Dynamics of the Psyche , A Dimensional Vectorization Model of Human Needs and Psychopathologies in Hilbert Space

This paper proposes a formal transposition of Abraham Maslow's humanistic hierarchy of needs to the mathematical framework of Hilbert space (ℋ). We postulate that an individual's motivational and clinical state does not inhabit a discrete, sequential rung but rather exists as a linear superposition of basis states, |Ψ⟩ = Σ cₙ|eₙ⟩, where the complex coefficients cₙ represent probability amplitudes associated with the presence, satisfaction, or pathological salience of each need dimension. The model rests on five core formal contributions: (i) the vectorization of the five Maslovian needs as an orthonormal basis {|eₙ⟩} satisfying ⟨eₙ|eₘ⟩ = δₙₘ; (ii) the introduction of the Riesz Representation Theorem as the mathematical foundation of the diagnostic act, understood as a bounded linear functional; (iii) the formulation of the clinical Born Rule, P(n) = |⟨eₙ|Ψ⟩|², which quantifies the probability that an individual will act motivated by a specific need; (iv) the Motivational Uncertainty Principle ΔS · ΔI ≥ ℏ/2, which establishes a complementarity between satisfaction certainty and impulse unpredictability; and (v) a Life Hamiltonian Ĥ governing the temporal evolution of the psychic state via a Schrödinger-type equation. We further develop a parallel vectorization of psychopathologies in five clinical dimensions and introduce additional tools —density operator ρ, Lindblad master equation, and Fourier transform— to address comorbidity, environmental decoherence, and the satisfaction/impulse duality. The model does not claim that the brain is a physically quantum system; rather, it adopts the mathematical structure of quantum functional analysis as a formal tool to describe psychological phenomena that the classical pyramid does not capture: motivational simultaneity, sudden regression, comorbidity, and temporal dynamics. We discuss implications for clinical practice, public health, behavioral economics, and social planning, as well as the epistemological limits of the model. The paper is offered as an open theoretical-formal framework, amenable to progressive empirical operationalization and articulation with current quantum cognition lines (Busemeyer, Bruza, Pothos). Keywords: Maslow · Hilbert space · mathematical psychology · quantum cognition · motivational superposition · uncertainty principle · Riesz theorem · psychic Hamiltonian · clinical vectorization · psychopathology · functional analysis · social decoherence.