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• W4366816465 abstract "Abstract In this article, I show how the Aharonov–Vaidman identity $$Aleft| psi rightrangle = leftlangle A rightrangle left| psi rightrangle + Delta A left| psi ^{perp }_A rightrangle $$ <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML> <mml:mrow> <mml:mi>A</mml:mi> <mml:mfenced> <mml:mi>ψ</mml:mi> </mml:mfenced> <mml:mo>=</mml:mo> <mml:mfenced> <mml:mi>A</mml:mi> </mml:mfenced> <mml:mfenced> <mml:mi>ψ</mml:mi> </mml:mfenced> <mml:mo>+</mml:mo> <mml:mi>Δ</mml:mi> <mml:mi>A</mml:mi> <mml:mfenced> <mml:msubsup> <mml:mi>ψ</mml:mi> <mml:mi>A</mml:mi> <mml:mo>⊥</mml:mo> </mml:msubsup> </mml:mfenced> </mml:mrow> </mml:math> 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 $$Delta A Delta B ge frac{1}{2} left| leftlangle [A,B] rightrangle right| $$ <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML> <mml:mrow> <mml:mi>Δ</mml:mi> <mml:mi>A</mml:mi> <mml:mi>Δ</mml:mi> <mml:mi>B</mml:mi> <mml:mo>≥</mml:mo> <mml:mfrac> <mml:mn>1</mml:mn> <mml:mn>2</mml:mn> </mml:mfrac> <mml:mfenced> <mml:mfenced> <mml:mo>[</mml:mo> <mml:mi>A</mml:mi> <mml:mo>,</mml:mo> <mml:mi>B</mml:mi> <mml:mo>]</mml:mo> </mml:mfenced> </mml:mfenced> </mml:mrow> </mml:math> 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." @default.
• W4366816465 created "2023-04-25" @default.
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• W4366816465 date "2023-04-24" @default.
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• W4366816465 title "Uncertainty from the Aharonov–Vaidman identity" @default.
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• W4366816465 doi "https://doi.org/10.1007/s40509-023-00301-8" @default.
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