[Paper Review] Nonperturbative Analysis of a Quantum Mechanical Model for Unstable Particles

U. Aglietti, P. M. Santini|arXiv (Cornell University)|Oct 28, 2010

Cold Atom Physics and Bose-Einstein Condensates1 references5 citations

TL;DR

This paper presents a nonperturbative analysis of a quantum mechanical model for unstable particles confined in a potential well with a delta-function barrier at the edge. Using exact asymptotic expansions in inverse time, it demonstrates that for weak coupling, decay is initially exponential (lifetime ~1/g²), transitioning to a power-law decay ~g⁴/t³ at late times, with similar behavior in both attractive and repulsive potentials—indicating resonance dynamics, not tunnelling, as the decay mechanism.

ABSTRACT

We present a detailed non-perturbative analysis of the time-evolution of a well-known quantum-mechanical system - a particle between potential walls - describing the decay of unstable states. For sufficiently high barriers, corresponding to unstable particles with large lifetimes, we find an exponential decay for intermediate times, turning into an asymptotic power decay. We explicitly compute such power terms in time as a function of the coupling in the model. The same behavior is obtained with a repulsive as well as with an attractive potential, the latter case not being related to any tunnelling effect.

Motivation & Objective

• To provide a nonperturbative analysis of time evolution in a quantum mechanical model of unstable particles, avoiding perturbative approximations.
• To investigate the decay behavior of a particle confined in a potential well with a delta-function barrier, focusing on the transition from exponential to power-law decay.
• To determine whether decay mechanisms are governed by tunnelling or resonance properties, particularly in the case of attractive potentials where tunnelling is absent.
• To compute the asymptotic power-law terms in time explicitly as a function of the coupling constant g.
• To clarify the role of non-diagonal pole contributions in the time evolution, especially for excited states, and their dominance over diagonal terms at intermediate times.

Proposed method

• Formulates a one-dimensional quantum system with a particle confined in a box (0 ≤ x ≤ π) and a δ-function potential at x = π, modeling unstable states via coupling strength g.
• Derives the Hamiltonian in dimensionless form, reducing the problem to a single coupling parameter g = ℏ²/(2mLλ), enabling nonperturbative analysis.
• Analyzes the spectrum by solving the Schrödinger equation with boundary conditions ψ(0) = 0 and discontinuity in derivative at x = π due to δ-function.
• Identifies poles in the complex k-plane corresponding to resonant states; computes their residues and complex energies (E = ω − iΓ/2) via analytic continuation.
• Constructs the time evolution of the wavefunction using a sum over poles in the complex k-plane, with asymptotic expansion in 1/t for large t.
• Applies saddle-point and steepest descent methods to derive the leading asymptotic behavior of the wavefunction, isolating power-law terms ~1/t³ and exponential decays ~e^{-Γt/2}.

Experimental results

Research questions

• RQ1 What is the nonperturbative time evolution of a quantum system modeling an unstable particle with a finite lifetime?
• RQ2 How does the decay law transition from exponential to power-law behavior in time, and what determines the crossover time?
• RQ3 Why does the same decay pattern (exponential followed by power-law) emerge in both attractive and repulsive potential configurations, despite the absence of tunnelling in the latter?
• RQ4 What is the quantitative dependence of the asymptotic power-law decay coefficient on the coupling constant g?
• RQ5 How do non-diagonal pole contributions (n ≠ l) affect the time evolution of excited states, and why are they significant at intermediate times?

Key findings

• For weak coupling (|g| ≪ 1), the system exhibits an exponential decay with lifetime τ ≈ 1/g², valid over a time range t ≲ log(1/g)/g².
• At asymptotically large times, the decay transitions to a power-law behavior ~g⁴/t³ for the probability density |ψ|², independent of the sign of the potential.
• The power-law decay arises from the asymptotic behavior of the wavefunction's Fourier transform, derived via saddle-point analysis in the complex k-plane.
• The same decay pattern is found for both attractive and repulsive potentials, indicating that decay is driven by resonance structure rather than tunnelling.
• Non-diagonal pole contributions (n ≠ l) dominate over diagonal ones at intermediate times for excited states (e.g., l = 2), due to slower decay rates (Γ ∝ n³), even though their coefficients are suppressed by powers of g.
• The first excited state (l = 2) shows a transient region where the first pole contribution (n = 1) dominates over the second pole (n = 2), with the power-law term taking over only at very late times.