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I'm reading Weinberg's Lectures on QM. On top of p23 it says $$i\hbar\frac{d}{dt}\int|\psi(x,t)|^2d^3x=i\hbar\int\psi^*(x,t)\frac{\partial}{\partial t}\psi(x)d^3x-i\hbar\int\left(\frac{\partial}{\partial t}\psi^*(x,t)\right)\psi(x)d^3x.$$

I'm confused about why the second term has a minus sign and why $\psi=\psi(x)$ only.

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    $\begingroup$ Comments to the question (v1): 1. There is no minus sign in the second term in Weinberg's 2013 book uploaded to Google books. 2. $\psi(x)$ instead of $\psi(x,t)$ is a typo. $\endgroup$ – Qmechanic Sep 1 '14 at 20:57
  • $\begingroup$ I agree. It's just strange that Weinberg makes such mistakes... $\endgroup$ – LorentzNoether Sep 1 '14 at 21:10
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    $\begingroup$ Related meta post: meta.physics.stackexchange.com/q/6111/2451 $\endgroup$ – Qmechanic Sep 2 '14 at 14:07
  • $\begingroup$ Didn't know that there's a new edition!... Apparently mine is dated 2013 as well. $\endgroup$ – LorentzNoether Sep 2 '14 at 21:45
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I think that both are typos.

The honest calculation should read \begin{align} i\hbar\frac{d}{dt}&\int|\psi(\mathbf x, t)|^2\, d^3x \\ &= i\hbar\int\frac{\partial}{\partial t}\Big(\psi^*(\mathbf x, t)\psi(\mathbf x, t)\Big)\, d^3x \\ &= i\hbar\int \left(\frac{\partial\psi^*}{\partial t}(\mathbf x,t)\psi(\mathbf x, t) + \psi^*(\mathbf x,t)\frac{\partial\psi}{\partial t}(\mathbf x,t)\right) \, d^3 x \\ &= \int \left(-i\hbar\frac{\partial\psi}{\partial t}(\mathbf x, t)\right)^*\psi(\mathbf x,t) \,d^3 x + \int\psi^*(\mathbf x,t)\left(i\hbar\frac{\partial\psi}{\partial t}(\mathbf x,t)\right)d^3x \\ &= -\int (H\psi)^*(\mathbf x,t)\psi(\mathbf x,t)\,d^3x + \int\psi^*(\mathbf x,t)(H\psi)(\mathbf x,t)\,d^3x \\ &= -\int\psi^*(\mathbf x,t)(H\psi)(\mathbf x,t)\,d^3x+ \int\psi^*(\mathbf x,t)(H\psi)(\mathbf x,t)\,d^3x \\ &= 0. \end{align}

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  • $\begingroup$ This looks good. It's just strange that Weinberg makes such mistakes... $\endgroup$ – LorentzNoether Sep 1 '14 at 21:10
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    $\begingroup$ @LorentzNoether Agreed. I have found other typos in the book as well; I think he was a bit sloppy with this one. In particular, I recall seeing some errors that were a lot worse than just incorrect signs in the section on constrained Hamiltonian mechanics. So beware! I actually emailed him about it, but never got a response. I almost feel like we should start a question that catalogs errors like this, but I don't know if such a thing is allowed on SE. $\endgroup$ – joshphysics Sep 1 '14 at 21:12

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