Path integral for the harmonic oscillator in one dimension with nonzero minimum position uncertainty

2006 ◽  
Vol 354 (5-6) ◽  
pp. 399-405 ◽  
Author(s):  
Khireddine Nouicer
Keyword(s):  
Harmonic Oscillator ◽  
Path Integral ◽  
One Dimension ◽  
Position Uncertainty ◽  
Minimum Position

Path integral for the damped harmonic oscillator coupled to its dual

1994 ◽  
Vol 35 (3) ◽  
pp. 1185-1191 ◽  
Author(s):  
L. Chetouani ◽  
L. Guechi ◽  
T. F. Hammann ◽  
M. Letlout
Keyword(s):  
Harmonic Oscillator ◽  
Path Integral ◽  
Damped Harmonic Oscillator

scholarly journals EXTENDED FEYNMAN FORMULA FOR THE HARMONIC OSCILLATOR BY THE DISCRETE TIME METHOD

2010 ◽  
Vol 25 (03) ◽  
pp. 179-188
Author(s):  
KUNIO FUNAHASHI
Keyword(s):  
Harmonic Oscillator ◽  
Discrete Time ◽  
Path Integral ◽  
Feynman Formula ◽  
Discrete Formulation ◽  
Rigorous Method

We revisit the extended Feynman formula for the harmonic oscillator beyond and at caustics. The extension has been made by some authors, however, it is not obtained by the discrete formulation of path integral, which we consider the most reliable regularization of it. We derive the result by, especially at caustics, more rigorous method than previous.

From Classical to Quantum Mechanics

2021 ◽  
pp. 207-219
Author(s):  
J. Iliopoulos ◽  
T.N. Tomaras
Keyword(s):  
Quantum Mechanics ◽  
Quantum Theory ◽  
Harmonic Oscillator ◽  
Path Integral ◽  
Classical Theory ◽  
Constrained Systems ◽  
Experimental Results ◽  
Basic Assumptions ◽  
Original Approach

In Chapter 2 we presented the method of canonical quantisation which yields a quantum theory if we know the corresponding classical theory. In this chapter we argue that this method is not unique and, furthermore, it has several drawbacks. In particular, its application to constrained systems is often problematic. We present Feynman’s path integral quantisation method and derive from it Schroödinger’s equation. We follow Feynman’s original approach and we present, in addition, more recent experimental results which support the basic assumptions. We establish the equivalence between canonical and path integral quantisation of the harmonic oscillator.

ENERGY-DEPENDENT POTENTIAL AND NORMALIZATION OF WAVE FUNCTION

2013 ◽  
Vol 28 (18) ◽  
pp. 1350079 ◽  
Author(s):  
A. BENCHIKHA ◽  
L. CHETOUANI
Keyword(s):  
Wave Function ◽  
Hydrogen Atom ◽  
Harmonic Oscillator ◽  
Path Integral ◽  
Spectral Decomposition ◽  
Wave Functions ◽  
Integral Approach ◽  
Path Integral Approach ◽  
Dependent Potential ◽  
Energy Dependent

The problem of normalization related to energy-dependent potentials is examined in the context of the path integral approach, and a justification is given. As examples, the harmonic oscillator and the hydrogen atom (radial) where, respectively the frequency and the Coulomb's constant depend on energy, are considered and their propagators determined. From their spectral decomposition, we have found that the wave functions extracted are correctly normalized.

Computational Power of Quantum Machines, Quantum Grammars and Feasible Computation

1998 ◽  
Vol 09 (02) ◽  
pp. 213-241 ◽  
Author(s):  
E. V. Krishnamurthy
Keyword(s):  
Harmonic Oscillator ◽  
Turing Machine ◽  
Path Integral ◽  
Quantum Mechanical ◽  
Polynomial Space ◽  
Computational Power ◽  
Computational Point ◽  
Quantum Dynamical ◽  
Special Cases ◽  
Quadratic Potentials

This paper studies the computational power of quantum computers to explore as to whether they can recognize properties which are in nondeterministic polynomial-time class (NP) and beyond. To study the computational power, we use the Feynman's path integral (FPI) formulation of quantum mechanics. From a computational point of view the Feynman's path integral computes a quantum dynamical analogue of the k-ary relation computed by an Alternating Turing machine (ATM) using AND-OR Parallelism. Hence, if we can find a suitable mapping function between an instance of a mathematical problem and the corresponding interference problem, using suitable potential functions for which FPI can be integrated exactly, the computational power of a quantum computer can be bounded to that of an alternating Turing machine that can solve problems in NP (e.g, factorization problem) and in polynomial space. Unfortunately, FPI is exactly integrable only for a few problems (e.g., the harmonic oscillator) involving quadratic potentials; otherwise, they may be only approximately computable or noncomputable. This means we cannot in general solve all quantum dynamical problems exactly except for those special cases of quadratic potentials, e.g., harmonic oscillator. Since there is a one to one correspondence between the quantum mechanical problems that can be analytically solved and the path integrals that can be exactly evaluated, we can say that the noncomputability of FPI implies quantum unsolvability. This is the analogue of classical unsolvability. The Feynman's path graph can be considered as a semantic parse graph for the quantum mechanical sentence. It provides a semantic valuation function of the terminal sentence based on probability amplitudes to disambiguate a given quantum description and obtain an interpretation in a linear time. In Feynman's path integral, the kernels are partially ordered over time (different alternate paths acting concurrently at the same time) and multiplied. The semantic valuation is computable only if the FPI is computable. Thus both the expressive power and complexity aspects quantum computing are mirrored by the exact and efficient integrability of FPI.

scholarly journals (1+1)-Dirac bound states in one dimension, with position-dependent Fermi velocity and mass

Open Physics ◽  
2013 ◽  
Vol 11 (4) ◽  
Author(s):  
Omar Mustafa
Keyword(s):  
Harmonic Oscillator ◽  
Bound States ◽  
Bound State ◽  
One Dimension ◽  
Fermi Velocity ◽  
Dirac Particles ◽  
The Continuum ◽  
Oscillator Models

AbstractWe extend Panella and Roy’s [17] work for massless Dirac particles with position-dependent (PD) velocity. We consider Dirac particles where the mass and velocity are both position-dependent. Bound states in the continuum (BIC)-like and discrete bound state solutions are reported. It is observed that BIC-like solutions are not only feasible for the ultra-relativistic (massless) Dirac particles but also for Dirac particles with PDmass and PD-velocity that satisfy the condition m(x) v F2 (x) = A, where A ≥ 0 is constant. Dirac Pöschl-Teller and harmonic oscillator models are also reported.

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