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American Journal of Physics

Applications of a simple quantum mechanical formula

Lior Goldenberg, Lev Vaidman A

Published Aug 1, 1996• 11 citations
Publisher

Abstract

New applications of the formula A|ψ〉=〈A〉|ψ〉+ΔA|ψ⊥〉 are discussed. Simple derivations of the Heisenberg uncertainty principle and of related inequalities are presented. It is also shown that a state corresponding to a maximal uncertainty of any given observable cannot be unique.
DOI
10.1119/1.18307

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New applications of the formula A∣ψ⟩=⟨A⟩∣ψ⟩+ΔA∣ψ⊥⟩A |ψ\rangle = \langle A \rangle |ψ\rangle + ΔA |ψ_{\perp} \rangleA∣ψ⟩=⟨A⟩∣ψ⟩+ΔA∣ψ⊥​⟩ are discussed. Simple derivations of the Heisenberg uncertainty principle and of related inequalities are presented. In addition, the formula is used in an instructive paradox which clarifies a fundamental notion in quantum mechanics.
Abstract In this article, I show how the Aharonov–Vaidman identity A∣ψ⟩=⟨A⟩∣ψ⟩+ΔA∣ψA⊥⟩A\left| \psi \right\rangle = \left\langle A \right\rangle \left| \psi \right\rangle + \Delta A \left| \psi ^{\perp }_A \right\rangle A∣ψ⟩=⟨A⟩∣ψ⟩+ΔA​ψA⊥​⟩ A ψ = A ψ + Δ A ψ A ⊥ can be used to prove relations between the standard deviations of observables in quantum mechanics. In particular, I review how it leads to a more direct and less abstract proof of the Robertson uncertainty relation ΔAΔB≥12∣⟨[A,B]⟩∣\Delta A \Delta B \ge \frac{1}{2} \left| \left\langle [A,B] \right\rangle \right| ΔAΔB≥21​∣⟨[A,B]⟩∣ Δ A Δ B ≥ 1 2 [ A , B ] than the textbook proof. I discuss the relationship between these two proofs and show how the Cauchy–Schwarz inequality can be derived from the Aharonov–Vaidman identity. I give Aharonov–Vaidman based proofs of the Maccone–Pati uncertainty relations and show how the Aharonov–Vaidman identity can be used to handle propagation of uncertainty in quantum mechanics. Finally, I show how the Aharonov–Vaidman identity can be extended to mixed states and discuss how to generalize the results to the mixed case.
New uncertainty relations in quantum mechanics are derived. They express restrictions imposed by quantum theory on probability distributions of canonically conjugate variables in terms of corresponding information entropies. The Heisenberg uncertainty relation follows from those inequalities and so does the Gross-Nelson inequality. The purpose of this paper is to derive a new, stronger version of the Heisenberg uncertainty relation in wave mechanics. This new uncertainty relation has a simple interpretation in terms of information theory. It is also closely related to newly discovered logarithmic Sobolev inequalities. The new uncertainty relation has the form (for wave functions normalized to unity)
The historical development of uncertainty relation, begining with first Heisenberg's thoughts of uncertainty principle is summed up in this thesis. After proving validity of Schwarz inequality general uncertainty relation for two hermitian operators is obtained, and from this general version the validity of Heisenberg uncertainty relation is than proved. The most important part of this work is the obtention of two new uncertainty relations, which are stronger then Heisenberg or Robertson-Schrödinger uncertainty relation, and their specific form for two examples - a free particle in a state discribed by the gaussian wave packet and the linear harmonic oscillator with a wave function in the shape of gaussian packet.