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<p>Only a few issues, such as the hydrogen atom, the harmonic oscillator, and square wells, provide an exact solution to the Schrödinger equation. Approximation methods can be used when exact solutions to the Schrödinger equation cannot be obtained. We develop a semiclassical Wentzel–Kramers–Brillouin (WKB) approximation for the quaternionic time-independent Schrödinger equation with purely imaginary quaternionic potentials. Using a generalized ansatz, the wavefunction separates into a real phase and a reduced quaternionic amplitude. The resulting equations yield a transport law that ensures the conservation of the probability current and a dispersion relation for the phase. Two formal solutions arise, but only one reduces correctly to the complex WKB limit, resolving the apparent multivaluedness of the quaternionic wavefunction. The formalism predicts corrections to semiclassical quantities at order <InlineEquation ID="IEq1"> <EquationSource Format="TEX">\(V_2^2 + V_3^2\)</EquationSource> </InlineEquation>, leading to modified momenta, tunneling probabilities, and quantization conditions. In interferometric setups, these corrections manifest themselves as phase shifts and visibility reductions. The consistency with complex quantum mechanics, together with identifiable predictions, establishes the quaternionic WKB approximation as a foundation for future theoretical and experimental studies.</p>

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Quaternionic WKB Approximation

  • 期刊论文

  • 发表日期：

    2025年11月11日

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  • Bhashkar Singh,
  • A. S. Rawat,
  • Seema Rawat

摘要

Only a few issues, such as the hydrogen atom, the harmonic oscillator, and square wells, provide an exact solution to the Schrödinger equation. Approximation methods can be used when exact solutions to the Schrödinger equation cannot be obtained. We develop a semiclassical Wentzel–Kramers–Brillouin (WKB) approximation for the quaternionic time-independent Schrödinger equation with purely imaginary quaternionic potentials. Using a generalized ansatz, the wavefunction separates into a real phase and a reduced quaternionic amplitude. The resulting equations yield a transport law that ensures the conservation of the probability current and a dispersion relation for the phase. Two formal solutions arise, but only one reduces correctly to the complex WKB limit, resolving the apparent multivaluedness of the quaternionic wavefunction. The formalism predicts corrections to semiclassical quantities at order \(V_2^2 + V_3^2\) , leading to modified momenta, tunneling probabilities, and quantization conditions. In interferometric setups, these corrections manifest themselves as phase shifts and visibility reductions. The consistency with complex quantum mechanics, together with identifiable predictions, establishes the quaternionic WKB approximation as a foundation for future theoretical and experimental studies.