This is not a homework problem, but advice on the presented examples would be helpful anyways.(adsbygoogle = window.adsbygoogle || []).push({});

Does anyone know of a good QM textbook or website that discusses the rules for applying derivatives in QM? In reviewing many of the derivations, I find many weird applications with no reasoning from the author. Here are two examples:

In trying to derive

[tex]\left\langle v \right\rangle = \frac{d \left\langle x \right\rangle}{dt}[/tex]

we see that

[tex]\frac{d \left\langle x \right\rangle}{dt} = \int x\frac{\partial}{\partial t}\left|\Psi\right|^{2}dx [/tex]

yet we can't pull the partial derivative over onto x--this is because the wave function may evolve over time and therefore change the expectation value (?).

In another example, in trying to derive

[tex]\left\langle p \right\rangle = m \frac{d \left\langle x \right\rangle}{dt} [/tex]

we eventually see that

[tex] m \frac{d \left\langle x \right\rangle}{dt} = \frac{i\hbar}{2}\int\left[ \Psi^{*}x\frac{\partial^{2}\Psi}{\partial x^{2}} - \frac{\partial^{2}\Psi^{*}}{\partial x^{2}}x\Psi\right] dx = \frac{i\hbar}{2}\int\left[ \Psi^{*}\frac{\partial^{2}(x\Psi)}{\partial x^{2}} - \frac{\partial^{2}\Psi^{*}}{\partial x^{2}}(x\Psi)\right] dx - i\hbar \int \Psi^{*}\frac{\partial}{\partial x}\Psi dx[/tex]

where the x is thrown into the partial derivative. I can't explain this one and the author offers no indication for his reasoning. Thanks.

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# Homework Help: Need info on rules for applying derivatives in QM equations

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