Nonexponential Decay of Wavefunctions and Scattering Resonances

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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.)
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Outline

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

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

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• 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
  EE(p).
• With proper normalization
• where ? is the decay constant and P0 is the
• interior probability at t0.

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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

t2?t
Time
2
1
0
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• 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).

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• 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.

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T/4
T/2
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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

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HO (cut at 100), ?(x,0)gaussian (width60, v00)
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HO (cut at 100), ?(x,0)gaussian (width60, v00)
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HO (cut at 100), ?(x,0)gaussian (width60, v00)
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HO (cut at 100), ?(x,0)gaussian (width60, v00)
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HO (cut at 100), ?(x,0)gaussian (width60, v00)
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HO (cut at 100), ?(x,0)ground (v01)
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HO (cut at 100), ?(x,0)ground (v01)
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HO (cut at 100), ?(x,0)ground (v01)
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LC (cut at 100), ?(x,0)gaussian (width60, v00)
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LC (cut at 100), ?(x,0)gaussian (width60, v00)
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LC (cut at 100), ?(x,0)gaussian (width60, v00)
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• 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.

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• 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).

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• Important observation at large times the
  probability DOES decay exponentially.

LC (cut at 25) ?(x,0)gaussian (width60, v00)
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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.

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• We fit Pin(t) with the function

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LC (cut at 100), ?(x,0)gaussian (width60, v00)
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.
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• 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.

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• The system is auto-correlated. Define

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Scattering Resonances

• One application Nuclear decay.
• Resonant scattering (production-decay of ?
  baryons in pion-proton scattering).
• In the CM frame Em (hc1). For large t

(Breit-Wigner curve)
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LC (cut at 100), ?(x,0)gaussian (width60, v00)
standard
modified
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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.