scholarly journals Studying Heisenberg-like Uncertainty Relation with Weak Values in One-dimensional Harmonic Oscillator

2022 ◽  
Vol 9 ◽  
Author(s):  
Xing-Yan Fan ◽  
Wei-Min Shang ◽  
Jie Zhou ◽  
Hui-Xian Meng ◽  
Jing-Ling Chen
Keyword(s):  
Lower Bound ◽  
Harmonic Oscillator ◽  
Coherent States ◽  
Uncertainty Relation ◽  
Quantum States ◽  
Weak Values ◽  
Mean Values ◽  
One Dimensional ◽  
The Mean ◽  
Arbitrary Quantum

As one of the fundamental traits governing the operation of quantum world, the uncertainty relation, from the perspective of Heisenberg, rules the minimum deviation of two incompatible observations for arbitrary quantum states. Notwithstanding, the original measurements appeared in Heisenberg’s principle are strong such that they may disturb the quantum system itself. Hence an intriguing question is raised: What will happen if the mean values are replaced by weak values in Heisenberg’s uncertainty relation? In this work, we investigate the question in the case of measuring position and momentum in a simple harmonic oscillator via designating one of the eigenkets thereof to the pre-selected state. Astonishingly, the original Heisenberg limit is broken for some post-selected states, designed as a superposition of the pre-selected state and another eigenkets of harmonic oscillator. Moreover, if two distinct coherent states reside in the pre- and post-selected states respectively, the variance reaches the lower bound in common uncertainty principle all the while, which is in accord with the circumstance in Heisenberg’s primitive framework.

PERELOMOV AND BARUT–GIRARDELLO SU(1, 1) COHERENT STATES FOR HARMONIC OSCILLATOR IN ONE-DIMENSIONAL HALF SPACE

2006 ◽  
Vol 21 (12) ◽  
pp. 2635-2644 ◽  
Author(s):  
Q. H. LIU ◽  
H. ZHUO
Keyword(s):  
Coherent State ◽  
Harmonic Oscillator ◽  
Half Space ◽  
Ground State ◽  
Classical Limit ◽  
Coherent States ◽  
Mean Values ◽  
One Dimensional ◽  
The Mean

The Perelomov and the Barut–Girardello SU(1, 1) coherent states for harmonic oscillator in one-dimensional half space are constructed. Results show that the uncertainty products ΔxΔp for these two coherent states are bound from below [Formula: see text] that is the uncertainty for the ground state, and the mean values for position x and momentum p in classical limit go over to their classical quantities respectively. In classical limit, the uncertainty given by Perelomov coherent does not vanish, and the Barut–Girardello coherent state reveals a node structure when positioning closest to the boundary x = 0 which has not been observed in coherent states for other systems.

scholarly journals A CLASS OF QUANTUM STATES WITH CLASSICAL-LIKE EVOLUTION

1994 ◽  
Vol 08 (29) ◽  
pp. 1823-1831 ◽  
Author(s):  
SALVATORE DE MARTINO ◽  
SILVIO DE SIENA ◽  
FABRIZIO ILLUMINATI
Keyword(s):  
Quantum Mechanics ◽  
Harmonic Oscillator ◽  
Coherent States ◽  
Quantum States ◽  
Time Dependent ◽  
Stationary States ◽  
Oscillator Potential ◽  
Classical Motion ◽  
Stochastic Formulation ◽  
New States

In the framework of the stochastic formulation of quantum mechanics we derive non-stationary states for a class of time-dependent potentials. The wave packets follow a classical motion with constant dispersion. The new states define a possible extension of the harmonic oscillator coherent states. As an explicit application, we study a sestic oscillator potential.

UNCERTAINTY RELATIONS FOR THE TIME-DEPENDENT SINGULAR OSCILLATOR

2006 ◽  
Vol 20 (10) ◽  
pp. 1211-1231 ◽  
Author(s):  
J. R. CHOI ◽  
I. H. NAHM
Keyword(s):  
Harmonic Oscillator ◽  
Excited States ◽  
Ground State ◽  
Coherent States ◽  
System Approach ◽  
Uncertainty Relation ◽  
Time Dependent ◽  
Uncertainty Relations ◽  
Number State ◽  
State Uncertainty

Uncertainty relations for the time-dependent singular oscillator in the number state and in the coherent state are investigated. We applied our developement to the Caldirola–Kanai oscillator perturbed by a singularity. For this system, the variation (Δx) decreased exponentially while (Δp) increased exponentially with time both in the number and in the coherent states. As k → 0 and χ → 0, the number state uncertainty relation in the ground state becomes 0.583216ℏ which is somewhat larger than that of the standard harmonic oscillator, ℏ/2. On the other hand, the uncertainty relation in all excited states become smaller than that of the standard harmonic oscillator with the same quantum number n. However, as k → ∞ and χ → 0, the uncertainty relations of the system approach the uncertainty relations of the standard harmonic oscillator, (n+1/2)ℏ.

scholarly journals A COHERENT STATE PATH INTEGRAL FOR ANYONS

1995 ◽  
Vol 10 (12) ◽  
pp. 985-989 ◽  
Author(s):  
J. GRUNDBERG ◽  
T.H. HANSSON
Keyword(s):  
Coherent State ◽  
Harmonic Oscillator ◽  
Path Integral ◽  
Coherent States ◽  
Integral Formula ◽  
Harmonic Potential ◽  
One Dimensional ◽  
Change Of Variables ◽  
The One ◽  
State Path

We derive an su (1, 1) coherent state path integral formula for a system of two one-dimensional anyons in a harmonic potential. By a change of variables we transform this integral into a coherent states path integral for a harmonic oscillator with a shifted energy. The shift is the same as the one obtained for anyons by other methods. We justify the procedure by showing that the change of variables corresponds to an su (1, 1) version of the Holstein-Primakoff transformation.

scholarly journals COMPLEXIFIER VERSUS FACTORIZATION AND DEFORMATION METHODS FOR GENERATION OF COHERENT STATES OF A 1D NLHO I: MATHEMATICAL CONSTRUCTION

2013 ◽  
Vol 10 (10) ◽  
pp. 1350056 ◽  
Author(s):  
R. ROKNIZADEH ◽  
H. HEYDARI
Keyword(s):  
Exact Solution ◽  
Schrödinger Equation ◽  
Harmonic Oscillator ◽  
Coherent States ◽  
Schrodinger Equation ◽  
Jacobi Polynomials ◽  
One Dimensional ◽  
The Difference ◽  
Mathematical Construction

Three methods: complexifier, factorization and deformation, for construction of coherent states are presented for one-dimensional nonlinear harmonic oscillator (1D NLHO). Since by exploring the Jacobi polynomials [Formula: see text], bridging the difference between them is possible, we give here also the exact solution of Schrödinger equation of 1D NLHO in terms of Jacobi polynomials.

EXACT SOLUTIONS, LADDER OPERATORS AND BARUT–GIRARDELLO COHERENT STATES FOR A HARMONIC OSCILLATOR PLUS AN INVERSE SQUARE POTENTIAL

2005 ◽  
Vol 19 (28) ◽  
pp. 4219-4227 ◽  
Author(s):  
SHI-HAI DONG ◽  
M. LOZADA-CASSOU
Keyword(s):  
Harmonic Oscillator ◽  
Exact Solutions ◽  
Coherent States ◽  
Factorization Method ◽  
Ladder Operators ◽  
Dynamical Group ◽  
One Dimensional ◽  
Hidden Symmetry ◽  
The One ◽  
Analytical Expressions

We present exact solutions of the one-dimensional Schrödinger equation with a harmonic oscillator plus an inverse square potential. The ladder operators are constructed by the factorization method. We find that these operators satisfy the commutation relations of the generators of the dynamical group SU(1, 1). Based on those ladder operators, we obtain the analytical expressions of matrix elements for some related functions ρ and [Formula: see text] with ρ=x2. Finally, we make some comments on the Barut–Girardello coherent states and the hidden symmetry between E(x) and E(ix) by substituting x→ix.

scholarly journals New mathematics for the nonadditive Tsallis’ scenario

2017 ◽  
Vol 31 (21) ◽  
pp. 1750151 ◽  
Author(s):  
G. L. Ferri ◽  
F. Pennini ◽  
A. plastino ◽  
M. C. Rocca
Keyword(s):  
Harmonic Oscillator ◽  
Coherent States ◽  
Probability Distributions ◽  
Mean Values

In this paper, we investigate quantum uncertainties in a Tsallis’ nonadditive scenario. To such an end we appeal to [Formula: see text]-exponentials (qEs), that are the cornerstone of Tsallis’ theory. In this respect, it is found that some new mathematics is needed and we are led to construct a set of novel special states that are the qE equivalents of the ordinary coherent states (CS) of the harmonic oscillator (HO). We then characterize these new Tsallis’ special states by obtaining the associated (i) probability distributions (PDs) for a state of momentum [Formula: see text], (ii) mean values for some functions of space an momenta and (iii) concomitant quantum uncertainties. The latter are then compared to the usual ones.

A COMPARISON BETWEEN WEHRL–LIEB AND VON NEUMANN ENTROPIES FOR TIME-EVOLVING SPIN-1/2 MIXED STATES

1994 ◽  
Vol 08 (16) ◽  
pp. 995-1006 ◽  
Author(s):  
S. S. MIZRAHI ◽  
V. V. DODONOV ◽  
D. OTERO
Keyword(s):  
Harmonic Oscillator ◽  
Coherent States ◽  
Quantum States ◽  
Alternative Formulation ◽  
Spin States ◽  
Continuous Representation ◽  
Commun Math Phys ◽  
Mixed States ◽  
Von Neumann ◽  
Math Phys

Years ago, A. Wehrl (Rev. Mod. Phys.50, 221 (1978)) introduced the concept of classicallike entropy of quantum states when a two-label continuous representation is used; for instance, the harmonic oscillator coherent states. Subsequently, E. H. Lieb (Commun. Math. Phys.62, 35 (1978)) extended that concept of entropy to the Bloch coherent spin states. Here, we consider spin-1/2 systems and calculate both the Wehrl–Lieb and von Neumann entropies, and then we compare the results and discuss the Wehrl–Lieb entropy as an alternative formulation to von Neumann's. As illustration, three examples are worked out: (i) the decoherence of a quantum state in a measurement process, (ii) the conservation of coherence, and (iii) the recoherence phenomena that appear in the solutions of a specific master equation that originates from a nonlinear Schrödinger equation.

scholarly journals Local States in One-Dimensional Symmetrical Quantum Systems

1979 ◽  
Vol 34 (12) ◽  
pp. 1452-1457 ◽  
Author(s):  
Jürgen Brickmann
Keyword(s):  
Half Space ◽  
Uncertainty Relation ◽  
Quantum Systems ◽  
Quantum States ◽  
Uncertainty Measure ◽  
One Dimensional ◽  
Quantum Dynamical ◽  
Local Quantum ◽  
Local Properties ◽  
Energy Moments

Abstract Local quantum states, which play an important role in quantum dynamical treatments, are expanded analytically with respect to a basis of eigen functions of a symmetrical Hamiltonian ℋ̂(x) = ℋ̂(- x). Exact local states (ELS) in one-dimensional symmetrical quantum systems are therein defined as quantum states which are local eigenstates of the Hamiltonian ℋ̂(x) on one half space ℝ+ or ℝ- and are identically equal to zero on the other half space. Local properties like the projection operator on one half space can be given in terms of ELS-basis, but it is shown that the energy moments 〈(〈ℋ̂ 〉 - 〈ℋ̂)k〉 with respect to the ELS do not converge. Consequently, if one uses the ELS as quasistationary initial states, as has been done recently by some authors [5], the lifetimes of these states cannot be estimated from time energy uncertainty relation using the second energy moment as an energy uncertainty measure. A harmonic oscillator system and a symmetrical double oscillator are treated as examples.

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