Proof of Hellmann Feynman Theorem for TD wavefunctions

[Pablolopez]

Dear users,

I am dealing with the proof of the Hellman Feynman-theorem for time-dependent wavefunctions given by the Wikipedia:

(http://en.wikipedia.org/wiki/Hellmann–Feynman_theorem#Proof_2)

I got stack:

$$\begin{align}<br /> &\frac{\partial}{\partial \lambda}\langle\Phi(\textbf{r},\textbf{R},t)|\hat{H}|\Phi(\textbf{r},\textbf{R},t)\rangle=<br /> \nonumber<br /> \\<br /> &=<br /> i\hbar \langle \frac{\partial}{\partial \lambda} \Phi(\textbf{r},\textbf{R},t)| \frac{\partial}{\partial t} \Phi(\textbf{r},\textbf{R},t)\rangle <br /> +<br /> \langle \Phi(\textbf{r},\textbf{R},t)|\frac{\partial}{\partial \lambda}\hat{H}|\Phi(\textbf{r},\textbf{R},t)\rangle -<br /> \nonumber<br /> \\<br /> &- i\hbar \langle \frac{\partial}{\partial t} \Phi(\textbf{r},\textbf{R},t)|\frac{\partial}{\partial \lambda} \Phi(\textbf{r},\textbf{R},t)\rangle<br /> =<br /> i\hbar\frac{d\lambda}{dt} \langle \frac{\partial}{\partial \lambda} \Phi(\textbf{r},\textbf{R},t)| \frac{\partial}{\partial \lambda} \Phi(\textbf{r},\textbf{R},t)\rangle<br /> +<br /> \nonumber<br /> \\<br /> &+ \langle \Phi(\textbf{r},\textbf{R},t)|\frac{\partial}{\partial \lambda}\hat{H}|\Phi(\textbf{r},\textbf{R},t)\rangle -i\hbar\frac{d\lambda}{dt} \langle \frac{\partial}{\partial \lambda} \Phi(\textbf{r},\textbf{R},t)| \frac{\partial}{\partial \lambda} \Phi(\textbf{r},\textbf{R},t)\rangle<br /> =<br /> \nonumber<br /> \\<br /> &=<br /> \langle\Phi(\textbf{r},\textbf{R},t)|\frac{\partial\hat{H}}{\partial\lambda}|\Phi(\textbf{r},\textbf{R},t)\rangle <br /> \end{align}$$

I cannot understand the step in which the total derivatives appear, why? could somebody help me?

Thanks in advance

 

[naima]

I think lambda is not supposed to depend on the other parameters (time, position)
so
$$d/d\lambda = \partial_\lambda$$

 

[Pablolopez]

Thanks naima,

I agree with that, however the step to transform:
[tex] \begin{equation}<br /> i\hbar \langle \frac{\partial}{\partial \lambda} \Phi(\textbf{r},\textbf{R},t)| \frac{\partial}{\partial t} \Phi(\textbf{r},\textbf{R},t)\rangle<br /> \end{equation}<br /> [\tex]<br /> <br /> and:<br /> <br /> [tex] \begin{equation}<br /> - i\hbar \langle \frac{\partial}{\partial t} \Phi(\textbf{r},\textbf{R},t)|\frac{\partial}{\par tial \lambda} \Phi(\textbf{r},\textbf{R},t)\rangle<br /> \end{equation}<br /> [\tex]<br /> <br /> into:<br /> [tex] \begin{equation}<br /> i\hbar\frac{d\lambda}{dt} \langle \frac{\partial}{\partial \lambda} \Phi(\textbf{r},\textbf{R},t)| \frac{\partial}{\partial \lambda} \Phi(\textbf{r},\textbf{R},t)\rangle<br /> \end{equation}<br /> [\tex]<br /> <br /> and:<br /> \begin{equation}<br /> $$-i\hbar\frac{d\lambda}{dt} \langle \frac{\partial}{\partial \lambda} \Phi(\textbf{r},\textbf{R},t)| \frac{\partial}{\partial \lambda} \Phi(\textbf{r},\textbf{R},t)\rangle<br /> \end{equation}<br /> [\tex]<br /> <br /> it is still not clear.<br /> <br /> Thanks for your help!$$[/tex][/tex][/tex]

 

[naima]

A problem with tex?
Why do you keep using $$\frac{d\lambda}{dt}[\tex] ?$$