Non-exponential Decay of Wavefunctions and Scattering Resonances

1. Non-exponential Decayof Wavefunctions and Scattering Resonances Athanasios Petridis Drake University COLLABORATORS: L. Staunton (Drake Univ.) M. Luban (Iowa State Univ.) J. Vermedahl (Drake Univ.) ACKNOWLEDGEMENTS: K. Bartschat (Drake Univ.)

2. Outline • Exponential decay • Numerical method • Case studies • Interpretation • Scattering resonances • Conclusions and outlook

3. V x x L L Exponential decay • A wavefunction initially inside a finite potential well will disperse through the walls if it is not an eigenfunction for this potential: it will decay. • Examples: escape of a particle from a well or decay of a composite object bound by V. V 0 0 Cut HO Cut LC

4. The probability for finding the particle inside the potential well is • This can also be expressed as

5. At times much greater than the longest normal-mode oscillation period in the potential it is often assumed that where  is the decay width of the system and E=E(p). • With proper normalization where  is the decay constant and P0 is the interior probability at t=0.

6. Numerical Method • We solve the time-dependent Shrödinger equation using the staggered leap-frog method. • We define a grid of n0 points with spacing x, and update the wavefunction at time intervals t. • We evaluate the function at using the stored values at t for every grid point n: t+2t Time 2 1 0

7. This method is VERY stable numerically but requires small time steps. • Normalization accuracy of 10-9 is achieved (10-12 per grid point). • Reflecting boundary conditions are used. This requires very large grid to avoid interference of the waves reflected on the wall with the wave inside the potential well. • The initial wavefunction can be “still” or have a group velocity (v0).

8. Testing the numerical method: • Verified that the square of the width of a minimum packet increases quadratically with time. • Verified that in a harmonic oscillator potential the magnitude squared of any linear combination of eigenfunctions with no initial group velocity will retrieve itself after exactly one classical period.

9. T/4 T/2

10. Case Studies • Cut harmonic oscillator potential (Cut HO). • Cut linear confining potential (Cut LC). • Initial wave functions: (1) general gaussian (2) HO ground state. • Initial group velocities: (1) zero (2) 1 unit. • Snapshots of |(x,t)|2 , Pin(t), dPin(t)/dt

11. HO (cut at 100), (x,0)=gaussian (width=60, v0=0)

12. HO (cut at 100), (x,0)=gaussian (width=60, v0=0)

13. HO (cut at 100), (x,0)=gaussian (width=60, v0=0)

14. HO (cut at 100), (x,0)=gaussian (width=60, v0=0)

15. HO (cut at 100), (x,0)=gaussian (width=60, v0=0)

16. HO (cut at 100), (x,0)=ground (v0=1)

17. HO (cut at 100), (x,0)=ground (v0=1)

18. HO (cut at 100), (x,0)=ground (v0=1)

19. LC (cut at 100), (x,0)=gaussian (width=60, v0=0)

20. LC (cut at 100), (x,0)=gaussian (width=60, v0=0)

21. LC (cut at 100), (x,0)=gaussian (width=60, v0=0)

22. General features of the results: • The wavefunction “breaths” inside the potential “exhaling” wavepackets that travel away in both directions. This is due to reflection and transmission of wave-components off the through the walls. • The probability, Pin, exhibits “plateaux” which appear to be periodic. They correspond to the “inhaling” mode of the function. • Following the “plateaux”-like behavior, the derivative of Pin fluctuates as well. • There is a transition time for the fluctuations to settle into a steady frequency.

23. Can this behavior be attributed to the sharpness of the potential (infinite classical force at the sharp edges)? • J. Vermedahl has rounded the corners of the cut HO potential and added a smooth drop to zero: the results are qualitatively the same! • However: the slope of the curve and the period and size of the fluctuations depend on the shape and magnitude of the potential and the Initial Conditions ((x,0), v0).

24. LC (cut at 25) (x,0)=gaussian (width=60, v0=0) • Important observation: at large times the probability DOES decay exponentially.

25. Interpretation • Generally, there is no complete analytical calculation. We are in the process of cut HO analytical calculations. • The probability function versus time appears to consist of a “median” curve with oscillations of fixed period around it. • The median deviates from a simple exponential at “short” times. • The amplitude of the oscillations appears to decrease with time. • There may be some initial “transition” interval.

26. We fit Pin(t) with the function

27. LC (cut at 100), (x,0)=gaussian (width=60, v0=0) K = 0.1089 N1 = 1.0015 N2 = 0.9892 W = 0.1911 C = 0.0084 L = 0.0046 Q = 0.7882 S(t) starts slightly negative and becomes positive with time.

28. There are FIVE time scales: 1. The overall (long time) decay constant. 2. The time it takes for the median curve to become a simple exponential. 3. The dominant period of oscillations. 4. The time it takes for the oscillations to damp out. 5. The initial oscillation transient time.

29. The system is auto-correlated. Define:

30. Scattering Resonances • One application: Nuclear decay. • Resonant scattering (production-decay of  baryons in pion-proton scattering). • In the CM frame E=m(ħ=c=1). For large t (Breit-Wigner curve)

31. LC (cut at 100), (x,0)=gaussian (width=60, v0=0) standard modified

32. Conclusions and outlook • There are deviations from exponential decay at small times.They can be connected with auto-correlations and “breathing” of the wavefunction. • They may be visible in time and energy domains. • Analytical understanding is needed. An analysis of the wavefunction on eigenfunctions would show oscillatory behavior. • The relativistic Dirac equation can be studied.