Finite-dimensional quantum mechanics of a particle

Jagannathan, R.; Santhanam, T. S.; Vasudevan, R.

doi:10.1007/bf00674253

Cited by 35 publications

(18 citation statements)

References 7 publications

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“…Quantum mechanics as it pertains to the harmonic oscillator connects the canonical variables, position and momentum, through the Fourier integral operator F via [4,3]:…”

Section: Introductionmentioning

confidence: 99%

On discrete Gauss–Hermite functions and eigenvectors of the discrete Fourier transform

Santhanam

2008

Signal Processing

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The problem of furnishing an orthogonal basis of eigenvectors for the discrete Fourier transform (DFT) is fundamental to signal processing. Recent developments in the area of discrete fractional Fourier analysis also rely upon the ability to furnish a basis of eigenvectors for the DFT or its centralized version. However, none of the existing approaches are able to furnish a commuting matrix where both the eigenvalue spectrum and the eigenvectors are a close match to corresponding properties of the continuous differential Gauss-Hermite operator. Furthermore, any linear combination of commuting matrices produced by existing approaches also commutes with the DFT, thereby bringing up the question of uniqueness. In this paper, inspired by concepts from quantum mechanics in finite dimensions, we present an approach that furnishes a basis of orthogonal eigenvectors for both versions of the DFT. This approach also furnishes a commuting matrix whose eigenvalue spectrum is a very close approximation to that of the Gauss-Hermite differential operator and consequently a framework for a unique definition of the discrete Gauss-Hermite operator.

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“…Quantum mechanics as it pertains to the harmonic oscillator connects the canonical variables, position and momentum, through the Fourier integral operator F via [4,3]:…”

Section: Introductionmentioning

confidence: 99%

On discrete Gauss–Hermite functions and eigenvectors of the discrete Fourier transform

Santhanam

2008

Signal Processing

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“…With the phase, shift, and inversion operator defined by Eqs. (11), (12), and (13) indexed by I ′′ = {0, 1, · · · , N − 1}, Eq. (28) can then be expressed as…”

Section: A Definition Of the Weyl Operator In Even-lattice Casementioning

confidence: 99%

Covariant projective representation of symplectic group on discrete phase space

Watanabe

Hashimoto

Horibe

et al. 2019

J. Phys.: Conf. Ser.

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The phase point operator ∆(q, p) is the quantum mechanical counterpart of the classical phase point (q, p). The discrete form of ∆(q, p) was formulated for an odd number of lattice points by Cohendet et al. and for an even number of lattice points by Leonhardt. Both versions have symplectic covariance, which is of fundamental importance in quantum mechanics. However, an explicit form of the projective representation of the symplectic group that appears in the covariance relation is not yet known. We show in this paper the existence and uniqueness of the representation, and describe a method to construct it using the Euclidean algorithm.

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“…We proceeded in a fashion similar to the approach described by Jagannathan and Vasudevan in [1]. In their paper they directly defined both their position and momentum operators and then detailed the relations between them and the finite Fourier transform.…”

Section: Operators In the Position Basismentioning

confidence: 99%

“…Musin also calculated a perturbative expression for the energy levels of this Hamiltonian, which depends on n B and n F , respectively the energy levels of our boson and fermion 1 : 1 There was a typo in the original paper: the g in the second term should have been g 2 . Equation 33 reflects this change.…”

Section: Supersymmetric Oscillatormentioning

confidence: 99%

Quantum Computation and Visualization of Hamiltonians using Discrete Quantum Mechanics and IBM QISKit

Miceli

McGuigan

2018

2018 New York Scientific Data Summit (NYSDS)

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Quantum computers have the potential to transform the ways in which we tackle some important problems. The efforts by companies like Google, IBM and Microsoft to construct quantum computers have been making headlines for years. Equally important is the challenge of translating problems into a state that can be fed to these machines. Because quantum computers are in essence controllable quantum systems, the problems that most naturally map to them are those of quantum mechanics. Quantum chemistry has seen particular success in the form of the variational quantum eigensolver (VQE) algorithm, which is used to determine the ground state energy of molecular systems. The goal of our work has been to use the matrix formulation of quantum mechanics to translate other systems so that they can be run through this same algorithm. We describe two ways of accomplishing this using a position basis approach and a Gaussian basis approach. We also visualize the wave functions from the eigensolver and make comparisons to theoretical results obtained with continuous operators.

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Finite-dimensional quantum mechanics of a particle

Cited by 35 publications

References 7 publications

On discrete Gauss–Hermite functions and eigenvectors of the discrete Fourier transform

On discrete Gauss–Hermite functions and eigenvectors of the discrete Fourier transform

Covariant projective representation of symplectic group on discrete phase space

Quantum Computation and Visualization of Hamiltonians using Discrete Quantum Mechanics and IBM QISKit

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