Numerical Solutions to the TimeDependent, Coupled Dirac Equation

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Numerical Solutions to the Time-Dependent,
Coupled Dirac Equation

• Athanasios Petridis
• and
• Khinlay Win
• Drake University

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The Dirac Equation

• Relativistic quantum equation for spin-1/2
  fermions, which are described by a 4-dimentional
  spinor ?.
• Including 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 next for s01. The method is
  stable. The accuracy is of order 10-10 per bin
  (?x0.01, ?t0.001).

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Position Expectation Value

• 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
  (Zitterbewegung).
• 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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Standard Deviation

• Standard deviation of the probability density, s,
  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 more pronounced
  as p increases and die out with time.
• s increases faster at lower p.

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Spin, z-component

• Expectation value of the z-component of the spin
  (perpendicular to the propagation direction)
  (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 high-frequency oscillations die out with time
  and are maximized at 2 s0 ?c.
• Results agree with J. W. Braun, Q. Su, and R.
  Grobe, Phys. Rev. A59, 604 (1999).

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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 width, 2a, is set equal to 2 s0 with s0 ?c
  1.0.
• Pin vs time (p 0) for V 0.1 (red), 1.0
  (green), 1.5 (blue), and 2.0 (purple) A and 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) B.
• Pin decays non-exponentially performing
  oscillations. This has also been observed in
  non-relativistic decays.
• The relativistic case includes a sudden increase
  in Pin for V gt 1 due to particle-antiparticle
  creation (Klein-paradox). The effect of p is
  small.

A
B
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Finite Quadratic Potential

• Harmonic potential a4x2/2, cut at xs01.
• Pin vs time for a 0.5, 2.5, and 3 (red, green,
  blue in A) and a 3.2, 3.6, and 4 (red, green,
  blue in B).
• Oscillations (often irregular ones) are observed.
• 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 (for medium t) near-resonance decay.

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Fermion-antifermion system

• A fermion (P) and an antifermion (N) coupled
  together may represent a meson.
• The minimal coupling scheme
• Fermions are coupled to retarded potentials
  produced by each other (here in 1 dimension,
  where A,j // x) and an external scalar potential

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• Initially the system is made of a positive energy
  and a negative energy spinor with gaussian
  spatial distribution. As p-gt0 and s0-gt8 they
  become spin up and down eigenstates.
• The probability densities for spinors, of
  initially positive and negative energy, are the
  same and, also, equal to the density for finding
  one OR the other particle at a location

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Technical Issues

• The presence of the retarded integrals decreases
  the stability of the staggered leap frog method.
• This requires smaller time steps which, in turn,
  require larger matrices for the currents,
  approaching or exceeding the compiler (or even
  the computer) capabilities.
• The problem is partially solved with dynamic
  memory allocation (in C).

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System x Expectation Value

• ltxgt vs time (red interacting fermion-antifermion
  , green single fermion)
• No external potential.
• Oscillations are hardly observed in the coupled
  system.

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System Standard Deviation

• Standard deviation vs time. Red p0, coupled,
  Green p0, single. Blue p1, coupled, Purple
  p1, single.
• The oscillations are hardly present in the case
  of coupled spinors.

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System Survival Probability

• Survival probability versus time for square-well
  potential of strength 0.1 and width 2 s0.
• The initial system is centered in the potential
  and has p0. Green coupled, Red single.
• The coupled system does not oscillate much.

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Conclusions

• The staggered leap-frog algorithm can be stable
  and accurate providing solutions to the
  time-dependent Dirac equation.
• Results are obtained in a few minutes on an
  average personal computer.
• Zitterbewegung 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 a single spinor relative to
  the Compton wavelength and the central momentum.
• Single 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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Conclusions (continued)

• Mutually interacting fernion-antifermion systems
  are harder to study because of increased
  numerical instability. A predictor-corrector step
  may help.
• They can model meson systems.
• They exhibit almost no Zitterbewegung but still
  do not decay exponentially.
• Decay studies in a periodic or quasi-periodic
  external scalar potential are in order.
• Internal strong interactions must be included
  next.