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 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.

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.

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 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 Generation of Time-Independent and Time-Dependent Harmonic Oscillator-Like Potentials Using Supersymmetric Quantum Mechanics

2018 ◽  
Vol 4 (1) ◽  
pp. 47-55
Author(s):  
Timothy Brian Huber
Keyword(s):  
Quantum Mechanics ◽  
Schrödinger Equation ◽  
Harmonic Oscillator ◽  
Mechanical System ◽  
Schrodinger Equation ◽  
Supersymmetric Quantum Mechanics ◽  
Time Dependent ◽  
Quantum Mechanical ◽  
Quantum Mechanical System ◽  
Intertwining Operator

The harmonic oscillator is a quantum mechanical system that represents one of the most basic potentials. In order to understand the behavior of a particle within this system, the time-independent Schrödinger equation was solved; in other words, its eigenfunctions and eigenvalues were found. The first goal of this study was to construct a family of single parameter potentials and corresponding eigenfunctions with a spectrum similar to that of the harmonic oscillator. This task was achieved by means of supersymmetric quantum mechanics, which utilizes an intertwining operator that relates a known Hamiltonian with another whose potential is to be built. Secondly, a generalization of the technique was used to work with the time-dependent Schrödinger equation to construct new potentials and corresponding solutions.

scholarly journals WKB FORMALISM AND A LOWER LIMIT FOR THE ENERGY EIGENSTATES OF BOUND STATES FOR SOME POTENTIALS

2007 ◽  
Vol 22 (35) ◽  
pp. 2675-2687 ◽  
Author(s):  
LUIS F. BARRAGÁN-GIL ◽  
ABEL CAMACHO
Keyword(s):  
Quantum Mechanics ◽  
Gravitational Field ◽  
Lower Bound ◽  
Harmonic Oscillator ◽  
Approximation Method ◽  
Bound States ◽  
The Other ◽  
Homogeneous Gravitational Field ◽  
Definition Of ◽  
Ground Energy

In this work the conditions appearing in the so-called WKB approximation formalism of quantum mechanics are analyzed. It is shown that, in general, a careful definition of an approximation method requires the introduction of two length parameters, one of them always considered in the textbooks on quantum mechanics, whereas the other is usually neglected. Afterwards we define a particular family of potentials and prove, resorting to the aforementioned length parameters, that we may find an energy which is a lower bound to the ground energy of the system. The idea is applied to the case of a harmonic oscillator and also to a particle freely falling in a homogeneous gravitational field, and in both cases the consistency of our method is corroborated. This approach, together with the so-called Rayleigh–Ritz formalism, allows us to define an energy interval in which the ground energy of any potential, belonging to our family, must lie.

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