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  • 1Faculty of Mathematics (NuHAG), University of Vienna, Oskar-Morgenstern-Platz 1, 1090 Vienna, Austria.

Entropy (Basel, Switzerland)
|December 3, 2020
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The Born-Jordan rule is essential for quantum mechanics equivalence between Heisenberg and Schrödinger pictures. New evidence supports this rule using short-time approximations to quantize classical Hamiltonians.

Keywords:
Born–Jordan quantizationVan Vleck determinantshort-time propagatorstime-slicing

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Area of Science:

  • Quantum Mechanics
  • Theoretical Physics

Background:

  • Equivalence between Heisenberg and Schrödinger pictures in quantum mechanics necessitates specific quantization rules.
  • Previous work identified the Born and Jordan quantization rules as crucial for this equivalence.

Purpose of the Study:

  • To provide further evidence supporting the Born-Jordan rule as the correct quantization scheme for quantum mechanics.
  • To demonstrate the validity of the Born-Jordan rule in quantizing classical Hamiltonians.

Main Methods:

  • Utilizing correct short-time approximations to the action functional, originally developed by Makri and Miller.
  • Applying these approximations to analyze the quantization of the classical Hamiltonian.

Main Results:

  • The application of short-time approximations to the action functional leads to the quantization of the classical Hamiltonian.
  • The results reinforce the Born-Jordan rule's role in establishing quantum mechanical picture equivalence.

Conclusions:

  • The Born-Jordan quantization rule is validated as the correct scheme for quantum mechanics.
  • Short-time approximations offer a viable method for demonstrating the Born-Jordan rule's efficacy in quantizing classical systems.