Timedependent Solutions to the Dirac Equation

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Time-dependent Solutions to the Dirac Equation

• Khinlay Win
• and
• Athanasios Petridis
• Drake University

2
The Dirac Equation

• Relativistic quantum equation for spin-1/2
  fermions, which are described by a 4-dimentional
  spinor ?.
• In the presence of an external scalar potential,
  V

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The Numerical Algorithm

• The staggered leap-frog algorithm is applied in a
  spatial grid of bin-size ?x (in 1 dimension) and
  with time step ?t
• The spatial derivatives are computed
  symmetrically.
• Reflecting boundary conditions are applied on a
  very large grid (running stops before reflections
  occur).

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Free Electron Propagation

• The initial spinor is (N normalization factor,
  m 1)
• The probability density ??? at t0 is Gaussian
  (s.d.s0).
• As s0?8,? becomes a positive energy plane wave,
  which for p0 is a spin 1/2 eigenstate.
• ?(x,t) is shown in Figure 1 for s01. The method
  is stable. The accuracy is of order 10-10 per bin
  (?x0.01, ?t0.001).

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Time unit 1/m 1.0
Figure 1.
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Position Expectation Value

• Figure 2 ltxgt vs time after subtraction of the
  drift velocity (red s0 0.5, p 0.01 green
  s0 0.5, p1.37 blue s01, p0.01, purple s0
  1.5, p1.0).
• High-frequency (2E) oscillations are observed
  (Zitterbewengung).
• The effect has a non-linear dependence on s0 and
  is maximized when 2 s0 ?c (Compton wavelength)
  for given p. It increases with p.

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Figure 2.
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Standard Deviation

• In Figure 3 the standard deviation of the
  probability density, s, is shown vs time (red s0
  0.5, p 0 green s0 0.5, p1.0 blue s0
  1.0, p 0 purple s0 1.0, p 1.0 light
  blue s0 1.5, p 0 yellow s0 1.5, p
  1.0).
• High-frequency oscillations are observed. They
  are more pronounced as p increases and die out
  with time.
• The wave-packet spreads faster at lower p.

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Figure 3.
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Spin, z-component

• In Figure 4, the expectation value of the
  z-component of the spin (perpendicular to the
  propagation direction) is shown (red s0 0.5, p
  0.01 green s0 0.5, p 1.37 blue s0
  1.0, p 0.01 purple s0 4.0, p 0.0).
• The observed high-frequency oscillations die out
  with time and are maximized at 2 s0 ?c.
• These results are in agreement with J. W. Braun,
  Q. Su, and R. Grobe, Phys. Rev. A59, 604 (1999).

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Figure 4.
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Decay and Survival Probability

• A decaying fermionic system can be described as a
  Dirac spinor initially set inside a potential
  well that tunnels through the potential walls.
• In a given reference frame, the survival
  probability of the system is defined as

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Finite Square Well Potential

• The strength, V, is varied and the width, 2a, is
  set equal to 2 s0 with s0 ?c 1.0.
• Pin vs time is shown in Figure 5a (p 0) for V
  0.1 (red), 1.0 (green), 1.5 (blue), and 2.0
  (purple) and Figure 5b for V 0.1, p 0.01
  (red), V 2.2, p 0.01 (green), V 0.9, p
  0.1 (blue) and V 2.2, p 0.1 (purple).
• Pin decays non-exponentially performing
  oscillations. This has also been observed in
  non-relativistic decays. It is due to reflections
  on the walls.
• The relativistic case, however, includes a sudden
  increase in Pin for V gt 1 due to
  particle-antiparticle creation (Klein-paradox)
  that slows the decay down. The effect of p is
  small.

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Figure 5a.
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Figure 5b.
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Finite Quadratic Potential

• Harmonic oscillator potential a4x2/2, cut at x
  s0 1.0.
• In Figures 6a and b Pin is shown vs time for a
  0.5, 2.5, and 3 (red, green, blue in 6a) and a
  3.2, 3.6, and 4 (red, green, blue in 6b).
• Oscillations (often irregular ones) are observed
  as before.
• As a increases the decay initially becomes faster
  due to pair production at the edges of the well
  off-resonance decay.
• As a becomes large the initial spinor approaches
  a resonance state of the potential (near the
  ground state of the harmonic oscillator). After
  some initial oscillations Pin is nearly
  exponential in time (for medium t)
  near-resonance decay.

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Figure 6a.
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Figure 6b.
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Conclusions

• The staggered leap-frog algorithm is stable and
  accurate, providing solutions to the
  time-dependent Dirac equation.
• Results are obtained in a few minutes on an
  average personal computer.
• Zitterbewengung is observed in the expectation
  value and standard deviation of the position and
  the expectation value of the spin, depending on
  the initial width of the spinor relative to the
  Compton wavelength and the central momentum.
• Spinors initially set in a potential well decay
  in an non-exponential manner exhibiting
  oscillations. Nearly exponential decay is
  observed near-resonances.

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Perspectives

• Three-dimensional problems can be addressed using
  this method.
• Electromagnetic 4-potential interactions can be
  inserted (external and/or internal).
• Relativistic decays of mesons to leptons can be
  studied in free space and in medium (the
  oscillations change the effective interactions
  with the medium).
• Relativistic spin-wave propagation can be
  explored.