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Proof of uncertainty principle in Nielsen and Chuang

Physics Asked by Paul Bonnot on December 20, 2020

I read the proof of uncertainty principle in Nielsen and Chuang.

There is something I don’t understand. They use operators $A$ and $B$ respectively as $P – langle P rangle$ and $Q – langle Q rangle$. As I understand it, $A$ and $B$ are the values of $P$ and $Q$ minus their expectation value. But for instance $A$ is an operator, and $P – langle P rangle$ is a scalar. How can we substract a scalar from an operator and get an operator? I think there is something I’m missing here.

One Answer

There is an implicitly written identity operator, e.g., $$P - langle P ranglehat {bf 1}, $$

Correct answer by Qmechanic on December 20, 2020

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