The document discusses the Van der Waals force and its relationship to the Casimir effect. It describes how the Van der Waals force can be derived from considerations of the Casimir effect without reference to zero-point energy. It also discusses how graphene provides a system to study quantum electrodynamics and Casimir forces between sheets of graphene. While zero-point energy is not required in the derivation, the band structure of graphene shares similarities with considerations of zero-point energy. The document examines the connections between Van der Waals forces, the Casimir effect, relativistic interactions, and graphene.

2. THE VAN DER WAALS FORCE
• The Van der Waals force acts in the range of nanometers
• It is weaker than covalent bonding, but stronger than ionic
interactions
• Thanks to VdW interaction we can explain solid water
properties, noble gas condensation, geko’s adhesion ability

3. • It was postulated in 1869 to explain inter-molecular forces
• According to IUPAC definition it is both repulsive or attractive
• It includes interactions such as:
dipole - dipole
dipole - induced dipole
forces between instantaneous dipoles
• It is one of the basic force used in AFM (Atomic Force
Microscopy)
Common forms of Van der Waals force:
 Sphere – Sphere
 Plane surfaces

4. • As we often use to do, we now
consider a system of two
harmonic oscillators.
• This system is equivalent to a two
coupled LC circuit with quantized
energy levels.

7. We derived the expression for the natural frequencies taking
into account the magnetic coupling. Infact…

8. • The minus sign indicates the attractive nature of the
electrical interaction.
• On the contrary, the magnetic interaction is repulsive.
• The k appearing here has a different value in respect of the
k used in the previous calculation.

9. Due to the geometry of the system, k is proportional to R-3.
Since it results
we have recovered the expression of the Van der Waals
interaction between two atoms.
By construction, this derivation of the Van der Waals force do
not describe interaction in terms of some dipole mechanics. It
brings the Casimir vacuum energy to the scale of atoms and
molecules.

10. THE CASIMIR FORCE
“In the case you have forgotten, is an attraction between
conducing planes that arises from the vacuum energy in the
intervening space.”
It is common opinion that the Casimir effect represents the
evidence of point zero energy in quantum field theory.
But, it can be calculated without reference to point zero energy,
simply taking into account some relativistic charge-current
interaction .
??

11. • The Graphene is a perfect playground for quantum
electrodynamics (QED)
• The E vs K chart, for low energy electrons in this two
dimensional space, is linear and it is described by Dirac-
like equation
Adjustments:
- the rest mass of the
electrons is zero
- c must be substituted by vF

12. Note, despite we required no point zero energy, if we look at
the band structure of Graphene we find something similar.
The Fermi level of the Graphene touches the Dirac points
allowing for the creation of low energy excitons starting from
the ground energy state, even without any solicitation.
Is this still a kind of point zero energy?
The Casimir force
for two sheets of Graphene

13. Conclusions
• We have introduced the Van der Waals forces
• We shown the derivability of Van der Waals force starting
with arguments about the Casimir effect
• It seems there’s no way to unbound Casimir effect and the
point zero energy
• Relativistic point of view lead to the same conclusion of
coherent states schema
1. D. Kleppner, “With Apologies to Casimir”, Phys. Today 43(10), 9 (1990)
2. A.K. Geim, K.S. Novoselov, “The Rise of Graphene”, Nat. Mat., 6 (2007)
3. A.H. Neto et. Al., “The Electronic Properties of Graphene”, Rev. Mod. Phys., 81 (2009)
and more…