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Need info on rules for applying derivatives in QM equations

  1. Aug 4, 2008 #1
    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

    [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.
     
  2. jcsd
  3. Aug 4, 2008 #2

    nrqed

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    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.
    See my comment above
     
  4. Aug 4, 2008 #3

    I know it is right, but suppose I said

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

    Since [tex]\partial x/\partial t = 0[/tex], then

    [tex]\frac{d \left\langle x \right\rangle}{dt} = 0 [/tex]

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

    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:

    [tex]\frac{\partial^{2}(x\Psi)}{\partial x^{2}}=x\frac{\partial^{2}\Psi}{\partial x^{2}}[/tex]

    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. =(
     
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