[Paper Review] Integrable wave function, describing space-time evolution of alpha-decay

A. Ya. Dzyublik|arXiv (Cornell University)|Jan 26, 2020

Radioactive Decay and Measurement Techniques7 references4 citations

TL;DR

This paper proposes an integrable wave function for alpha-decay by modeling the alpha-particle as a time-evolving wave packet, avoiding the non-normalizable solutions of standard Gamov theory. It derives a decay rate consistent with the quasiclassical Gamov formula only far from resonance conditions, showing that the Bohr-Sommerfeld quantization rule does not apply to quasibound levels in alpha-decay, and validates the use of Moshinsky's approach for wave front distortions.

ABSTRACT

In the framework of decay theory of Goldberger and Watson we treat $α$-decay of nuclei as a transition caused by a residual interaction between the initial unperturbed bound state and the scattering states with alpha-particle. The integrable wave function for the $α$-decay is derived. The alpha-particle is described by the wave packet, having small amplitude inside the nucleus and exponentially growing in external region up to the $α$-wave front. The Moshinsky's distortions of the alpha-wave front are analyzed. It is found that the energy of the decaying level does not satisfy commonly accepted Bohr-Sommerfeld quantization rule for the quasibound levels. Only far from this condition the decay rate turns out to be determined by the Gamov's factor for the barrier penetrability. The derived general expression for the decay rate is approximated by the familiar quasiclassical formula.

Motivation & Objective

• To resolve the inconsistency of non-normalizable Gamov wave functions in alpha-decay by constructing a physically consistent, square-integrable time-dependent wave function.
• To challenge the conventional use of the Bohr-Sommerfeld quantization rule for quasibound levels in alpha-decay, showing it does not apply to the energy of the decaying state.
• To derive a decay rate expression based on residual interaction between bound and scattering states, consistent with experimental decay behavior.
• To analyze wave front distortions using Moshinsky’s formalism and show they are negligible in typical experimental settings.

Proposed method

• Formulates the alpha-decay process as a transition from an initial bound state φₐ to continuum scattering states φ₊ᵦ via a residual interaction V′, following Goldberger-Watson decay theory.
• Constructs a time-dependent wave packet Φ(r,t) as a superposition of partial waves wₗ(κ,R), ensuring normalization and physical consistency.
• Imposes a sharp wave front at r_f = v_d t to truncate the exponentially growing wave function, preserving unit norm.
• Applies Moshinsky’s method to model wave front distortions, showing they occur over a narrow time interval Δt and are negligible experimentally.
• Derives the decay width Γ from the squared amplitude Cl² of the wave function, linking it to the WKB action Sₗ.
• Uses the rigged Hilbert space formalism implicitly to justify the physical interpretation of non-Hermitian Hamiltonians and complex eigenvalues.

Experimental results

Research questions

• RQ1 Does the energy of the decaying level in alpha-decay satisfy the Bohr-Sommerfeld quantization rule, as commonly assumed?
• RQ2 Can a physically normalizable, integrable wave function be constructed to describe the space-time evolution of alpha-decay without diverging wave functions?
• RQ3 How do wave front distortions, arising from Moshinsky’s approach, affect the decay dynamics and experimental observability?
• RQ4 What is the correct expression for the decay width Γ when the system is near or far from resonance, and how does it relate to the Gamov factor?
• RQ5 Why does the standard Gamov approach fail to describe the decay rate accurately near resonance, and how can this be corrected?

Key findings

• The energy of the decaying level εₐ does not satisfy the Bohr-Sommerfeld quantization rule, contradicting a long-standing assumption in alpha-decay theory.
• The decay width Γ is proportional to the squared amplitude Cl² of the wave function, with Cl² ∼ e⁻²ˢˡ far from resonance and Cl² ∼ e²ˢˡ near resonance.
• Only far from the resonance condition does the decay rate reduce to the familiar Gamov formula λ ∼ e⁻²ˢ, while near resonance the decay becomes effectively instantaneous.
• Wave front distortions calculated via Moshinsky’s method occur over a very narrow time interval Δt and are negligible in typical experimental observations.
• The derived wave packet Φ(r,t) is integrable, normalized to unity, and describes the alpha-particle as emerging from the nucleus with a sharp front at r_f = v_d t.
• The model shows that the standard complex-energy approach leads to non-normalizable states, while the wave packet approach provides a consistent, probabilistically interpretable solution.