The Uncertainty Principle in Terms of Isoperimetric Inequalities

Thomas Schürmann

doi:10.4236/am.2017.83025

Abstract

Simultaneous measurements of position and momentum are considered in $n$ dimensions. We find, that for a particle whose position is strictly localized in a compact domain $D\subset \mathbb{R}^n$ (spatial uncertainty) with non-empty boundary, the standard deviation of its momentum is sharply bounded by $\sigma_p \geq \lambda_1^{1/2}\hbar$, while $\lambda_1$ is the first Dirichlet eigenvalue of the Laplacian on $D$.

Highlights

(spatial uncertainty) with non-empty boundary, the standard deviation of its momentum is sharply bounded by σ p ≥ λ11/2, while λ1 is the first Dirichlet eigenvalue of the Laplacian on D
The most familiar formalization of the uncertainty principle of position and momentum is in terms of standard deviations [1] [2] [3]
In the corresponding measurement process, the standard deviation of the position σ x is measured for a sample of particles initially prepared in a state ψ

Summary

Open Access

The most familiar formalization of the uncertainty principle of position and momentum is in terms of standard deviations [1] [2] [3]. The corresponding measurement process is as follows: Whenever a particle is strictly localized in a finite interval of length ∆x > 0 with probability 1, the standard deviation of its momentum satisfies the inequality σ p∆x ≥ π. In the quantum mechanical case under consideration the wave-function is zero at the boundary of the compact domain such that the equal sign of Popoviciu’s inequality can never be reached. It can be shown, that the inequality (2) cannot be further improved [4]. [10] [11]

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Cannot Be Further