Momentum Space Analysis of the Infinite Square Well: Operator Domains and Physical Tails
Year: 2026
Type: preprint
Abstract: The infinite square well is a paradigm of energy quantization, but defining its momentum distribution requires specifying an operational measurement protocol. We consider the sudden release of a confined eigenstate followed by free expansion: extending the spatial state by zero onto the real line yields a continuous momentum distribution via the Fourier transform. This continuous distribution is conceptually distinct from the internal Fourier sine series; in particular, p n = nπℏ/L is a characteristic Fourier scale rather than a momentum eigenvalue. Because the first-derivative operator on the interval with Dirichlet boundary conditions lacks a canonical self-adjoint realization, the strictly confined state possesses no intrinsic momentum distribution. The released distribution exhibits shifted probability lobes and an oscillatory p-4 asymptotic tail, yielding a finite second moment ⟨p 2 ⟩ = 2mE n but a divergent fourth moment ⟨p 4 ⟩. These spectral properties directly reflect the Sobolev regularity of the zero-extended state, ψ n ∈ H 1 (R) \ H 2 (R). Finally, we connect this mathematical structure to the variational formulation: the energy quadratic form is well-defined on the form domain H 1 0 (0, L), whereas the operator domain is restricted to H 2 (0, L) ∩ H 1 0 (0, L). This distinction elucidates why finite-energy trial states need not satisfy second-derivative regularity and clarifies the role of orthogonality constraints in variational approximations of excited states.
Source: SSRN Electronic Journal
DOI: 10.2139/ssrn.7682796
Author: Amir Kahrom
Language: English
Subfield: Mathematical Physics
Field: Mathematics
Domain: Physical Sciences
Open Access status: green
OpenAlex ID: W7225410772