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Journal of Physics A Mathematical and General

A variation of the Heisenberg uncertainty relation involving an average

Shunlong Luo

Published Apr 5, 2001• 6 citations
Publisher

Abstract

We derive a parameter-based position-momentum uncertainty relation involving the distance for a state to be displaced to become an orthogonal state and the average rather than the variance of the momentum. This may be useful when the variance of the momentum is infinite and thus the conventional uncertainty relations involving variances provide no useful information.

DOI
10.1088/0305-4470/34/15/307

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Position--momentum uncertainty relation for the average confidence width

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We introduce the average confidence width $Δ_{a} x=\int_0^1Δ_{c} x (θ)\,\mathrm{d}θ$: the confidence width $Δ_{c} x(θ)$---the measure of the smallest region of position space carrying probability $θ$---averaged over all confidence levels. It is the first moment of the decreasing rearrangement of the position density, an $L^1$ (mean-absolute-deviation) measure of localization, and $Δ_{a} x\,Δ_{a} p$ is dilation invariant. We study the uncertainty relation $Δ_{a} x\,Δ_{a} p\ge c^\ast\hbar$ for pure and mixed states. A mean--entropy argument with the Bialynicki-Birula--Mycielski relation gives $c^\ast\geπ/e$, while the ground state $ψ_0$ of the Fourier-invariant operator $|x|+|p|$, with $E_0=1.1040744$, gives $c^\ast\le E_0^2\approx1.2190<4/π$: Gaussian states, which saturate the Heisenberg--Kennard and entropic relations, are not optimal here. We prove that $ψ_0$ is nondegenerate, nodeless, even, self-dual, and symmetric decreasing, with algebraic tails, and that the sharp hybrid relations $Δ_{a} x \cdot 2 \langle |p-p_0| \rangle \ge E_0^2\hbar$ and $2\langle|x-x_0| \rangle \cdotΔ_{a} p\ge E_0^2\hbar$ hold for all states, so $Δ_{a} x\,Δ_{a} p\ge E_0^2\hbar$ whenever either density is symmetric unimodal; the second variation of $Δ_{a} x\,Δ_{a} p$ at $ψ_0$ is positive definite apart from the symmetry directions. Extensive numerical minimization, including coherent states, never finds a product below $E_0^2\hbar$ and consistently returns $ψ_0$, supporting the conjecture $c^\ast=E_0^2$. In two dimensions the relation is sharp and saturated by Gaussians, identifying the dimension as the origin of the non-Gaussian optimizer in one dimension. Finally, we discuss other complementary pairs and the relation of the width to interferometric predictability and visibility.

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We derive a parameter-based position-momentum uncertainty relation involving the distance for a state to be displaced to become an orthogonal state and the average rather than the variance of the momentum. This may be useful when the variance of the momentum is infinite and thus the conventional uncertainty relations involving variances provide no useful information.
The mathematical content of the uncertainty principle is contained in special states akin to, and including, states of minimum uncertainty. Specific uncertainty relations are produced by averaging observables in these states. The approach is shown to give more flexible and complete expressions of indeterminacy than the familiar generalized Heisenberg inequality: statistical measures other than variances can be employed, the method is not limited to describing uncertainties in terms of products of statistical dispersions, and the resulting inequalities have the most stringent bounds attainable independent of the system state. Applications are then made to find uncertainty relations for position-momentum average displacements from the means, time-energy indeterminacy, and uncertainties in angular momentum components.
Recently, S\'anchez-Moreno et al. [New J. Phys. 8, 330 (2006)] have shown that the three-dimensional Heisenberg uncertainty relation ${\ensuremath{\sigma}}_{r}{\ensuremath{\sigma}}_{p}\ensuremath{\ge}3\ensuremath{\hbar}/2$ might be significantly sharpened if the relevant quantum state describes the particle in a central potential. I extend that result to the case of states which are not the eigenstates of the square of the angular momentum operator. I derive a lower bound for ${\ensuremath{\sigma}}_{r}{\ensuremath{\sigma}}_{p}$ which involves the mean value and the variance of the square of the angular momentum operator.
We introduce the average confidence width Δax=∫01Δcx(θ) dθΔ_{a} x=\int_0^1Δ_{c} x (θ)\,\mathrm{d}θΔa​x=∫01​Δc​x(θ)dθ: the confidence width Δcx(θ)Δ_{c} x(θ)Δc​x(θ)---the measure of the smallest region of position space carrying probability θθθ---averaged over all confidence levels. It is the first moment of the decreasing rearrangement of the position density, an L1L^1L1 (mean-absolute-deviation) measure of localization, and Δax ΔapΔ_{a} x\,Δ_{a} pΔa​xΔa​p is dilation invariant. We study the uncertainty relation Δax Δap≥c∗ℏΔ_{a} x\,Δ_{a} p\ge c^\ast\hbarΔa​xΔa​p≥c∗ℏ for pure and mixed states. A mean--entropy argument with the Bialynicki-Birula--Mycielski relation gives c∗≥π/ec^\ast\geπ/ec∗≥π/e, while the ground state ψ0ψ_0ψ0​ of the Fourier-invariant operator ∣x∣+∣p∣|x|+|p|∣x∣+∣p∣, with E0=1.1040744E_0=1.1040744E0​=1.1040744, gives c∗≤E02≈1.2190<4/πc^\ast\le E_0^2\approx1.2190<4/πc∗≤E02​≈1.2190<4/π: Gaussian states, which saturate the Heisenberg--Kennard and entropic relations, are not optimal here. We prove that ψ0ψ_0ψ0​ is nondegenerate, nodeless, even, self-dual, and symmetric decreasing, with algebraic tails, and that the sharp hybrid relations Δax⋅2⟨∣p−p0∣⟩≥E02ℏΔ_{a} x \cdot 2 \langle |p-p_0| \rangle \ge E_0^2\hbarΔa​x⋅2⟨∣p−p0​∣⟩≥E02​ℏ and 2⟨∣x−x0∣⟩⋅Δap≥E02ℏ2\langle|x-x_0| \rangle \cdotΔ_{a} p\ge E_0^2\hbar2⟨∣x−x0​∣⟩⋅Δa​p≥E02​ℏ hold for all states, so Δax Δap≥E02ℏΔ_{a} x\,Δ_{a} p\ge E_0^2\hbarΔa​xΔa​p≥E02​ℏ whenever either density is symmetric unimodal; the second variation of Δax ΔapΔ_{a} x\,Δ_{a} pΔa​xΔa​p at ψ0ψ_0ψ0​ is positive definite apart from the symmetry directions. Extensive numerical minimization, including coherent states, never finds a product below E02ℏE_0^2\hbarE02​ℏ and consistently returns ψ0ψ_0ψ0​, supporting the conjecture c∗=E02c^\ast=E_0^2c∗=E02​. In two dimensions the relation is sharp and saturated by Gaussians, identifying the dimension as the origin of the non-Gaussian optimizer in one dimension. Finally, we discuss other complementary pairs and the relation of the width to interferometric predictability and visibility.