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The Quantum Mellin transform
J. Twamley, G. J. Milburn
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 $s=1/2-ip_\eta$. 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.
- oai_identifier:
- oai:arXiv.org:quant-ph/0702107
- categories:
- quant-ph
- comments:
- 23 pages, 6 Figures
- doi:
- 10.1088/1367-2630/8/12/328
- arxiv_id:
- quant-ph/0702107
- journal_ref:
- New J Phys 8 328 (2006)
- created:
- 2007-02-12
Full article ▸
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