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In the Casimir effect, after performing the regularization, it is found that the zero point energy between two conducting plates in a distance $L$ from eachother is (in the 1D case), $$E=-\frac{\hbar c \pi}{24 L}.$$ From this result, it is argued that the force felt between the two plates containig the quantized modes is $$F=-\frac{\partial E}{\partial L}\propto-\frac{1}{L^2},$$ which means that this force is attractive, and thus, the Casimir effect emerges. However, it is not clear to me, as from my understanding, the force of a conservative system is calculated with respect to the gradient of the potential energy, and not the total energy (also, in the situation of the Casimir effect, the system is modeled as an infinite well potential, so that there is zero potential energy between these plates).

Then, how can this derivative with respect to the total energy be interpreted as an atractive force between the plates?

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  • $\begingroup$ Where do you see a calculation of the total energy of the system? $\endgroup$ May 4, 2023 at 19:05
  • $\begingroup$ look at this version of casimir forces en.wikipedia.org/wiki/… $\endgroup$
    – anna v
    May 5, 2023 at 3:57
  • $\begingroup$ @FlatterMann do you mean that its only the derivative of the vacuum state? This is correct, just as it says in the wikipedia link in the other comment by anna v. However, I'm afraid my confusion still stands. Taking this link as reference, I understand that the energy of this vacuum level is dependent on the separation of the plates, and that energy decreases as this separation gets smaller. But where does this concept of force at any point in the plates comes from? (And thank you for your answers!) $\endgroup$ May 5, 2023 at 8:46
  • $\begingroup$ A force is a classical measurement. It means the same as it always has: that which accelerates a mass, in this case the two plates would start accelerating towards each other unless held back by a reaction force. The vacuum energy is, most likely, an ill-defined concept. The plates in this system are made of the same vacuum as the empty space between them. QED simply can't model them because the nuclei in the plates require quantum-chormodynamics at this level. So you are left with a semi-classical exploration of the quantized em field in a (hypothetical) ideal cavity. $\endgroup$ May 5, 2023 at 15:48

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