1930, pp. 54-73. [4] J. J. Sakurai, “Modern Quantum Mechanics,” The Benja- min-Cummings Publishing Company, 1985. [5] C. Dullemond, “Fermi’s ‘Golden Rule’ and Non-Expo- nential Decay,” arXiv:quant-ph/0202105, 2003. [6] V. P. Oleinik and J. D. Arepjev, “On the Tunneling of Electrons out of the Potential Well in an Electric Field,” Journal of Physics A, Vol. 17, No. 9, 1984, pp. 1817-1827. [7] M. I. Shirokov, “Exponential Character of Decay Laws,” Soviet Journal of Nuclear Physics, Vol. 21, 1975, pp. 347-353. [8] V. V. Flambaum and F. M. Izrailev, Unconventional decay law for excited states in closed many-body systems. Physical Review E, Vol. 64, No. 2, 2001, pp. 026124- 026130. [9] P. G. Silvestrov, “Stretched Exponential Decay of a Quasiparticle in a Quantum Dot,” Physical Review B, Vol. 64, No. 11, 2001, pp. 113309-113313. [10] L. Accardi, S. V. Kozyrev and I. V. Volovich, “Non- Exponential Decay for Polaron Model,” Physics Letters A, Vol. 260, No. 1-2, 1999, pp. 31-38. [11] G. Wilk and Z. Wlodarczyk, “Nonexponential Decays and Nonextensivity,” Physics Letters A, Vol. 290, No. 1-2, 2001, pp. 55-58. [12] A. N. Petridis, L. P. Staunton, M. Luban and J. Vermedahl, Talk Given at the Fall Meeting of the Division of Nuclear Physics of the American Physical Society, Tucson, Arizona, Unpublished, 2003. [13] S. R.Wilkinson, et al., “Experimental Evidence for Non- Exponential Decay in Quantum Tunnelling,” Nature, Vol. 387, 1997, pp. 575-577. [14] N. G. Kelkar, M. Nowakowski and K. P. Khemchandani, “Hidden Evidence of Nonexponential Nuclear Decay,” Physical Review C, Vol. 70, No. 2, 2004, pp. 24601- 24605. [15] C. Rothe, S. I Hintschich and A. P. Monkman, “Violation of the Exponential-Decay Law at Long Times,” Physical Review Letters, Vol. 96, No. 16, 2006, pp. 163601- 163604. [16] B. L. Burrows and M. Cohen, “Exact Time-Dependent Solutions for a Double-Well Model,” Journal of Physics A, Vol. 36, No. 46, 2003, pp. 11643-11653. [17] R. M. Cavalcanti, P. Giacconi and R. Soldati, “Decay in a Uniform Field: An Exactly Solvable Model,” Journal of Physics A, Vol. 36, No. 48, 2003, pp. 12065-12080. [18] A. Messiah, “Quantum Mechanics,” Dover Publishers, Mineola, 1999. [19] The expressions for this wavefunction and the probability density are very long and complicated. They are available from the authors upon request.