The Quaternionic Quantum Mechanics

Arbab, Arbab I.

doi:10.5539/apr.v3n2p160

Cited by 37 publications

(39 citation statements)

References 11 publications

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“…Further, the differential equations (20) and (21) are invariant under the same transformation (22). Therefore, it is seen, that the correspondence between the symmetry of differential equations and the mathematical nature [in the concept of the number theory] of the quantities, incoming in given equations seems to be taking place.…”

Section: Symmetry Of Differential Equations Andmentioning

confidence: 87%

“…In the case of the invariance of differential equations under transformation (22), that is, in the case of the invariance under transformations of multiplicative group of the linear space of complex numbers over the field of real numbers, the proof seems to be evident. Really, the possibility of multiplication of full set of field functions on complex number means, that the field functions themselves have to be complex-defined functions.…”

Section: Differential Equations Which Are Invariant Under Transformamentioning

confidence: 98%

“…At the same time the energy conservation law and the conservation of Poynting vector will be full filled, if the given transformation is applied to EM-field potentials [14,15]. However, the energy conservation law and the conservation of Poynting vector, that is mathematical constructions, to which enter , E H − −   vector functions, will not be full filled by the transformation (22) at an arbitrary β. The charge remains to be Lorentz invariant quantity [14], at the same time both the field characteristics, the energy and the impulse (determined by Poynting vector) are not Lorentz invariant quantities [14].…”

Section: Symmetry Of Differential Equations Andmentioning

confidence: 99%

“…It seems to be substantial that differential equations for the dynamics of the nonrelativistic classical mechanics are invariant under transformation (22) too. Really, the dynamics of classical mechanics systems is described by Lagrange equations or by equivalent canonical Hamilton equations.…”

Section: Differential Equations Which Are Invariant Under Transformamentioning

confidence: 99%

“…It is understandable, that the transformation of field functions by the relation (22) is equivalent to the multiplication of field functions by an arbitrary complex number. The attention was drawn in [14], that the relation (22) gives the automorphism of the space of field functions. It is known, that the automorphism of any linear space leads to a number of useful properties of the objects, which belong to the given space.…”

Section: Symmetry Of Differential Equations Andmentioning

confidence: 99%

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To Principles of Quantum Theory Construction

Yerchuck¹,

Dovlatova²,

Alexandrov³

2014

IJP

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New insight to the principles of the quantum theory construction is given. It is based on the symmetry study of main differential equations of mechanics and electrodynamics. It has been shown, that differential equations, which are invariant under transformations of groups, which are symmetry groups of mathematical numbers (considered within the frames of the number theory) determine the mathematical nature of the quantities, incoming in given equations. The substantial consequence of the given consideration is the proof of the main postulate of quantum mechanics, that is, the proof of the statement, that to any quantum mechanical quantity can be set up into the correspondence the Hermitian matrix. It is shown, that a non-abelian character of the multiplicative group of the quaternion ring leads to the nonapplicability of the quaternion calculus for the construction of new versions of quantum mechanics directly. The given conclusion seems to be actual, since there is a number of modern publications with the development of the quantum mechanics theory using the quaternions with the standard basis {e, i, j, k}. The correct way for the construction of new versions of quantum mechanics on the quaternion base is discussed in the paper presented. It is realized by means of the representation of the quaternions through the basis of the linear space of complex numbers over the field of real numbers, under the multiplicative group of which the equations of the dynamics of mechanical systems are invariant. At the same time the quaternion calculus is applicable in electrodynamics, at that the new versions of quantum electrodynamics can be constructed by an infinite number of the ways corresponding to an infinite number of the matrix representations of the standard quaternion basis {e, i, j, k}. The given conclusion is the consequence of the high symmetry of Maxwell equations.

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Section: Symmetry Of Differential Equations Andmentioning

confidence: 87%

Section: Differential Equations Which Are Invariant Under Transformamentioning

confidence: 98%

Section: Symmetry Of Differential Equations Andmentioning

confidence: 99%

Section: Differential Equations Which Are Invariant Under Transformamentioning

confidence: 99%

Section: Symmetry Of Differential Equations Andmentioning

confidence: 99%

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To Principles of Quantum Theory Construction

Yerchuck¹,

Dovlatova²,

Alexandrov³

2014

IJP

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Quaternionic Quantum Particles

Giardino

2019

Adv. Appl. Clifford Algebras

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If Ψ is a quaternionic wave function, then iΨ = Ψi. Thus, there are two versions of the quaternionic Schrödinger equation (QSE). In this article, we present the second possibility for solving the QSE, following on from a previous article. After developing the general methodology, we present the quaternionic free particle solution and the scattering of the quaternionic particle through a scalar barrier.

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