The quantum Mellin transform

J. Twamley; G. J. Milburn

doi:10.1088/1367-2630/8/12/328

The quantum Mellin transform

J. Twamley^*, G. J. Milburn

^*Corresponding author for this work

Research output: Contribution to journal › Article › peer-review

31 Citations (Scopus)

6 Downloads (Pure)

Abstract

We uncover a new type of unitary operation for quantum mechanics on the half-line which yields a transformation to 'hyperbolic phase space' (η, p_η). 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 wavef unction via a Mellin transform on to the critical line s = 1 /2 -ip_η. 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.

Original language	English
Article number	328
Pages (from-to)	1-20
Number of pages	20
Journal	New Journal of Physics
Volume	8
DOIs	https://doi.org/10.1088/1367-2630/8/12/328
Publication status	Published - 20 Dec 2006

Bibliographical note

Copyright 2006 IOP Publishing Ltd. Reprinted from New journal of physics. This material is posted here with the permission of IOP Publishing Ltd and the authors. Use of this material is permitted for personal, research and non-commercial uses. Further information regarding the copyright applicable to this article can be viewed at http://www.iop.org/EJ/journal/-page=extra.copyright/NJP.

Access to Document

10.1088/1367-2630/8/12/328

Publisher version (open access).pdfFinal published version, 794 KB

Cite this

@article{17a80bb766e04799a9c54b1a360a12ba,

title = "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' (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 wavef unction 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.",

author = "J. Twamley and Milburn, {G. J.}",

note = "Copyright 2006 IOP Publishing Ltd. Reprinted from New journal of physics. This material is posted here with the permission of IOP Publishing Ltd and the authors. Use of this material is permitted for personal, research and non-commercial uses. Further information regarding the copyright applicable to this article can be viewed at http://www.iop.org/EJ/journal/-page=extra.copyright/NJP.",

year = "2006",

month = dec,

day = "20",

doi = "10.1088/1367-2630/8/12/328",

language = "English",

volume = "8",

pages = "1--20",

journal = "New Journal of Physics",

issn = "1367-2630",

publisher = "Institute of Physics",

}

TY - JOUR

T1 - The quantum Mellin transform

AU - Twamley, J.

AU - Milburn, G. J.

N1 - Copyright 2006 IOP Publishing Ltd. Reprinted from New journal of physics. This material is posted here with the permission of IOP Publishing Ltd and the authors. Use of this material is permitted for personal, research and non-commercial uses. Further information regarding the copyright applicable to this article can be viewed at http://www.iop.org/EJ/journal/-page=extra.copyright/NJP.

PY - 2006/12/20

Y1 - 2006/12/20

N2 - We 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 wavef unction 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.

AB - We 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 wavef unction 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.

UR - http://www.scopus.com/inward/record.url?scp=33845912950&partnerID=8YFLogxK

U2 - 10.1088/1367-2630/8/12/328

DO - 10.1088/1367-2630/8/12/328

M3 - Article

AN - SCOPUS:33845912950

VL - 8

SP - 1

EP - 20

JO - New Journal of Physics

JF - New Journal of Physics

SN - 1367-2630

M1 - 328

ER -

The quantum Mellin transform

Abstract

Bibliographical note

Access to Document

Other files and links

Fingerprint

Cite this