Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Log inSign up
Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Blog
Job Board
Q3AS

© 2026 Aqora Quantum S.A.S.

TermsPrivacyLegal Notice

Quaternionic representation of a one-dimensional wave packet

R. Deepika, Krishnan Muthunagai

Published Jun 6, 2024• 0 citations
Publisher

Abstract

The introduction of complex numbers marked a significant leap in mathematics, introducing the imaginary unit i to represent the square root of -1. This innovative concept proved invaluable in solving equations involving square roots of negative numbers. The extension to quaternions involved introducing additional imaginary units denoted as j and k . A quaternion in an interesting concept that extends complex numbers to 4-D.The manuscript is about using quaternions to calculate wave packets in one-dimension, and anti-hermitian operators to obtain the results in quaternionic form including expectation values of position, momentum and energy. The results are compared to the exisiting results on complex wave packets in one-dimension.

DOI
10.22541/au.171767024.40491431/v1

Related papers

Triangular potential and one-dimensional wave packet in quaternionic quantum mechanics

R. Deepika, Krishnan Muthunagai

A quaternion is an interesting concept that extends complex numbers to 4-D. This paper is about using quaternions to represent triangular potential and wave packets in one-dimension. Using, wave packets in one-dimension and anti-hermitian operators results are obtained in quaternionic form including expectation values of position, momentum and energy.

International Journal of Geometric Methods in Modern Physics•Feb 19, 2026•0 citations

Generalization of adding angular momenta and circular potential in quaternionic quantum mechanics

R. Deepika, Krishnan Muthunagai

Complex numbers were created by introducing the imaginary unit i to represent the square root of -1, allowing solutions to equations that involved square roots of negative numbers. Complex numbers were further extended to quaternionic numbers by introducing additional imaginary units, namely " j " and " k ". Quaternions are non-commutative and involve four components, with each component being a real number or a multiple of an imaginary unit. Usually complex numbers are used to represent wave functions in quantum mechanics. Solutions using quaternions to square well, spin and angular momentum, Dirac equations have been obtained by many researchers. In this article, we have made use of quaternions to study the generalization of adding angular momenta, the digital signal processing of a quaternionic function and circular potential of a particle in real Hilbert space and have obtained quaternionic solutions in terms of Bessel functions.

Heliyon•Feb 1, 2024•4 citations

Quaternionic bound states

Stefano De Leo, Gisele Cristina Ducati

We study the bound-state solutions of vanishing angular momentum in a quaternionic spherical square-well potential of finite depth. As in standard quantum mechanics, such solutions occur for discrete values of energy. At first glance, it seems that the continuity conditions impose a very restrictive constraint on the energy eigenvalues and, consequently, no bound states were expected for energy values below the pure quaternionic potential. Nevertheless, a careful analysis shows that pure quaternionic potentials do not remove bound states. It is also interesting to compare these new solutions with the bound state solutions of the trial-complex potential. The study presented in this paper represents a preliminary step towards a full understanding of the role that quaternionic potentials could play in quantum mechanics. Of particular interest for the authors is the analysis of confined wave packets and tunnelling times in this new formulation of quantum theory.

arXiv:math-ph/0504025•Journal of Physics A Mathematical and General•Mar 31, 2005•36 citations

Imagination and real quantization

Stanislav Ordin

The paper shows that the use of mathematical imaginaries, complex numbers, and even more so quaternions in the Quantum Theory to increase the dimension of solutions of one-dimensional equations is, in principle, incorrect. As a result, Einstein's condition was violated: SOME equations of Classical Physics can be rewritten in operator form. The use of correct multidimensional models makes it possible, using the example of the Harmonic Oscillator, to obtain the Planck-Einstein Quantization for da Broglie waves.

International Journal of Physics and Applications•Jan 1, 2022•0 citations

Nonrelativistic quaternionic quantum mechanics in one dimension

A. J. Davies, Bruce H. J. McKellar

We present a formalism for treating one-dimensional problems in quaternionic quantum mechanics. As an example, we derive an explicit form for the T matrix for scattering from a square (quaternionic) barrier, and use this result to calculate the transmission and reflection coefficients. We show that the qualitative form of these coefficients is the same as in complex quantum mechanics, even when the barrier has a nonzero value for the quaternionic components of the potential.

Physical Review A•Oct 1, 1989•80 citations

On the Necessity of Complex Numbers in Quantum Mechanics

Marco Oppio

In principle, the lattice of elementary propositions of a generic quantum system admits a representation in real, complex or quaternionic Hilbert spaces as established by Solèr’s theorem (1995) closing a long standing problem that can be traced back to von Neumann’s mathematical formulation of quantum mechanics. However up to now there are no examples of quantum systems described in Hilbert spaces whose scalar field is different from the set of complex numbers. We show that elementary relativistic systems cannot be described by irreducible strongly-continuous unitary representations of SL(2, C) on real or quaternionic Hilbert spaces as a consequence of some peculiarity of the generators related with the theory of polar decomposition of operators. Indeed such a ”naive” attempt leads necessarily to an equivalent formulation on a complex Hilbert space. Although this conclusion seems to give a definitive answer to the real/quaternionic-quantum-mechanics issue, it lacks consistency since it does not derive from more general physical hypotheses as the complex one does. Trying a more solid approach, in both situations we end up with three possibilities: an equivalent description in terms of a Wigner unitary representation in a real, complex or quaternionic Hilbert space. At this point the ”naive” result turns out to be a definitely important technical lemma, for it forbids the two extreme possibilities. \nIn conclusion, the real/quaternionic theory is actually complex. This improved approach is based upon the concept of von Neumann algebra of observables. Unfortunately, while \nthere exists a thorough literature about these algebras on real and complex Hilbert spaces, an analysis on the notion of von Neumann algebra over a quaternionic Hilbert space is \ncompletely absent to our knowledge. There are several issues in trying to define such a mathematical object, first of all the inability to construct linear combination of operators \nwith quaternionic coeffients. Restricting ourselves to unital real *-algebras of operators we are able to prove the von Neumann Double Commutant Theorem also on quaternionc \nHilbert spaces. Clearly, this property turns out to be crucial.

•Jan 1, 2018•1 citation
The introduction of complex numbers marked a significant leap in mathematics, introducing the imaginary unit i to represent the square root of -1. This innovative concept proved invaluable in solving equations involving square roots of negative numbers. The extension to quaternions involved introducing additional imaginary units denoted as j and k . A quaternion in an interesting concept that extends complex numbers to 4-D.The manuscript is about using quaternions to calculate wave packets in one-dimension, and anti-hermitian operators to obtain the results in quaternionic form including expectation values of position, momentum and energy. The results are compared to the exisiting results on complex wave packets in one-dimension.
Complex numbers were created by introducing the imaginary unit i to represent the square root of -1, allowing solutions to equations that involved square roots of negative numbers. Complex numbers were further extended to quaternionic numbers by introducing additional imaginary units, namely " j " and " k ". Quaternions are non-commutative and involve four components, with each component being a real number or a multiple of an imaginary unit. Usually complex numbers are used to represent wave functions in quantum mechanics. Solutions using quaternions to square well, spin and angular momentum, Dirac equations have been obtained by many researchers. In this article, we have made use of quaternions to study the generalization of adding angular momenta, the digital signal processing of a quaternionic function and circular potential of a particle in real Hilbert space and have obtained quaternionic solutions in terms of Bessel functions.
We study the bound-state solutions of vanishing angular momentum in a quaternionic spherical square-well potential of finite depth. As in standard quantum mechanics, such solutions occur for discrete values of energy. At first glance, it seems that the continuity conditions impose a very restrictive constraint on the energy eigenvalues and, consequently, no bound states were expected for energy values below the pure quaternionic potential. Nevertheless, a careful analysis shows that pure quaternionic potentials do not remove bound states. It is also interesting to compare these new solutions with the bound state solutions of the trial-complex potential. The study presented in this paper represents a preliminary step towards a full understanding of the role that quaternionic potentials could play in quantum mechanics. Of particular interest for the authors is the analysis of confined wave packets and tunnelling times in this new formulation of quantum theory.
The paper shows that the use of mathematical imaginaries, complex numbers, and even more so quaternions in the Quantum Theory to increase the dimension of solutions of one-dimensional equations is, in principle, incorrect. As a result, Einstein's condition was violated: SOME equations of Classical Physics can be rewritten in operator form. The use of correct multidimensional models makes it possible, using the example of the Harmonic Oscillator, to obtain the Planck-Einstein Quantization for da Broglie waves.
In principle, the lattice of elementary propositions of a generic quantum system admits a representation in real, complex or quaternionic Hilbert spaces as established by Solèr’s theorem (1995) closing a long standing problem that can be traced back to von Neumann’s mathematical formulation of quantum mechanics. However up to now there are no examples of quantum systems described in Hilbert spaces whose scalar field is different from the set of complex numbers. We show that elementary relativistic systems cannot be described by irreducible strongly-continuous unitary representations of SL(2, C) on real or quaternionic Hilbert spaces as a consequence of some peculiarity of the generators related with the theory of polar decomposition of operators. Indeed such a ”naive” attempt leads necessarily to an equivalent formulation on a complex Hilbert space. Although this conclusion seems to give a definitive answer to the real/quaternionic-quantum-mechanics issue, it lacks consistency since it does not derive from more general physical hypotheses as the complex one does. Trying a more solid approach, in both situations we end up with three possibilities: an equivalent description in terms of a Wigner unitary representation in a real, complex or quaternionic Hilbert space. At this point the ”naive” result turns out to be a definitely important technical lemma, for it forbids the two extreme possibilities. \nIn conclusion, the real/quaternionic theory is actually complex. This improved approach is based upon the concept of von Neumann algebra of observables. Unfortunately, while \nthere exists a thorough literature about these algebras on real and complex Hilbert spaces, an analysis on the notion of von Neumann algebra over a quaternionic Hilbert space is \ncompletely absent to our knowledge. There are several issues in trying to define such a mathematical object, first of all the inability to construct linear combination of operators \nwith quaternionic coeffients. Restricting ourselves to unital real *-algebras of operators we are able to prove the von Neumann Double Commutant Theorem also on quaternionc \nHilbert spaces. Clearly, this property turns out to be crucial.