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Register a new userAlgebraic Solution of the Harmonic Oscillator With Minimal Length Uncertainty Relations
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In quantum mechanics with minimal length uncertainty relations the Heisenberg-Weyl algebra of the one-dimensional harmonic oscillator is a deformed SU(1,1) algebra. The eigenvalues and eigenstates are constructed algebraically and they form the infinite-dimensional representation of the deformed SU(1,1) algebra. Our construction is independent of prior knowledge of the exact solution of the Schrodinger equation of the model. The approach can be generalized to the $D$-dimensional oscillator with non-commuting coordinates.
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Using operator ordering techniques based on BCH-like relations of the su(1,1) Lie algebra and a time-splitting approach,we present an alternative method of solving the dynamics of a time-dependent quantum harmonic oscillator for any initial state. We find an iterative analytical solution given by simple recurrence relations that are very well suited for numerical calculations. We use our solution to reproduce and analyse some results from literature in order to prove the usefulness of the method and, based on these references, we discuss efficiency in squeezing, when comparing the parametric resonance modulation and the Janszky-Adam scheme.
We consider a harmonic oscillator (HO) with a time dependent frequency which undergoes two successive abrupt changes. By assumption, the HO starts in its fundamental state with frequency omega_{0}, then, at t = 0, its frequency suddenly increases to omega_{1} and, after a finite time interval tau, it comes back to its original value omega_{0}. Contrary to what one could naively think, this problem is a quite non-trivial one. Using algebraic methods we obtain its exact analytical solution and show that at any time t > 0 the HO is in a squeezed state. We compute explicitly the corresponding squeezing parameter (SP) relative to the initial state at an arbitrary instant and show that, surprisingly, it exhibits oscillations after the first frequency jump (from omega_{0} to omega_{1}), remaining constant after the second jump (from omega_{1} back to omega_{0}). We also compute the time evolution of the variance of a quadrature. Last, but not least, we calculate the vacuum (fundamental state) persistence probability amplitude of the HO, as well as its transition probability amplitude for any excited state.
We solve the generalized relativistic harmonic oscillator in 1+1 dimensions in the presence of a minimal length. Using the momentum space representation, we explore all the possible signs of the potentials and discuss their bound-state solutions for fermion and antifermions. Furthermore, we also find an isolated solution from the Sturm-Liouville scheme. All cases already analyzed in the literature, are obtained as particular cases.
In the framework of the generalized uncertainty principle, the position and momentum operators obey the modified commutation relation $[X,P]=ihbarleft(1+beta P^2right)$ where $beta$ is the deformation parameter. Since the validity of the uncertainty relation for the Shannon entropies proposed by Beckner, Bialynicki-Birula, and Mycieslki (BBM) depends on both the algebra and the used representation, we show that using the formally self-adjoint representation, i.e., $X=x$ and $P=tanleft(sqrt{beta}pright)/sqrt{beta}$ where $[x,p]=ihbar$, the BBM inequality is still valid in the form $S_x+S_pgeq1+lnpi$ as well as in ordinary quantum mechanics. We explicitly indicate this result for the harmonic oscillator in the presence of the minimal length.
We re-derive the Schr{o}dinger-Robertson uncertainty principle for the position and momentum of a quantum particle. Our derivation does not directly employ commutation relations, but works by reduction to an eigenvalue problem related to the harmonic oscillator, which can then be further exploited to find a larger class of constrained uncertainty relations. We derive an uncertainty relation under the constraint of a fixed degree of Gaussianity and prove that, remarkably, it is saturated by all eigenstates of the harmonic oscillator. This goes beyond the common knowledge that the (Gaussian) ground state of the harmonic oscillator saturates the uncertainty relation.
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