A closed formula for the barrier transmission coefficient in quaternionic quantum mechanics

A thorough approach to quantum mechanics should rely on an algebraic operation of multiplication based on proper algebraic structures, i.e. division algebras. Quaternions, with their commutation rules isomorphic to SU(2), offer such a framework. In this work, a quaternion algebraic approach derived from the finite groups is used to solve the spatially invariant time-dependent Pauli–Schrödinger equation and derive the Rabi formula and the principle of nuclear magnetic resonance and its generalization. The spatial and time-dependent Pauli–Schrödinger equation is then written using left-chirality quaternions, i.e. by expressing the Pauli spinor, its time derivative, and the Hamiltonian as elements of the same division algebra. The quaternion formalism is used to derive the dispersion relation of a free electron and of an electron in a constant magnetic field. Finally, the formalism is applied to calculate the transmission probabilities through a one-dimensional delta scatterer and through a resonant tunneling structure composed of two one-dimensional quaternionic delta-scatterers in series. The results are compared to the transmission probabilities calculated using traditional (complex) quantum mechanics.

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“…To the left of the scatterer, solution to equation (139) is given by in its simplectic form by [7,[24][25][26][27]…”

Section: Energy Dispersion Relationmentioning

confidence: 99%

On the quaternionic form of the Pauli–Schrödinger equation

Cahay

Morris²

2019

Phys. Scr.

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“…The least squares problem appears in many fields, such as computational mathematics, linear regression, machine learning, cybernetics and image processing. In addition, quaternions and quaternion matrices play an increasingly important role in quantum mechanics [1,2], special relativity [3,4,5], signal processing [6,7,8], relativistic dynamics [9,10], computer graphics [11,12,13] and other application fields.…”

Section: Introductionmentioning

confidence: 99%

The solutions of the quaternion multiple equality constrained least squares problem

Zhang

Liu

2023

Preprint

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Many data models of practical problems are related to the quaternion multiple equality constrained least squares (QMELS) problem, which is a generalization of the quaternion equality constrained least squares (QELS) problem. In this paper, by utilizing the MP inverse, the quaternion QR factorization, the quaternion CS factorization and some known results of the QELS problem, we first transform the QMELS problem into an unconstrained least squares (LS) problem. Next, we give a representation formula for solutions of the QMELS problem. After that, we present a real structure-preserving algorithm of its minimum norm solution. Finally, a example is proposed to indicate the validity.

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“…Driven by its theoretical significance and potential practical applications, extensive investigations have explored how matter waves propagate through various potential distributions [7,8]. Such efforts span a diverse array of quantum mechanical domains, including non-Hermitian quantum mechanics (NHQM) [9][10][11][12], space-fractional quantum mechanics (SFQM) [13][14][15][16][17][18], and quaternionic quantum mechanics (QQM) [19][20][21][22][23][24][25]. With an array of contributions over the past century, quantum tunneling and the associated analytical calculations of scattering coefficients continue to be a focal point of quantum research [1][2][3][4][5][6][7][8][26][27][28][29][30].…”

Section: Introductionmentioning

confidence: 99%

Tunneling from general Smith–Volterra–Cantor potential

Singh

Umar

Hasan

et al. 2023

Journal of Mathematical Physics

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We study the tunneling problem from the general Smith–Volterra–Cantor (SVC) potential of finite length L characterized by the scaling parameter ρ and stage G. We show that the SVC( ρ) potential of stage G is a special case of the super periodic potential (SPP) of order G. By using the SPP formalism developed by us earlier, we provide the closed form expression of the tunneling probability T G( k) with the help of the q-Pochhammer symbol. The profile of T G( k) with wave vector k is found to saturate with increasing stage G. Very sharp transmission resonances are found to occur in this system, which may find applications in the design of sharp transmission filters.

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A closed formula for the barrier transmission coefficient in quaternionic quantum mechanics

Cited by 13 publications

References 23 publications

On the quaternionic form of the Pauli–Schrödinger equation

On the quaternionic form of the Pauli–Schrödinger equation

The solutions of the quaternion multiple equality constrained least squares problem

Tunneling from general Smith–Volterra–Cantor potential

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