Exact lower bound of the uncertainty principle product for the harmonic oscillator with position-momentum coupling

• URL: http://arxiv.org/abs/2402.07842v1
• Date: Mon, 12 Feb 2024 17:52:47 GMT
• Title: Exact lower bound of the uncertainty principle product for the harmonic oscillator with position-momentum coupling
• Authors: Yamen Hamdouni
• Abstract summary: We show that the uncertainty principle product for the position and momentum operators for a system described by the Hamiltonian $ hat H= frachatp22m +frac12 m omega2 hatx2+fracmu2(hat x hat p+ hat p hat x)$ where $muomega$ reads $Delta x Delta pgefrachbar omega2sqrtomega2-mu
• Score: 0.0
• License: http://creativecommons.org/licenses/by/4.0/
• Abstract: We show that the uncertainty principle product for the position and momentum operators for a system described by the Hamiltonian $ \hat H= \frac{\hat{p}^2}{2m} +\frac{1}{2} m \omega^2 \hat{x}^2+\frac{\mu}{2}(\hat x \hat p+ \hat p \hat x)$ where $\mu<\omega$ reads $\Delta x \Delta p\ge\frac{\hbar \omega}{2\sqrt{\omega^2-\mu^2}}$. All the values bellow this lower bound are thus quantum-mechanically forbidden. We construct the annihilation and creation operators for this system and we calculate the expectation values of the operators $\hat p$ and $\hat x$ with respect to the corresponding coherent states.

Related papers

• Generalized quantum Zernike Hamiltonians: Polynomial Higgs-type algebras and algebraic derivation of the spectrum [0.0]
  We consider the quantum analog of the generalized Zernike systems given by the Hamiltonian.<n>This two-dimensional quantum model exhibits higher-order integrals of motion within the enveloping algebra of the Heisenberg algebra.
  arXiv  Detail & Related papers  (2025-02-04T17:06:54Z)
• Isochronous oscillator with a singular position-dependent mass and its quantization [6.412262542272846]
  We present an analysis of the equation $ddotx - (1/2x) dotx2 + 2 omega2 x - 1/8x = 0$, where $omega > 0$ and $x = x(t)$ is a real-valued variable. The dynamics is exactly solvable for both the branches and so for definiteness, we stick to the $x > 0$ branch.
  arXiv  Detail & Related papers  (2025-01-14T20:33:16Z)
• Fermi's golden rule in tunneling models with quantum waveguides perturbed by Kato class measures [0.0]
  We show that the resolvent of the Hamiltonian has the second pole which reproduces the resonance at $z(rho)$ with the perturbations $z(rho)=mathcal E_beta ; n+mathcal O Big(frac exp(-sqrt2 |mathcal E_beta ;n| rho )rho Big)$ for $rho$ large and with the resonant energy $mathcal E_beta ;n$.
  arXiv  Detail & Related papers  (2024-12-16T17:37:27Z)
• Generalized Liénard systems with momentum-dependent mass: Isochronicity and bound states [6.412262542272846]
  We explore some classical and quantum aspects of the nonlinear Li'enard equation $ddotx + k x dotx + omega2 x + (k2/9) x3 = 0$. We demonstrate that such an equation could be derived from an equation of the Levinson-Smith kind which is of the form $ddotz + J(z) dotz2 + F(z) dotz + G(z) = 0$.
  arXiv  Detail & Related papers  (2024-12-12T09:27:57Z)
• Dimension Independent Disentanglers from Unentanglement and Applications [55.86191108738564]
  We construct a dimension-independent k-partite disentangler (like) channel from bipartite unentangled input. We show that to capture NEXP, it suffices to have unentangled proofs of the form $| psi rangle = sqrta | sqrt1-a | psi_+ rangle where $| psi_+ rangle has non-negative amplitudes.
  arXiv  Detail & Related papers  (2024-02-23T12:22:03Z)
• On the $O(\rac{\sqrt{d}}{T^{1/4}})$ Convergence Rate of RMSProp and Its Momentum Extension Measured by $\ell_1$ Norm [54.28350823319057]
  This paper considers the RMSProp and its momentum extension and establishes the convergence rate of $frac1Tsum_k=1T.<n>Our convergence rate matches the lower bound with respect to all the coefficients except the dimension $d$.<n>Our convergence rate can be considered to be analogous to the $frac1Tsum_k=1T.
  arXiv  Detail & Related papers  (2024-02-01T07:21:32Z)
• Small-time controllability for the nonlinear Schr\"odinger equation on $\mathbb{R}^N$ via bilinear electromagnetic fields [55.2480439325792]
  We address the small-time controllability problem for a nonlinear Schr"odinger equation (NLS) on $mathbbRN$ in the presence of magnetic and electric external fields. In detail, we study when it is possible to control the dynamics of (NLS) as fast as desired via sufficiently large control signals.
  arXiv  Detail & Related papers  (2023-07-28T21:30:44Z)
• Rigorous derivation of the Efimov effect in a simple model [68.8204255655161]
  We consider a system of three identical bosons in $mathbbR3$ with two-body zero-range interactions and a three-body hard-core repulsion of a given radius $a>0$.
  arXiv  Detail & Related papers  (2023-06-21T10:11:28Z)
• Contactium: A strongly correlated model system [0.0]
  We study the one-particle description of a system that comprises two fermions or bosons in a confinement interacting through the Fermi--Huang pseudopotential. A detailed analysis of the one-particle description reveals several peculiarities that are not encountered in conventional model systems.
  arXiv  Detail & Related papers  (2023-03-27T08:27:33Z)
• Concentration Phenomenon for Random Dynamical Systems: An Operator Theoretic Approach [10.051309746913512]
  This paper formalizes the concentration phenomenon for a given observable. It turns out that even if the observable/ reward function is unbounded, but for some. for some $q>2$, $|er|_q. The role of emphreversibility in concentration phenomenon is demystified.
  arXiv  Detail & Related papers  (2022-12-07T14:36:46Z)
• On parametric resonance in the laser action [91.3755431537592]
  We consider the selfconsistent semiclassical Maxwell--Schr"odinger system for the solid state laser. We introduce the corresponding Poincar'e map $P$ and consider the differential $DP(Y0)$ at suitable stationary state $Y0$.
  arXiv  Detail & Related papers  (2022-08-22T09:43:57Z)
• Some Remarks on the Regularized Hamiltonian for Three Bosons with Contact Interactions [77.34726150561087]
  We discuss some properties of a model Hamiltonian for a system of three bosons interacting via zero-range forces in three dimensions. In particular, starting from a suitable quadratic form $Q$, the self-adjoint and bounded from below Hamiltonian $mathcal H$ can be constructed. We show that the threshold value $gamma_c$ is optimal, in the sense that the quadratic form $Q$ is unbounded from below if $gammagamma_c$.
  arXiv  Detail & Related papers  (2022-07-01T10:01:14Z)
• From quartic anharmonic oscillator to double well potential [77.34726150561087]
  It is shown that by taking uniformly-accurate approximation for anharmonic oscillator eigenfunction $Psi_ao(u)$, obtained recently, it is possible to get highly accurate approximation for both the eigenfunctions of the double-well potential and its eigenvalues.
  arXiv  Detail & Related papers  (2021-10-30T20:16:27Z)
• On connection between perturbation theory and semiclassical expansion in quantum mechanics [0.0]
  Perturbation Theory in powers of the coupling constant $g$ and the semiclassical expansion in powers of $hbar1/2$ for the energies coincide. The equations which govern dynamics in two spaces, the Riccati-Bloch equation and the Generalized Bloch equation, respectively, are presented.
  arXiv  Detail & Related papers  (2021-02-09T03:13:56Z)
• Note on a Product Formula Related to Quantum Zeno Dynamics [0.0]
  We prove that $lim_nrightarrow infty (P,mathrme-itH/nP)n = mathrme-itH_PP$ holds in the strong operator topology. We derive modifications of this product formula and its extension to the situation when $P$ is replaced by a strongly continuous projection-valued function satisfying $P(0)=P$.
  arXiv  Detail & Related papers  (2020-12-30T07:04:41Z)
• Anharmonic oscillator: a solution [77.34726150561087]
  The dynamics in $x$-space and in $(gx)-space corresponds to the same energy spectrum with effective coupling constant $hbar g2$. A 2-classical generalization leads to a uniform approximation of the wavefunction in $x$-space with unprecedented accuracy.
  arXiv  Detail & Related papers  (2020-11-29T22:13:08Z)
• Unitary unfoldings of Bose-Hubbard exceptional point with and without particle number conservation [0.0]
  Non-Hermitian but $cal PT-$symmetric quantum system of an $N-$plet of bosons described by Bose-Hubbard Hamiltonian $H(gamma,v,c)$ is picked up. Non-Hermitian but $cal PT-$symmetric quantum system of an $N-$plet of bosons described by the three-parametric Bose-Hubbard Hamiltonian $H(gamma,v,c)$ is picked up.
  arXiv  Detail & Related papers  (2020-08-28T21:06:17Z)
• $\mathcal{PT}$ symmetry of a square-wave modulated two-level system [23.303857456199328]
  We study a non-Hermitian two-level system with square-wave modulated dissipation and coupling. Based on the Floquet theory, we achieve an effective Hamiltonian from which the boundaries of the $mathcalPT$ phase diagram are captured.
  arXiv  Detail & Related papers  (2020-08-17T03:18:36Z)

This list is automatically generated from the titles and abstracts of the papers in this site.

This site does not guarantee the quality of this site (including all information) and is not responsible for any consequences.