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:

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

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:
<br /> \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 /> &lt;br /&gt; \begin{equation}&lt;br /&gt; - 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&lt;br /&gt; \end{equation}&lt;br /&gt; [\tex]&lt;br /&gt; &lt;br /&gt; into:&lt;br /&gt; &amp;lt;br /&amp;gt; \begin{equation}&amp;lt;br /&amp;gt; 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&amp;lt;br /&amp;gt; \end{equation}&amp;lt;br /&amp;gt; [\tex]&amp;lt;br /&amp;gt; &amp;lt;br /&amp;gt; and:&amp;lt;br /&amp;gt; \begin{equation}&amp;lt;br /&amp;gt; &amp;amp;lt;br /&amp;amp;gt; -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&amp;amp;lt;br /&amp;amp;gt; \end{equation}&amp;amp;lt;br /&amp;amp;gt; [\tex]&amp;amp;lt;br /&amp;amp;gt; &amp;amp;lt;br /&amp;amp;gt; it is still not clear.&amp;amp;lt;br /&amp;amp;gt; &amp;amp;lt;br /&amp;amp;gt; Thanks for your help!

 

[naima]

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