Momentum Space Analysis of the Infinite Square Well: Operator Domains and Physical Tails

The infinite square well is a paradigm of energy quantization, but defining its momentumdistribution requires specifying an operational measurement protocol. We consider the sud-den release of a confined eigenstate followed by free expansion: extending the spatial stateby zero onto the real line yields a continuous momentum distribution via the Fourier trans-form. This continuous distribution is conceptually distinct from the internal Fourier sineseries; in particular, pn = nπℏ/L is a characteristic Fourier scale rather than a momentumeigenvalue. Because the first-derivative operator on the interval with Dirichlet boundaryconditions lacks a canonical self-adjoint realization, the strictly confined state possesses nointrinsic momentum distribution. The released distribution exhibits shifted probability lobesand an oscillatory p−4 asymptotic tail, yielding a finite second moment ⟨p2⟩ = 2mEn but adivergent fourth moment ⟨p4⟩. These spectral properties directly reflect the Sobolev regular-ity of the zero-extended state, ψen ∈ H1(R) \ H2(R). Finally, we connect this mathematicalstructure to the variational formulation: the energy quadratic form is well-defined on theform domain H10(0, L), whereas the operator domain is restricted to H2(0, L) ∩ H10(0, L).This distinction elucidates why finite-energy trial states need not satisfy second-derivativeregularity and clarifies the role of orthogonality constraints in variational approximations ofexcited states.