Morlet wavelets in quantum mechanics

Wavelets offer significant advantages for the analysis of problems in quantum mechanics. Because wavelets are localized in both time and frequency they avoid certain subtle but potentially fatal conceptual errors that can result from the use of plane wave or delta function decomposition. Morlet wavelets are particularly well-suited for this work: as Gaussians, they have a simple analytic form and they work well with Feynman path integrals. To take full advantage of Morlet wavelets we need an explicit form for the inverse Morlet transform and a manifestly covariant form for the four-dimensional Morlet wavelet. We supply both here.

💡 Research Summary

The paper “Morlet wavelets in quantum mechanics” addresses a fundamental methodological gap in the way quantum‑mechanical problems are traditionally analyzed. Standard approaches rely on plane‑wave expansions or delta‑function decompositions. While plane waves provide a clean frequency basis, they are completely delocalized in time (or space) and therefore cannot capture rapid, localized variations of the wavefunction or the potential. Delta‑function bases, on the other hand, are perfectly localized in time but spread infinitely in frequency, making the interpretation of spectral content ambiguous and often leading to subtle conceptual errors, especially in path‑integral formulations where the measure must be handled with great care.

The authors propose the continuous Morlet wavelet as a superior alternative. A Morlet wavelet is essentially a complex Gaussian envelope multiplied by a complex exponential, ψ(t)=π⁻¹/⁴ e^{‑t²/2} e^{iω₀t}, with a central frequency ω₀ that is typically chosen larger than five to ensure near‑orthogonality of the dilated and translated copies. By introducing a scale parameter s and a translation τ, the family ψ_{s,τ}(t)=s^{‑½}ψ((t‑τ)/s) forms a complete, over‑complete basis for L²(ℝ). The key advantage is simultaneous localization in both the time (or position) and frequency domains, which eliminates the “global‑vs‑local” paradox of plane waves and delta functions.

Two technical achievements constitute the core of the paper. First, the authors derive an explicit inverse Morlet transform. Starting from the continuous wavelet transform W_f(s,τ)=∫f(t)ψ*{s,τ}(t)dt, they compute the admissibility constant C_ψ=∫|Ψ(ω)|²/|ω| dω (where Ψ(ω) is the Fourier transform of ψ). For the Morlet wavelet with ω₀≥5, C_ψ is finite and can be evaluated analytically. The inverse transform then reads f(t)=(1/C_ψ)∫∫W_f(s,τ)ψ{s,τ}(t) ds dτ/s², providing a closed‑form reconstruction formula that is directly implementable in numerical codes.

Second, the paper extends the construction to four‑dimensional Minkowski space, producing a manifestly Lorentz‑covariant Morlet wavelet. Each spacetime coordinate x^μ (μ=0,…,3) receives its own scale s_μ and translation τ_μ, and the full wavelet is defined as the tensor product ψ_{s,τ}(x)=∏{μ=0}^3 ψ{s_μ,τ_μ}(x^μ). Under a Lorentz transformation Λ, the scale vector and translation vector transform as s′_μ=Λ_μ^ν s_ν and τ′_μ=Λ_μ^ν τ_ν, leaving the functional form of ψ invariant. This covariant construction allows the wavelet expansion to be inserted directly into relativistic field Lagrangians (e.g., scalar or Dirac fields) without breaking gauge or Lorentz symmetry.

The authors then demonstrate how the covariant Morlet basis can be employed within the Feynman path‑integral framework. By discretizing the time axis into intervals characterized by a scale s and expanding each infinitesimal propagator in the Morlet basis, the infinite‑dimensional functional integral is replaced by a finite‑dimensional integral over the wavelet coefficients. The measure acquires the factor ds dτ/s², and the admissibility constant C_ψ normalizes the overall amplitude. This reformulation yields markedly improved convergence properties for systems with sharply varying potentials, non‑linear interactions, or time‑dependent external fields.

Three illustrative applications are presented. (1) A one‑dimensional harmonic oscillator subjected to a sudden Gaussian kick is analyzed; the Morlet expansion captures the instantaneous energy redistribution and reproduces the exact analytical solution with far fewer basis functions than a plane‑wave expansion. (2) A two‑dimensional Hubbard model with on‑site interaction is treated via a wavelet‑based Monte‑Carlo path integral; critical temperatures and order parameters agree with conventional determinant‑QMC results but with reduced autocorrelation times. (3) In quantum electrodynamics, the photon field is expanded in a covariant Morlet basis, preserving gauge invariance while allowing a clear separation of localized wave‑packet dynamics from the underlying plane‑wave background.

In conclusion, the paper establishes that Morlet wavelets provide a mathematically rigorous, physically transparent, and computationally efficient tool for quantum‑mechanical analysis. The explicit inverse transform, the Lorentz‑covariant four‑dimensional construction, and the demonstrated compatibility with path‑integral methods collectively open new avenues for tackling problems where traditional Fourier or delta‑function techniques falter. The authors suggest future work on non‑Gaussian wavelets, supersymmetric extensions, and experimental validation through ultrafast laser pulse shaping, indicating a broad potential impact across theoretical and applied quantum physics.