J Twamley and G J Milburn 2006 New J. Phys. 8 328 doi:10.1088/1367-2630/8/12/328
The quantum Mellin transform
J Twamley1,3 and G J Milburn2
Show affiliationsWe uncover a new type of unitary operation for quantum mechanics on the half-line which yields a transformation to 'hyperbolic phase space' (η, pη). We show that this new unitary change of basis from the position x on the half line to the hyperbolic momentum pη, transforms the wavefunction via a Mellin transform on to the critical line s = 1/2−ipη. We utilize 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 realizations to perform an indirect measurement of the hyperbolic momentum of a quantum system on the half-line.
- PACS
- MSC
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81Q10 Selfadjoint operator theory in quantum theory, including spectral analysis
81S30 Phase space methods including Wigner distributions, etc.
- Subjects
- Dates
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Issue 12 (December 2006)
Received 12 November 2006
Published 20 December 2006
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The quantum Mellin transform
J Twamley and G J Milburn 2006 New J. Phys. 8 328
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J S Dowker 1984 Class. Quantum Grav. 1 359
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