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In Weinberg's "Lecture on Quantum Mechanics" (2nd edition, page 79) in equation 3.5.11, about the momentum operator acting on states definite position, the minus sign is missing. Is this just a typo or am I missing something? Note that this is repeated in page 193, in the footnote.

Note that in the first edition it is the same equation, page 75, and the footnote is in page 187.

Edit: The equation question reads

$$P_j \Phi_\mathbf{x} ~=~ \color{\red}{+}i \hbar \frac{\partial}{\partial x^j} \Phi_\mathbf{x}.\tag{3.5.11}$$

Qmechanic
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Konstantinos Kyritsis
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1 Answers1

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Weinberg is stating the fact that the momentum operator $\hat{p}$ acts on a position eigen-ket$^1$ $|x\rangle$ with the opposite sign $$\hat{p}|x\rangle ~=~\color{\red}{+}i\hbar\frac{\partial |x\rangle}{\partial x}$$ as compared to a position eigen-bra $\langle x |$ or a wave function $\psi(x)\equiv\langle x |\psi\rangle$: $$\langle x |\hat{p} ~=~\color{\red}{-}i\hbar\frac{\partial \langle x |}{\partial x}, \qquad \hat{p}\psi(x) ~=~\color{\red}{-}i\hbar\frac{\partial\psi(x)}{\partial x}. $$ This is e.g. explained in my Phys.SE answer here.

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$^1$ Weinberg's notation for a position eigen-ket is $|x\rangle\equiv\Phi_x$.

Qmechanic
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