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Wave packet propagation across barriers by semiclassical initial value methods.

Jakob Petersen1, Kenneth G Kay2

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Summary
This summary is machine-generated.

New semiclassical initial value representation (IVR) formulas accurately describe barrier tunneling for wave packets. This method improves upon traditional Van Vleck-Gutzwiller (VVG) and Generalized Gaussian Wave Packet Dynamics (GGWPD) treatments, especially near the classical transmission threshold.

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Area of Science:

  • Quantum mechanics
  • Physical chemistry
  • Computational physics

Background:

  • Semiclassical initial value representation (IVR) formulas struggle to accurately model quantum tunneling through potential barriers.
  • Existing methods like Van Vleck-Gutzwiller (VVG) require complex initial conditions and time-dependent paths for tunneling, which can be computationally intensive and prone to inaccuracies.
  • The classical limit of these semiclassical formulas does not inherently capture the necessary complex trajectories for barrier tunneling.

Purpose of the Study:

  • To derive a novel, simple initial value representation (IVR) expression for the propagator that correctly describes one-dimensional barrier transmission in the classical limit.
  • To develop an IVR formula for the transmitted wave packet that reduces to the Generalized Gaussian Wave Packet Dynamics (GGWPD) expression in the classical limit.
  • To present uniform semiclassical versions of these IVR formulas and simplify them using real trajectories and WKB penetration factors.

Main Methods:

  • Derivation of a new IVR expression for the propagator applicable to one-dimensional barrier transmission.
  • Development of an IVR formula for the transmitted wave packet, validated against the GGWPD method.
  • Application of uniform semiclassical approximations and simplification using real trajectories and WKB penetration factors.
  • Numerical simulations of wave packet transmission through Eckart and Gaussian barriers.

Main Results:

  • The derived IVR formulas exhibit the correct tunneling behavior in the classical limit.
  • Uniform semiclassical IVR treatments provide accurate results for wave packet transmission through Eckart and Gaussian barriers.
  • The new IVR method outperforms VVG and GGWPD treatments, particularly when the wave packet's mean energy is near the classical transmission threshold.
  • Contour integral formulations of the IVR expressions are presented and analyzed using steepest descent, confirming their relationship to GGWPD and semiclassical validity.

Conclusions:

  • The developed IVR approach offers a significant improvement for describing quantum tunneling phenomena in one-dimensional systems.
  • This method provides accurate and reliable predictions for wave packet transmission, overcoming limitations of previous semiclassical techniques.
  • The generalized contour integral formulation opens avenues for applying this method to a broader range of complex quantum systems.