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 (11)

• Encoding a qubit in an oscillator
• Self-adjoint extensions of operators and the teaching of quantum mechanics
• Entangling movable mirrors in a double cavity system
• Error Analysis For Encoding A Qubit In An Oscillator
• Constructing finite dimensional codes with optical continuous variables
• Preparing encoded states in an oscillator
• Radiation Pressure Induced Einstein-Podolsky-Rosen Paradox
• Continuous variable encoding by ponderomotive interaction
• Generating continuous variable quantum codewords in the near-field atomic lithography
• Classical Aspects of Quantum Walls in One Dimension
• Quantum distribution functions for radial observables

Cited by in corpus (22)

• Physical meaning of the radial index of Laguerre-Gauss beams
• Physics of the Riemann Hypothesis
• The H=xp model revisited and the Riemann zeros
• H = x p with interaction and the Riemann zeros
• Will a physicist prove the Riemann Hypothesis?
• A quantum mechanical model of the Riemann zeros
• The Riemann zeros as energy levels of a Dirac fermion in a potential built from the prime numbers in Rindler spacetime
• The Riemann zeros as spectrum and the Riemann hypothesis
• General covariant xp models and the Riemann zeros
• A self-adjoint arrival time operator inspired by measurement models
• Canonical quantization on the half-line and in an interval based upon a new concept for the momentum in a space with boundaries
• Quantum dynamics and relaxation in comb turbulent diffusion
• Horizon temperature on the real line
• An $xp$ model on $Ad{S}_2$ spacetime
• Optimal Covariant Measurement of Momentum on a Half Line in Quantum Mechanics
• Duality between the quantum inverted harmonic oscillator and inverse square potentials
• On the Quantum Reconstruction of the Riemann zeros
• Asymptotic analysis on a pseudo-Hermitian Riemann-zeta Hamiltonian
• Distribution of xp in some molecular rotational states
• Factorization with a logarithmic energy spectrum of a central potential
• Hamiltonian for the Hilbert-Pólya Conjecture
• Biorthogonal systems on unit interval and zeta dilation operators