Pablolopez
- 2
- 0
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 /> &\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}<br />
I cannot understand the step in which the total derivatives appear, why? could somebody help me?
Thanks in advance
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 /> &\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}<br />
I cannot understand the step in which the total derivatives appear, why? could somebody help me?
Thanks in advance
Last edited: