Revivals and quantum carpets for the relativistic Schrödinger equation

TL;DR

This work addresses wavepacket revivals for a relativistic spinless particle in a 1D infinite well governed by the Salpeter equation $i \hbar \partial_t \psi = \sqrt{m^2 c^4 + c^2 \hat p^2}\psi + V(x)\psi$, showing that well-defined revivals persist despite relativistic nonlocality. The authors solve in momentum space to obtain eigenfunctions that match the non-relativistic case inside the well, with a relativistically corrected energy spectrum $E_n=\sqrt{m^2 c^4+(\hbar c k_n)^2}$ and $k_n=n\pi/L$, enabling explicit wavepacket construction and the analysis of revival times; they derive $T_{cl}^R=\dfrac{2L}{v_{n_0}}$ and $T_{rev}^R=(2n)\,\dfrac{2L}{v_{n_0}}\gamma^2= T_{rev}^{NR}\gamma$, illustrating how revivals fade as $v_{n_0}\to c$. By computing quantum carpets across non-relativistic, intermediate, and ultra-relativistic regimes, the paper reveals Schrödinger-like revivals, purely classical bouncing, and an intricate interplay in between, along with numerical methods and energy-spacing analyses that validate the results. The findings clarify how relativistic corrections modify coherence phenomena in bounded quantum systems and provide a robust benchmark for numerical approaches in relativistic quantum dynamics. Overall, the Salpeter framework emerges as a consistent and insightful model for relativistic wavepacket evolution in simple geometries, bridging quantum and classical behavior.

Abstract

We investigate wavepacket dynamics for a relativistic particle in a box evolving according to the relativistic Schrödinger (also known as the Salpeter) equation. We derive the solutions for an infinite well -- which contrary to the standard relativistic wave equations (such as the Klein-Gordon or Dirac equations) -- are well defined, and use these solutions to construct wavepackets. We obtain expressions for the wavepacket revival times and explore the corresponding quantum carpets (the space-time probability density plots) for different dynamical regimes. We further analyze level spacing statistics as the dynamics goes from the non-relativistic regime to the ultra-relativistic limit.

Figures (5)

• Figure 1: Variation of the different revival times for the Salpeter particle in a box for $L=800 \, \lambda_C$. We are able to identify two regimes, the low energy regime ($n$ between 1 and about 50) where the revivals times corresponds to the non-relativistic particle in a box and the high energy regime (around $n$ greater than 500) where the velocity tends to $c$.
• Figure 2: Quantum carpet for the Salpeter equation in a well. All initial Gaussian wave-packet start with $\Delta x= L/20$ and no initial velocity. Yet, depending on the initial position, we observe new revivals. Figure (a) starts with a wave packet having no clear symmetry and thus we simply observe regular revivals. On the other hand, figure (b) and (c) starts with Gaussian centered at $2L/3$ and $L/2$ respectively thus exhibiting new revivals at fractions of the usual revival time.
• Figure 3: Propagation of a wave-packet in the relativistic regime. The propagation is affected by the light cone as the wavepacket cannot spread beyond it. As a consequence the interference effects are minimal here, the wave packet bounces in a classical-like manner on the walls of the well. The typical revival time is simply given by the classical time $T_{cl}=2L/v$ which is here close to $2L/c$.
• Figure 4: Quantum carpet in the intermediate regime with a Gaussian wave packet starting in the middle of the box with an initial velocity of $p_0=1.2 \hbar/L$ and width of $L/25$. We observe in (a) the appearance of interference patterns forming the usual ridges and canals but blurred out. Figure (b) shows a zoom of the quantum carpet around $t=T_{rev}/2$. On the zoom, we see that the interference pattern is actually the effect of many rebounds corresponding to the particle bouncing inside the well.
• Figure 5: Population distribution $|a_n|^2$ for an initial Gaussian wave packet with $p_0=0$. (a) $x_0=2L/3$, (b) $x_0=L/2$. In both cases, the coefficient distribution follows a Gaussian curve centered at $n=0$. Due to the symmetry of the initial wave packet in the well, a third (a) or a half (b) of the coefficients vanish. The coefficients are computed using the diagonalization method.