The Quantum Mellin transform

Abstract

We 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 $p_{\eta }$, transforms the wavefunction via a Mellin transform on to the critial line $s=1/2-i{p}_{\eta }$. 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 Figures

References in corpus (1)

• Continuous variable encoding by ponderomotive interaction

Cited by in corpus (4)

• H = x p with interaction and the Riemann zeros
• A quantum mechanical model of the Riemann zeros
• Optimal Covariant Measurement of Momentum on a Half Line in Quantum Mechanics
• On the Quantum Reconstruction of the Riemann zeros