Need info on rules for applying derivatives in QM equations

[buffordboy23]

This is not a homework problem, but advice on the presented examples would be helpful anyways.

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

$$\left\langle v \right\rangle = \frac{d \left\langle x \right\rangle}{dt}$$

we see that

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

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

$$\left\langle p \right\rangle = m \frac{d \left\langle x \right\rangle}{dt}$$

we eventually see that

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

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.

 

[nrqed]

This is not a homework problem, but advice on the presented examples would be helpful anyways.

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

$$\left\langle v \right\rangle = \frac{d \left\langle x \right\rangle}{dt}$$

we see that

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

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 (?).

I am not sure I understand your comment. The above equality is correct because x does not depend on time so you can pass it throught the derivative with respect to time.

In another example, in trying to derive

$$\left\langle p \right\rangle = m \frac{d \left\langle x \right\rangle}{dt}$$

we eventually see that

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

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.

See my comment above

 

[buffordboy23]

I am not sure I understand your comment. The above equality is correct because x does not depend on time so you can pass it throught the derivative with respect to time.

I know it is right, but suppose I said

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

Since $$\partial x/\partial t = 0$$, then

$$\frac{d \left\langle x \right\rangle}{dt} = 0$$

which is wrong because it neglects the evolution of the wave-function with time. Basically, I don't want to fall into such traps.

See my comment above

So for the second example, it sounds like you are saying that x does not depend on the position in space? Therefore, x is a constant and is treated as such when thrown into the partial derivative:

$$\frac{\partial^{2}(x\Psi)}{\partial x^{2}}=x\frac{\partial^{2}\Psi}{\partial x^{2}}$$

This doesn't make sense to me because of the extra integral. Do you have any recommendations on what I can do to obtain better clarity in applying these strange techniques? I would likely never thought of it unless I saw it somewhere else first. How has your experience been in deriving equations in QM? It seems like a lot of trial and error in my efforts so far--there are many directions in which to manipulate some equation but the direction is not always clear, so I choose one direction and keep chugging away until a page later I see that the road is blocked, and I must try a new direction...time is wasted and I want to be more efficient. =(