2026-07-072026-07-07http://salesiana.dossiersoluciones.com/handle/123456789/123829We uncover a new type of unitary operation for quantum mechanics on the half-line which yields a transformation to ``Hyperbolic phase space''. We show that this new unitary change of basis from the position x on the half line to the Hyperbolic momentum $p_η$, transforms the wavefunction via a Mellin transform on to the critial line $s=1/2-ip_η$. We utilise this new transform to find quantum wavefunctions whose Hyperbolic momentum representation approximate a class of higher transcendental functions, and in particular, approximate the Riemann Zeta function. We finally give possible physical realisations to perform an indirect measurement of the Hyperbolic momentum of a quantum system on the half-line.23 pages, 6 FiguresQuantum Physicstext