scholarly journals Erlangen Programme at Large 3.2 Ladder Operators in Hypercomplex Mechanics

10.14311/1402 ◽  
2011 ◽  
Vol 51 (4) ◽  
Author(s):  
V. V. Kisil
Keyword(s):  
Quantum Mechanics ◽  
Representation Theory ◽  
Harmonic Oscillator ◽  
Correspondence Principle ◽  
Free Particle ◽  
Hypercomplex Numbers ◽  
Ladder Operators ◽  
Elliptic Case ◽  
Hyperbolic Case ◽  
Parabolic Case

We revise the construction of creation/annihilation operators in quantum mechanics based on the representation theory of the Heisenberg and symplectic groups. Besides the standard harmonic oscillator (the elliptic case) we similarly treat the repulsive oscillator (hyperbolic case) and the free particle (the parabolic case). The respective hypercomplex numbers turn out to be handy on this occasion. This provides a further illustration to the Similarity and Correspondence Principle.

scholarly journals SOME ASPECTS OF QUANTUM MECHANICS AND FIELD THEORY IN A LORENTZ INVARIANT NONCOMMUTATIVE SPACE

2013 ◽  
Vol 28 (07) ◽  
pp. 1350017 ◽  
Author(s):  
EVERTON M. C. ABREU ◽  
M. J. NEVES
Keyword(s):  
Field Theory ◽  
Quantum Mechanics ◽  
Harmonic Oscillator ◽  
Spectral Representation ◽  
Free Particle ◽  
Transition Amplitude ◽  
Noncommutative Space ◽  
Moyal Product ◽  
Massive Scalar Field ◽  
Scalar Action

We obtained the Feynman propagators for a noncommutative (NC) quantum mechanics defined in the recently developed Doplicher–Fredenhagen–Roberts–Amorim (DFRA) NC background that can be considered as an alternative framework for the NC space–time of the early universe. The operators' formalism was revisited and we applied its properties to obtain an NC transition amplitude representation. Two examples of DFRA's systems were discussed, namely, the NC free particle and NC harmonic oscillator. The spectral representation of the propagator gave us the NC wave function and energy spectrum. We calculated the partition function of the NC harmonic oscillator and the distribution function. Besides, the extension to NC DFRA quantum field theory is straightforward and we used it in a massive scalar field. We had written the scalar action with self-interaction ϕ4 using the Weyl–Moyal product to obtain the propagator and vertex of this model needed to perturbation theory. It is important to emphasize from the outset, that the formalism demonstrated here will not be constructed by introducing an NC parameter in the system, as usual. It will be generated naturally from an already existing NC space. In this extra dimensional NC space, we presented also the idea of dimensional reduction to recover commutativity.

Summation over Feynman histories: the free particle and the harmonic oscillator

1957 ◽  
Vol 53 (1) ◽  
pp. 199-205 ◽  
Author(s):  
H. Davies
Keyword(s):  
Quantum Mechanics ◽  
Harmonic Oscillator ◽  
Free Particle ◽  
Phase Factor ◽  
Arbitrary Phase

ABSTRACTUsing a particular parametrization of paths, the free particle and the harmonic oscillator are treated by the Feynman method of summation over histories of the system. The propagators obtained are, apart from an arbitrary phase factor, those of conventional quantum mechanics.

scholarly journals Explicit Formulas for All Scator Holomorphic Functions in the (1+2)-Dimensional Case

Symmetry ◽  
2020 ◽  
Vol 12 (9) ◽  
pp. 1550
Author(s):  
Jan L. Cieśliński ◽  
Dzianis Zhalukevich
Keyword(s):  
Vector Space ◽  
Special Relativity ◽  
Holomorphic Functions ◽  
Lorentz Symmetry ◽  
Dimensional Case ◽  
Hypercomplex Numbers ◽  
Explicit Formulas ◽  
Elliptic Case ◽  
Hyperbolic Case ◽  
All Solutions

Scators form a vector space endowed with a non-distributive product, in the hyperbolic case, have physical applications related to some deformations of special relativity (breaking the Lorentz symmetry) while the elliptic case leads to new examples of hypercomplex numbers and related notions of holomorphicity. Until now, only a few particular cases of scator holomorphic functions have been found. In this paper we obtain all solutions of the generalized Cauchy–Riemann system which describes analogues of holomorphic functions in the (1+2)-dimensional scator space.

Alternative formulation of quantum mechanics

1965 ◽  
Vol 61 (4) ◽  
pp. 917-921
Author(s):  
Brian Knight
Keyword(s):  
Quantum Mechanics ◽  
Harmonic Oscillator ◽  
Free Particle ◽  
Alternative Formulation ◽  
Operator Identity

AbstractThe two formulations of quantum mechanics in q-space only are compared by the construction of Schrödinger's operator identity. The examination in detail of the examples of the harmonic oscillator and free particle shows clearly the role of the re-normalization term in both formulations.

QUANTUM MECHANICS WITH p-ADIC NUMBERS

1989 ◽  
Vol 04 (19) ◽  
pp. 5133-5147 ◽  
Author(s):  
YANNICK MEURICE
Keyword(s):  
Hilbert Space ◽  
Quantum Mechanics ◽  
Harmonic Oscillator ◽  
Heisenberg Group ◽  
Canonical Transformation ◽  
Free Particle ◽  
Complex Hilbert Space ◽  
Canonical Transformations ◽  
Linear Canonical Transformation

We discuss unitary realizations of the Heisenberg group and the linear canonical transformations over a complex Hilbert space but with dynamical variables on a p-adic field Qp. For all p, an appropriate choice of phase turns the realization of the linear canonical transformation into a representation up to a sign of SL (2, Qp). We give the spectra of the subgroups corresponding to the free particle and the harmonic oscillator. We discuss briefly the possibility of an adelic interpretation.

Sound as a transverse wave

2017 ◽  
Vol 13 (1) ◽  
pp. 4522-4534
Author(s):  
Armando Tomás Canero
Keyword(s):  
Quantum Mechanics ◽  
Gravitational Waves ◽  
Longitudinal Wave ◽  
Transverse Wave ◽  
Sound Propagation ◽  
Correspondence Principle ◽  
Classical Mechanics ◽  
Wave Model ◽  
Scientific Experiments

This paper presents sound propagation based on a transverse wave model which does not collide with the interpretation of physical events based on the longitudinal wave model, but responds to the correspondence principle and allows interpreting a significant number of scientific experiments that do not follow the longitudinal wave model. Among the problems that are solved are: the interpretation of the location of nodes and antinodes in a Kundt tube of classical mechanics, the traslation of phonons in the vacuum interparticle of quantum mechanics and gravitational waves in relativistic mechanics.

scholarly journals IMPLICATIONS OF A QUANTUM MECHANICAL TREATMENT OF THE UNIVERSE

1998 ◽  
Vol 13 (05) ◽  
pp. 347-351 ◽  
Author(s):  
MURAT ÖZER
Keyword(s):  
Quantum Mechanics ◽  
Wave Packet ◽  
Early Universe ◽  
Mechanical Treatment ◽  
Correspondence Principle ◽  
Scale Factor ◽  
Quantum Mechanical ◽  
Quantum Mechanical Treatment ◽  
The Standard Model ◽  
The Universe

We attempt to treat the very early Universe according to quantum mechanics. Identifying the scale factor of the Universe with the width of the wave packet associated with it, we show that there cannot be an initial singularity and that the Universe expands. Invoking the correspondence principle, we obtain the scale factor of the Universe and demonstrate that the causality problem of the standard model is solved.

scholarly journals Chaotic dynamics of a free particle interacting linearly with a harmonic oscillator

2005 ◽  
Vol 208 (1-2) ◽  
pp. 96-114 ◽  
Author(s):  
Stephan De Bièvre ◽  
Paul E. Parris ◽  
Alex Silvius
Keyword(s):  
Harmonic Oscillator ◽  
Chaotic Dynamics ◽  
Free Particle

QUANTUM SUPERSPACE, q-EXTENDED SUPERSYMMETRY AND PARASUPERSYMMETRIC QUANTUM MECHANICS

1993 ◽  
Vol 08 (28) ◽  
pp. 2657-2670 ◽  
Author(s):  
K. N. ILINSKI ◽  
V. M. UZDIN
Keyword(s):  
Quantum Mechanics ◽  
Extended Supersymmetry ◽  
Harmonic Oscillator ◽  
Unit Circle ◽  
Canonical Quantization ◽  
Matrix Representations ◽  
Dimensional Matrix ◽  
Continuous Parameter ◽  
Superspace Formalism

We describe q-deformation of the extended supersymmetry and construct q-extended supersymmetric Hamiltonian. For this purpose we formulate q-superspace formalism and construct q-supertransformation group. On this basis q-extended supersymmetric Lagrangian is built. The canonical quantization of this system is considered. The connection with multi-dimensional matrix representations of the parasupersymmetric quantum mechanics is discussed and q-extended supersymmetric harmonic oscillator is considered as a simplest example of the described constructions. We show that extended supersymmetric Hamiltonians obey not only extended SUSY but also the whole family of symmetries (q-extended supersymmetry) which is parametrized by continuous parameter q on the unit circle.

scholarly journals Relativistic quantum mechanics

1932 ◽  
Vol 136 (829) ◽  
pp. 453-464 ◽  
Keyword(s):  
Quantum Mechanics ◽  
Quantum Theory ◽  
Classical Theory ◽  
Correspondence Principle ◽  
Relativistic Quantum ◽  
Classical Mechanics ◽  
Hamiltonian Function ◽  
Relativistic Quantum Mechanics ◽  
Classical Electrodynamics ◽  
Quantum Phenomena

The steady development of the quantum theory that has taken place during the present century was made possible only by continual reference to the Correspondence Principle of Bohr, according to which, classical theory can give valuable information about quantum phenomena in spite of the essential differences in the fundamental ideas of the two theories. A masterful advance was made by Heisenberg in 1925, who showed how equations of classical physics could be taken over in a formal way and made to apply to quantities of importance in quantum theory, thereby establishing the Correspondence Principle on a quantitative basis and laying the foundations of the new Quantum Mechanics. Heisenberg’s scheme was found to fit wonderfully well with the Hamiltonian theory of classical mechanics and enabled one to apply to quantum theory all the information that classical theory supplies, in so far as this information is consistent with the Hamiltonian form. Thus one was able to build up a satisfactory quantum mechanics for dealing with any dynamical system composed of interacting particles, provided the interaction could be expressed by means of an energy term to be added to the Hamiltonian function. This does not exhaust the sphere of usefulness of the classical theory. Classical electrodynamics, in its accurate (restricted) relativistic form, teaches us that the idea of an interaction energy between particles is only an approxi­mation and should be replaced by the idea of each particle emitting waves which travel outward with a finite velocity and influence the other particles in passing over them. We must find a way of taking over this new information into the quantum theory and must set up a relativistic quantum mechanics, before we can dispense with the Correspondence Principle.

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