Quantum uncertainty relation saturated by the eigenstates of the harmonic oscillator

Details Quantum uncertainty relation saturated by the eigenstates of the harmonic oscillator Quantum uncertainty relation saturated by the eigenstates of the harmonic oscillator

2012-01-02

arxiv.org/abs/1201.0453v1

Physics

Quantum Physics

6 pages, 1 figure

Scientific paper

We re-derive the Schr\"odinger-Robertson uncertainty principle for the position and momentum of a quantum particle in one dimension. Our derivation does not directly employ commutation relations, but works by reduction to an eigenvalue problem, 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 generalizes the well-known property that the (Gaussian) ground state of the harmonic oscillator saturates the uncertainty relation.

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