# Van der Waals and Casimir forces

Does one need to invoke quantum mechanics to explain Casimir or van der Waals forces? I see that textbooks show derivations of van der Waal forces with no QM but the Casimir force is typically described within QM.

Additional questions I have are: Is there a difference between van der Waals and Casimir forces? Are there distinct examples of these two forces in real life? Is there a way to prove a given force is van der Waal and not Casimir or vice versa?

The Casimir effect and the Van der Waals force between two conducting plates are one and the same thing.

To see this, consider the boundary conditions postulated for the Casimir effect. The electric field has to be exactly zero at the plates. Because of this, it is said, the zero point energy of the vacuum is lower in between the plates than outside, which causes the interaction. But these references to the vacuum and virtual particles are mere heuristics. What does it mean for the electric field to be zero at the plates? The charges in the plate will have to redistribute and polarize the plate to generate a corresponding field.

But the interactions between fluctuating polarizations are precisely the dispersion forces that are responsible for the Van der Waals interaction.

Thus they are two explanation of the same phenomenon.

• Right answer! If some speak of only vacuum electric field fluctuations, it is not clear why they make plates attract (interact) if the corresponding electric field (which is a force) is zero at the plates. – Vladimir Kalitvianski Jun 25 '11 at 22:17
• @KMM, does one need to invoke QM to explain casimir/VdW force at all ? – New Horizon Jun 27 '11 at 18:14
• @New Horizon: Intuitively, I would say no, as the idea of fluctuating dipoles exists at any scale. I don't have the time to do an exact calculation (exams) but I think that the classical approximation of the effect would be extremely weak and I don't think you would get the same power law. In short: No, you can use classical electrodynamics to get a qualitative but I think it would just be a (bad) approximation to the quantum mechanical result. – Kasper Meerts Jun 27 '11 at 18:43
• @kmm thanks. Just clarifying. So are you saying the correct power law/strength can be derived only using QM ? did i understand properly ? – New Horizon Jun 27 '11 at 18:58
• @New Horizon: Although I lack a real calculation, I'm convinced you need QM to get the correct power law. No problem at all, I'm happy if I helped you. – Kasper Meerts Jun 27 '11 at 19:32

Here is an argument that Casimir force is really van der Waals force, and not a force that originates from vacuum energy: http://lanl.arxiv.org/abs/1605.04143

In short, vacuum energy originates from the pure electromagnetic term in the Hamiltonian, which does not have any explicit dependence on matter degrees of freedom and hence cannot generate any forces on matter. It does have an implicit dependence on matter degrees (the distance between the Casimir plates) originating from the solution of equations of motion, but the general principles of classical and quantum mechanics tell us that it is not legitimate to use such an implicit dependence to calculate the force.

• I guess you are the author of said article? Maybe you could give a small sketch of the argument for the nontechnical reader? Link-only answers are discouraged. Otherwise this is certainly interesting and valid for the question! – Martin May 18 '16 at 14:38
• The premise of the heuristic idea in the paper seems wrong. The Hamiltonian $H_{em}$ does depend on $y$ since $y$ determines integration bounds in the space. I can't see why differentiating EM energy with respect to $y$ to get force is wrong, since this provably works in case of charged capacitor plates, where the field between the plates is electrostatic and the energy is proportional to $y$. – Ján Lalinský May 18 '16 at 21:36
• Jan Lalinsky, just because a method works sometimes does not mean that it always works. See the mismatch between Eqs. (33) and (36) in the paper. If a method does not always work, then this method cannot be fundamental. – Hrvoje May 19 '16 at 9:34

I think people tend to equate both and present them as "alternative explanations of the same phenomenon" simply because both forces tend to have the same order of magnitude and dimensional dependency.

I haven't heard a proper argument why this should be so. Casimir is related to missing modes lowering the inter-layer vacuum pressure, while Van Der Waals are dipole-dipole interactions. So it seems to me they are different effects, which very similar behaviours in the same orders of magnitude, so probably the experimental measurements are actually subtle contributions from both.

I don't know of any practical way of discerning between both macroscopically, i would be glad to hear what others have to say on this

• one common interpretation is to treat casimir force as a retarded vander-wal force. I hope someone clarifies. Thanks for the thoughts – New Horizon Jun 24 '11 at 20:33
• I think the argument of missing modes and the concept of differences in zero-point energy are just heuristics. It is postulated that all modes of the electric field must be zero on the plates. Why is that? Because the atoms in the plates will polarize and form a corresponding charge density to cancel the electric field. This means that the Casimir force can be explained as the interaction between polarizable molecules, which is exactly the Van der Waals force. – Kasper Meerts Jun 24 '11 at 21:49