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arXiv:2402.07842

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

Yamen Hamdouni

Published Feb 12, 2024• 0 citations
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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 ω^2 \hat{x}^2+\fracμ{2}(\hat x \hat p+ \hat p \hat x)$ where $μ<ω$ reads $Δx Δp\ge\frac{\hbar ω}{2\sqrt{ω^2-μ^2}}$. 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.

DOI
10.48550/arxiv.2402.07842

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This is a pedagogical paper where we present a physically motivated approach to introduce the coherent states of a harmonic oscillator from which it is simple to rigorously derive their mathematical definition. We do this in two different ways that turn out to be equivalent but emphasize two related but different aspects of classicality. First, we analyze which are the quantum states that are the closest one can get to a point in phase space and demonstrate the validity of the following theorem: (i) The product of the uncertainty in position and that of momentum saturates the bound imposed by Heisenberg uncertainty relations for all times if and only if the state is an eigenstate of the annihilation operator. Second, we analyze the way in which the difference between the expectation value of the energy and the energy associated with the expectation values of position and momentum depends on the state, and show the validity of the following theorem (ii) the difference between the expectation value of the energy and the energy associated with the expectation values is minimal if and only if the state is an eigenstate of the annihilation operator. We also show that the reason why coherent states are chosen as the most classical ones by the decoherence process induced by coupling the particle to an environment in the standard Quantum Brownian motion model, is precisely due to the validity of the two above theorems.
Some of the most enduring questions in physics--including the quantum measurement problem and the quantization of gravity--involve the interaction of a quantum system with a classical environment. Two linearly coupled harmonic oscillators provide a simple, exactly soluble model for exploring such interaction. Even the ground state of a pair of identical oscillators exhibits effects on the quantum nature of one oscillator, e.g., a diminution of position uncertainty, and an increase in momentum uncertainty and uncertainty product, from their unperturbed values. Interaction between quantum and classical oscillators is simulated by constructing a quantum state with one oscillator initially in its ground state, the other in a coherent or Glauber state. The subsequent wave function for this state is calculated exactly, both for identical and distinct oscillators. The reduced probability distribution for the quantum oscillator, and its position and momentum expectation values and uncertainties, are obtained from this wave function. The oscillator acquires an oscillation amplitude corresponding to a beating between the normal modes of the system; the behavior of the position and momentum uncertainties can become quite complicated. For oscillators with equal unperturbed frequencies, i.e., at resonance, the uncertainties exhibit a time-dependent quantum squeezing which can be extreme.
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Calculates expectation values and uncertainties of the radial coordinate and momentum, and the azimuthal angle and angular momentum components for a three-dimensional harmonic oscillator in a coherent state. The operators corresponding to powers of the angular variable are defined by their Fourier expansions. For large distances of the particle from the origin, where the oscillator becomes classical, radial and angular uncertainty products approach the value h(cross)/ square root 2, except for the point of discontinuity, where the angular uncertainty becomes infinite. For small distances, where the oscillator tends to its quantum ground state, the radial uncertainty stays finite and the angular uncertainty product tends to zero. It is also shown that the concept of a polar momentum operator is meaningless.
The Heisenberg uncertainty relation, concerning the product of the standard deviations of position and momentum, is studied for a quantum oscillator characterized by deformed boson algebra. Special forms of the deformed number operator are determined such that the ground state and any deformed coherent state of the oscillator under study exhibit minimum uncertainty, as the canonical boson algebra provides, but the product of the corresponding standard deviations is less than the canonical lower bound, ℏ/2. Above all, each standard deviation and, consequently, the uncertainty become arbitrarily small for the ground state, other states of the Fock basis, and deformed coherent states with arbitrarily small magnitude of the label, if the parameters defining the special deformed number operator are properly chosen.