Quantum mechanical Gaussian wavepackets of single relativistic particles

The gist: The minimal initial width of a wavepacket for a good Gaussian approximation in position space is about the Compton wavelength of the particle divided by its Lorentz factor at the initial moment, which can be considered as a manifestation of relativistic length contraction

Relativistic Wavepacket Evolution

The study investigates how selected quasi-(1+1) dimensional wavepacket solutions to the Klein-Gordon equation for a relativistic charged particle in uniform motion or accelerated by a uniform electric field in Minkowski space. The research explores how good the charge density of a Klein-Gordon wavepacket can be approximately described by a Gaussian state with the single-particle interpretation. A key finding is that for free relativistic particles in uniform motion, the minimal initial width for the KG charge density of a Gaussian KG wavefunction evolving like a Gaussian wavepacket centered at its classical trajectory is about λC /γ0, where γ0 = 1/p1 − (v0/c)2 is the Lorentz factor of the particle initially at speed v0.

Gaussian Approximation Justification

The paper addresses a question regarding the range of validity or the cutoff for an effective theory like Refs. [6, 7] for single electrons moving in EM fields, specifically whether a Gaussian probability density of the wavefunctions can be justified to mimic a nearly Gaussian charge density of the KG wavefunction or Dirac spinor. The authors construct KG wavepackets initially Gaussian for free particles at constant speed and for accelerated particles in a uniform electric field. They find that if the initial width is well below the Compton wavelength of the particle, i.e., cϑ = 0.1 in Figure 1, the charge density behaves very non-Gaussian.

Similarity to Gaussian Functions

The authors calculate the projection GΨ and estimate it by calculating Gρ (Eq. (14)) to see how close the KG charge densities are to a normalized Gaussian function of half-width σ centered at x = ¯x(t). They observe that for the same wavepacket solution, GΨ drops faster than Gρ as t increases because the phase of Ψ becomes highly nonlinear in x for t > 0. The half-widths σ for Ψ and ρ tend to overlap as t → ∞.

Momentum Spectrum Analysis

The peak-splitting of the charge density with initial width well below its Compton wavelength indicates that the momentum spectrum of the corresponding wavepacket is so broad that many modes of high positive and negative momenta are included. When cϑ mc = ħ/λ¯C involved in the wavepacket, which are moving at speeds close to c and would constructively interfere around the lightcone in the t-x diagram.

Uniform Electric Field Dynamics

For a charged particle initially at rest, they find that ρ can be close to a Gaussian function with the value of σ0 as small as 3. For a charged particle initially at rest, the minimal initial width for a long-lasting Gaussian charge density goes down to O(1/γ0) of the Compton wavelength as shown in Figure 1. The spreading rate of ρ is lower for a particle moving faster when both speed and field strength are small, and this can be considered a manifestation of length contraction.

Conclusion on Cutoff

The minimal initial width of ρ we found in this paper actually implies that, to keep the single-particle interpretation and the Gaussianity, the charge density of a nearly Gaussian KG wavepacket at each moment t can at most be squeezed to a width about λC /γ¯(t). This suggests that the UV cutoff of the electron-photon interaction could be about λC /γ¯(t) in an effective theory quantized in Minkowski coordinates for single electrons interacting with EM fields [6, 7].

Appendix on Classical Action

The classical action for a charged particle moving in EM fields in the Minkowskitime gauge reads Scl = Z dt (−mc2r1 − 1/c2dzi/dt dzi/dt + qcA0(t, z(t)) + qdzi/dt Ai(t, z(t))). For a free particle in uniform motion at speed v0, one has Aµ = 0 and Scl = Z dt (−mc2r1 − v20c2). The phase of wavefunction (10) evaluated along the particle’s classical trajectory, namely, ϕ(t, x¯(t)), behaves similarly to the particle’s classical action Scl in (A.2) for sufficiently large t. For a charged particle moving in a uniform electric field considered in Section 3, Scl ∼ −mc2αt2/2 = −cFt 2/2. The phase ϕ(t, x(t)) of wavefunction (23) with (22) and (14) also behaves like Scl in (A.4) for sufficiently large t.

References

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1. Improved modeling of quantum charge density in effective theories allows AI systems to approximate wavefunctions using Gaussian states when initial width is above O(lambda C)/gamma0, providing a justified description for single-particle interpretation in charged particle dynamics.

2. Enhanced ability to predict the evolution of relativistic charged particle wavepackets under uniform motion or acceleration by utilizing the minimal initial width derived as about lambda C divided by its Lorentz factor at the initial moment.

3. Development of a predictive framework where non-Gaussian features in KG charge density, such as double-peak structures, are identified when initial width is below lambda C, enabling AI to flag instances where single-particle interpretation fails and particle-antiparticle production is indicated.

4. Capability to determine the range of validity for linearized effective theories describing single electrons in quantum electromagnetic fields by exploring parameter space based on the minimal initial width requirement for a good Gaussian approximation.

We study the evolutions of selected quasi-(1+1) dimensional wavepacket solutions to the Klein-Gordon equation for a relativistic charged particle in uniform motion or accelerated by a uniform electric field in Minkowski space. We explore how good the charge density of a Klein-Gordon wavepacket can be approximately described by a Gaussian state with the single-particle interpretation. We find that the minimal initial width of a wavepacket for a good Gaussian approximation in position space is about the Compton wavelength of the particle divided by its Lorentz factor at the initial moment. This can be considered as a manifestation of relativistic length contraction, which is also observed numerically in the spreading of the wavepacket's charge density.