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An Analytical Approach to Studying Non-exponential Decay

Outline. Introduction to the ProblemAn Analytical Approach Infinite Wall and Delta-function PotentialEven PotentialsStatistical BehaviorConclusions and Perspectives. Introduction to the Problem. It is generally expected that wavefunctions initially set inside potential wells will decay exponen

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An Analytical Approach to Studying Non-exponential Decay

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    1. An Analytical Approach to Studying Non-exponential Decay

    2. Outline Introduction to the Problem An Analytical Approach Infinite Wall and Delta-function Potential Even Potentials Statistical Behavior Conclusions and Perspectives

    3. Introduction to the Problem It is generally expected that wavefunctions initially set inside potential wells will decay exponentially if they are not eigenfunctions. The Breit-Wigner energy distribution leads to exponential decay of resonances. Numerical studies of the time-dependent Schrödinger equation show that this is not generally true.

    6. An Analytical Approach Solve the time-independent Schrödinger equation subject to boundary conditions to obtain . Use completeness for the given Hilbert space and locality to obtain time-dependent solutions: The spectral function, ?(E), must yield convergent energy (E) integrals and square-integrable wavefunctions for all times.

    7. Infinite Wall and Delta-function Potential

    12. Even Potentials

    13. Sine and Cosine solutions project out parts of the spectral function which are even or odd in the momentum (p). Their coefficients may differ. We consider the solution as a superposition of “incoming” and “outgoing” plane waves; e.g. in region (I):

    14. In the case of overall even symmetry in x we select the “incoming” wave spectral function to have one or more poles with negative imaginary part:

    15. The integral that includes the pole is done with contour integration and using the residue theorem. The negative imaginary part ensures that the wavefunction dies-out at large distances. The other integral is also convergent and yields a square-integrable function as well. The probability density has an oscillatory cross-term.

    18. Statistical Behavior At large times the survival probability decays essentially like an exponential. For many-particle systems the “starting time” for the decay is not the same for all members. Therefore, there is smearing of the observed deviations from exponential decay (work in progress).

    19. Conclusions and Perspectives Non-exponential decay of wavefunctions has been established ANALYTICALLY and NUMERICALLY (presentation by Jon Vermedahl). The details depend on the potential and the spectral function. Extension to many (independent) particle systems is under-way.

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