Proof of Hellmann Feynman Theorem for TD wavefunctions

Pablolopez
Messages
2
Reaction score
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 /> &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
 
Last edited:
Physics news on Phys.org
I think lambda is not supposed to depend on the other parameters (time, position)
so
d/d\lambda = \partial_\lambda
 
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!
 
Last edited:
A problem with tex?
Why do you keep using \frac{d\lambda}{dt}[\tex] ?
 
Insights auto threads is broken atm, so I'm manually creating these for new Insight articles. Towards the end of the first lecture for the Qiskit Global Summer School 2025, Foundations of Quantum Mechanics, Olivia Lanes (Global Lead, Content and Education IBM) stated... Source: https://www.physicsforums.com/insights/quantum-entanglement-is-a-kinematic-fact-not-a-dynamical-effect/ by @RUTA
If we release an electron around a positively charged sphere, the initial state of electron is a linear combination of Hydrogen-like states. According to quantum mechanics, evolution of time would not change this initial state because the potential is time independent. However, classically we expect the electron to collide with the sphere. So, it seems that the quantum and classics predict different behaviours!

Similar threads

Back
Top