The Quantum Mellin transform

Details The Quantum Mellin transform The Quantum Mellin transform

2007-02-12

arxiv.org/abs/quant-ph/0702107v1

New J Phys 8 328 (2006)

Physics

Quantum Physics

23 pages, 6 Figures

Scientific paper

10.1088/1367-2630/8/12/328

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 $s=1/2-ip_\eta$. 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.

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