The wording here I think is wrong it is not a more rigorous definition it is simply a more general definition. Nothing is wrong with the definition using the derivative except it does not allow for functionals defined by integrals of generalized functions to be differentiated or any function for that matter in which the exchange of integration and differentiation isn't allowed over the region of interest, while the more general definition allows for both. I think the wording here is actually important it is like saying that the definition of a Riemann integral is not rigorous because it is not the most general definition of what is meant by the integration of function just because you have a more general definition of something does not make the previous definition wrong or less rigorous only less general. I may be wrong but these are my thoughts. Also this is still the best video I found on this topic on TH-cam
Is delta at 2:20 really Kroneker's delta? It seems to be more like a Dirac's delta. Kroneker's delta wouldn't even change the functional most of the time.
Yes, it's the Dirac delta-distribution that is meant. You can view it as the "continuous" Kronecker's delta, as it slaughters your integral, while the discrete Kronecker's delta does practically the same to a sum. Therefore, the terminology is often mixed up. Hope that clears the confusion!
@@jacquessmeets4427 Best thing I've found so far: Density-Functional Theory of Atoms an Molecules, Robert G. Parr - Appendix A and en.wikipedia.org/wiki/Functional_derivative
And how would the functional derivative of F[Ψ] = ⟨Ψ(x,T)|Ô|Ψ(x,T)⟩ be? I've been trying for many days to decipher how the result could be δF[Ψ]/δΨ(x',t) = O Ψ*(x',t) δ(T - t). Ô is any unitary operator. Can you please give me a hint? I watched your video, but it didn't help much now.
This yields a fairly natural way to understand where the Euler-Lagrange equation comes from. Great video!
Thank you!
The wording here I think is wrong it is not a more rigorous definition it is simply a more general definition. Nothing is wrong with the definition using the derivative except it does not allow for functionals defined by integrals of generalized functions to be differentiated or any function for that matter in which the exchange of integration and differentiation isn't allowed over the region of interest, while the more general definition allows for both. I think the wording here is actually important it is like saying that the definition of a Riemann integral is not rigorous because it is not the most general definition of what is meant by the integration of function just because you have a more general definition of something does not make the previous definition wrong or less rigorous only less general. I may be wrong but these are my thoughts. Also this is still the best video I found on this topic on TH-cam
Is delta at 2:20 really Kroneker's delta?
It seems to be more like a Dirac's delta.
Kroneker's delta wouldn't even change the functional most of the time.
Yes, it's the Dirac delta-distribution that is meant. You can view it as the "continuous" Kronecker's delta, as it slaughters your integral, while the discrete Kronecker's delta does practically the same to a sum. Therefore, the terminology is often mixed up.
Hope that clears the confusion!
Excellent video. When will a new video be uploaded?
Thank you! We are currently writing a textbook. When this is finished, there will be time for new videos again. Sorry for the delay!
@@tutorialsontheoreticalphys8971 tell us the title when its ready
still waiting on that functional integration video..
could you recommend a textbook to study this subject?
I'm interested too, did you get any suggestions to share?
@@marllos1028 not yet.
@@jacquessmeets4427 Best thing I've found so far:
Density-Functional Theory of Atoms an Molecules, Robert G. Parr - Appendix A
and en.wikipedia.org/wiki/Functional_derivative
I am not aware of a really good textbook on this topic.
not sure but Calculus of Variations 1 by Giaquita and Hildebrandt has some idea of this
Gosh!😳🥶
And how would the functional derivative of F[Ψ] = ⟨Ψ(x,T)|Ô|Ψ(x,T)⟩ be? I've been trying for many days to decipher how the result could be δF[Ψ]/δΨ(x',t) = O Ψ*(x',t) δ(T - t). Ô is any unitary operator. Can you please give me a hint? I watched your video, but it didn't help much now.
I’m facing the same problem if you get the answer let me know
@@michaeladjei5174 Do you still need it? How can I send a manuscript to you?