# Van der Waals Forces

#### Gavroy

hi.
first of all i am a little bit confused about the difference between casimir and van der waals forces?
isn't it so, that both forces are consequences of Lifgarbagez's theory?
so how can you distinguish between them?

another question I asked myself is, how do Casimir and Van der Waals force depend on the curvature of the spacetime.
is there an expansion of Lifgarbagez's theory that deals with this topic?

besides, i do not understand how it is possible to calculate the van der waals force in quantum mechanics? i thought that fluctuations are quantum electrodynamics, but in quantum mechanics (cohen-tannoudji )they actually calculate this force between 2 hydrogen atoms?

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#### alxm

Did you see https://www.physicsforums.com/showthread.php?t=481430" at explaining the distinction there.

The short answer is that I'd say they're both consequences of the same physical phenomenon, which is the (relatively) long-range correlation of electronic motion. Or viewed from another angle, the higher-order effects of the polarizability of atoms/molecules. So the Casimir and London (dispersion) forces are the same thing at the fundamental level. ("van der Waals" is an ambiguous term that may refer to other interactions, like permanent dipole interactions).

The difference here is that London's original calculation (and probably the one in Cohen-Tannodji, although I don't have a copy at-hand) was done using the ordinary, non-relativistic Schrödinger equation with a classical field. So this description of the force does not take into account the fact that the electromagnetic force is mediated by photons and travels at finite speed (retarded potential), and the quantization of the field, which are things that become significant at close range. And this change is what the Casimir effect is. So you can consider it either what the London force becomes at close range, or the difference between the London force's r-6 potential and the more accurate Casimir-Polder potential.

I once had some problems trying to figure all the same question: "If the Casimir effect is supposedly the short-range version of the London force, how come non-relativistic theory reproduces London forces?" But as I explained above, it turns out that the answer is almost semantic: The Casimir effect is essentially defined to be the part of the London force/dispersion interaction that can't be described non-relativistically.

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#### Q-reeus

The short answer is that I'd say they're both consequences of the same physical phenomenon, which is the (relatively) long-range correlation of electronic motion. Or viewed from another angle, the higher-order effects of the polarizability of atoms/molecules. So the Casimir and London (dispersion) forces are the same thing at the fundamental level...
Having no expertise in this subject am nevertheless fascinated by the dilemma that seems to exist. Most pop science articles on Casimir force describe it as evidence for and based purely upon vacuum ZPF, but in a 'groundbreaking' paper http://arxiv.org/abs/hep-th/0503158 Jaffe essentially demolished that view and supports your own:
"In discussions of the cosmological constant, the Casimir effect is often invoked as decisive evidence that the zero point energies of quantum ﬁelds are “real”. On the contrary, Casimir effects can be formulated and Casimir forces can be computed without reference to zero point energies.They are relativistic, quantum forces between charges and currents."

"As evidence of the “reality” of the quantum ﬂuctuations of ﬁelds in the vacuum, theorists often point to the Casimir effect [6]. For example,Weinberg in his introduction to the cosmological constant problem, writes[5], “Perhaps surprisingly, it was along time before particle physicists began seriously to worry about [quantum zero point ﬂuctuation contributions to λ] despite the demonstration in the Casimir effect of the reality of zero-point
energies.” More recent examples can be found in the widely read reviews by Carroll[7], “ ... And the vacuum ﬂuctuations themselves are very real, as evidenced by the Casimir effect.” and by Sahni and Starobinsky [8],[9] “The existence of zero-point vacuum ﬂuctuations has been spectacularly demonstrated by the Casimir effect.”"

"The object of this paper is to point out that the Casimir effect gives no more (or less) support for the “reality” of the vacuum energy of ﬂuctuating quantum ﬁelds than any other one-loop effect in quantum electrodynamics, like the vacuum polarization contribution to the Lamb shift, for example. The Casimir force can be calculated without reference to vacuum ﬂuctuations, and like all other observable effects in QED, it vanishes as the ﬁne structure constant, α, goes to zero. There is a long history and large literature surrounding the question whether the zero point ﬂuctuations of quantized ﬁelds are “real”[13]. Schwinger, in particular, attempted to formulate QED without reference to zero point ﬂuctuations[14]. In contrast Milonni has recently reformulated all of QED from the point of view of zero point ﬂuctuations[13]. The question of whether zero point ﬂuctuations of the vacuum are or are not “real” is beyond the scope of this paper. Instead I address only the narrower question of whether the Casimir effect can be considered evidence in their favor."

Finally, in the Conclusion:
"I have presented an argument that the experimental conﬁrmation of the Casimir effect does not establish the reality of zero point ﬂuctuations. Casimir forces can be calculated without reference to the vacuum and, like any other dynamical effect in QED, vanish as α→0. The vacuum-to-vacuum graphs (See Fig. 1) that deﬁne the zero point energy do not enter the calculation of the Casimir force, which instead only involves graphs with external lines. So the concept of zero point ﬂuctuations is a heuristic and calculational aid in the description of the Casimir effect, but not a necessity. The deeper question remains: Do the zero point energies of quantum ﬁelds contribute to the energy density of the vacuumand, mutatis mutandis, to the cosmological constant? Certainly there is no experimental evidence for the “reality” of zero point energies in quantum ﬁeld theory (without gravity). Perhaps there is a consistent formulation of relativistic quantum mechanics in which zero point energies never appear. I doubt it. Schwinger intended source theory to provide such a formulation. However, to my knowledge no one has shown that source theory or another S-matrix based approach can provide a complete description of QED to all orders. In QCD conﬁnement would seem to present an insuperable challenge to an S-matrix based approach, since quarks and
gluons do not appear in the physical S-matrix. Even if one could argue away quantum zero point contributions to the vacuum energy, the problem of spontaneous symmetry breaking remains: condensates that carry energy appear at many energy scales in the StandardModel. So there is good reason to be skeptical of attempts to avoid the standard formulation of quantum ﬁeld theory and the zero point energies it brings with it. Still, no known phenomenon, including the Casimir effect, demonstrates that zero point energies are “real”."

Not sure from that if Jaffe is having an each way bet or just being diplomatic. Has an embarrassing dilemma been quietly swept under the carpet? Given the popular viewpoint Casimir force arises as a direct consequence of suppression of longer wavelength modes between perfectly conducting surfaces, there is the question of how real such fluctuations can be. It's not like one can split the difference. Might have it all wrong, but seems clear Casimir force arizes from inter-atomic forces alone, or vacuum ZPF alone - both give the same answer!

#### Gavroy

okay, but why is it actually possible to calculate the london forces in quantum mechanics.

imho, where do these fluctuations come from. i thought that this concept would be an invention of quantum electrodynamics?

and is it always true that london force is proportional to 1/r^6 ?

i saw, that Landau Lifgarbagez § 89 and Cohen-Tannoudji argued, that there can be other cases where it happens to be 1/r³ or something like that? or is this just a consequence of using pertubation theory?

#### alxm

Not sure from that if Jaffe is having an each way bet or just being diplomatic. Has an embarrassing dilemma been quietly swept under the carpet? Given the popular viewpoint Casimir force arises as a direct consequence of suppression of longer wavelength modes between perfectly conducting surfaces, there is the question of how real such fluctuations can be. It's not like one can split the difference. Might have it all wrong, but seems clear Casimir force arizes from inter-atomic forces alone, or vacuum ZPF alone - both give the same answer!
I'm not sure there's a dilemma here. I believe it is the majority opinion that virtual particles aren't really 'real', and our own A Neumaier has a whole section in http://www.mat.univie.ac.at/~neum/physfaq/physics-faq.html" [Broken] dedicated to that issue. Now, the Casimir and London forces are certainly 'real' by whatever standard you have. And the physical property that gives rise to them - polarizability, is certainly 'real' as well (although it's a bulk property and at the microscopic scale has some limits to its usefulness, but I'll get to that). And the quantization of the EM field is of course real too, nobody doubts photons exist and can act as particles.

Classically, two neutral-but-polarizable spheres (e.g. a model atom or molecule) won't spontaneously attract each other at a distance. Quantum-mechanically they do, due to the London/dispersion force. You can describe that in a number of ways. One is that it's basically the uncertainty principle applied to polarizable matter, giving this instantaneous induced dipole. A more concrete but equivalent description (in the case of atoms/molecules) is that it's the long-range correlation of electronic motion. A less concrete view preferred by some quantum-information theorists is that it's long-range electron 'continuous-variable entanglement'. These are all just different ways of describing the same thing with different models.

The Casimir effect, is effectively the same force (IMO), since it's caused by the same underlying things. It's basically defined as the corrections to the London force that occur once you take into account the quantized and relativistic field (=QED effects). Those corrections are always either smaller in energy or occur at short range. Basically, the Casimir effect is the additional effect you get once you also take into account that the uncertainty principle/QM applies to the EM field itself.

So there's no real dispute about what the Casimir effect is. The discussion of the "reality" of virtual particles is something else, IMO, which is only related insofar the Casimir effect has specifically been cited as 'proof' of this. I can't say I know why the Casimir effect would be any different in this regard from any other QED effect. It is different in the sense in that it, together with the Lamb shift, belong to the tiny set of directly-observable QED effects. That's proof that QED works and describes real things correctly. But I don't see how it's proof that the theoretical constructs of perturbation theory are 'real'. (And neither does Jaffe or Neumeier, etc) I won't bother repeating the arguments, though.

But since there's only one electromagnetic field in reality, and the classical field is only an approximation of it, it follows that the London force is just a semi-classical approximation of the Casimir effect, since all else with the model is equal. Of course, the same model also predicts other forces - ordinary dipole interactions and such. But those exist classically, hence the distinction. (Although the term 'van der Waals forces' in its broader sense does include those) To get back to something I mentioned earlier, in the case of the Casimir effect, the model has some constraints though. Real things aren't made of nice polarizable continuums. (a flat metal plate makes for a decent approximation of one though) Real things are made of atoms and molecules with electrons. So you have inhomogenities, but at close range you also have the exchange interaction, which a simple polarizability model neglects.

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#### alxm

imho, where do these fluctuations come from. i thought that this concept would be an invention of quantum electrodynamics?
As I said above, the uncertainty in the location/momentum of the electrons, basically.
and is it always true that london force is proportional to 1/r^6 ?
Asymptotically at high distances, yes.
i saw, that Landau Lifgarbagez § 89 and Cohen-Tannoudji argued, that there can be other cases where it happens to be 1/r³ or something like that?
Yes, but then you're talking about other interactions. The dipole-dipole attraction is proportional to r-3, but that's not the dispersion force.

#### Q-reeus

I'm not sure there's a dilemma here. I believe it is the majority opinion that virtual particles aren't really 'real',...
Thanks for the explanation - imho you have it all correct, but it means a lot of articles/authors are flat-out wrong. For instance, the 'ZPF is real' POV of Weinberg, Carrol etc mentioned by Jaffe is repeated by these Wikipedia entries: http://en.wikipedia.org/wiki/Vacuum_state "In many situations, the vacuum state can be defined to have zero energy, although the actual situation is considerably more subtle. The vacuum state is associated with a zero-point energy, and this zero-point energy has measurable effects. In the laboratory, it may be detected as the Casimir effect. In physical cosmology, the energy of the vacuum state appears as the cosmological constant. In fact, the energy of a cubic centimeter of empty space has been calculated to be one trillionth of an erg.[7] An outstanding requirement imposed on a potential Theory of Everything is that the energy of the vacuum state must explain the physically observed cosmological constant."
And http://en.wikipedia.org/wiki/Zero-point_energy "Vacuum energy is the zero-point energy of all the fields in space, which in the Standard Model includes the electromagnetic field, other gauge fields, fermionic fields, and the Higgs field. It is the energy of the vacuum, which in quantum field theory is defined not as empty space but as the ground state of the fields. In cosmology, the vacuum energy is one possible explanation for the cosmological constant.[3] The variation in zero-point energy as the boundaries of a region of vacuum move leads to the Casimir effect, which is observable in nanoscale devices. A related term is zero-point field, which is the lowest energy state of a particular field."
So where does that leaves ZPF? A quite contrary blog entry to the above Wikipedia stuff, by Jack Sarfatti here http://sci.tech-archive.net/Archive/sci.physics.relativity/2004-09/3372.html specifically finds the ZPF has zero net contribution. Hard to understand it heuristically but it seems ZPF can be allowed to exist in some sense but have no physical effect.

#### alxm

Sure, plenty of them are wrong. But they also tend to be popular-scientific accounts.

BTW, Jack Sarfatti is a notorious crank and so Puthoff, who he's talking about. I wouldn't be reading these guys usenet posts to try to better understand anything.

#### cgk

Thanks for the explanation - imho you have it all correct, but it means a lot of articles/authors are flat-out wrong.
That is also my impression of the situation. Imo, what we are seeing here is a huge assortment of beliefs and points of view which cannot be backed by direct experimental evidence... so instead it is backed by people cross-citing each other as having the same opinion. But even if a huge majority thinks the same way it does not necessarily imply that its point of view is correct.

There are actually plenty of wrong statements which are made again and again.
For example, during my physics studies I learned that wave functions of molecules are too complicated objects to be calculated, based on some dimensionality argument (if we stored only ten points per spatial dimension...). The professor apparently read that in a book, and never bothered to check... he would have come across an entire field of science which's main topic is dealing with algorithms for calculating exactly such wave functions. The same (wrong) argument is also made in many lectures of EKU Gross, a big figure in time-dependent density functional theory, despite him being aware of wave function methods for calculating excited states.

I guess sometimes people just like an argument so much that they don't care whether it actually reflects the entire truth. Or an argument is so popular that they don't bother for looking up the details.

#### dm4b

So, would I be summing this up correctly in saying it this way:

Two phenomenon have often been cited as "proof" that ZPF is real, or has real observable effects. They are:

(1) Casmir Effect
(2) Cosmological Constant Problem

It has been shown that (1) does not prove ZPF is "real", or even needed to calculate and understand the Casmir Effect.

The jury is still out on (2).

Is that pretty much it in a nutshell?

#### cgk

yes, it is.

And, as a side remark: When people start to seriously consider the cosmological constant problem as proof of anything, than that already says a lot.

#### Born2bwire

Gold Member
okay, but why is it actually possible to calculate the london forces in quantum mechanics.

imho, where do these fluctuations come from. i thought that this concept would be an invention of quantum electrodynamics?

and is it always true that london force is proportional to 1/r^6 ?

i saw, that Landau Lifgarbagez § 89 and Cohen-Tannoudji argued, that there can be other cases where it happens to be 1/r³ or something like that? or is this just a consequence of using pertubation theory?
I've been away from the forums so I haven't seen these discussions so I apologize if this has already been stated.

London did not specify where the fluctuations came from. Basically he derived his London Force (or dispersive force as he called it) by looking what was then the new theory of quantum mechanics. He just took a simplified view of an electron on a spring, which in terms of quantum mechanics states that the ground state for a harmonic oscillator has non-zero energy. He did not clarify, at least not in his early publications of the topic that I have read, what he thought would be the physical interpretation or cause of these fluctuations. It should be noted though that I believe that this fluctuating polarization is independent of the quantum model of the atom. That is, I do not think you can just simply attribute it to the fact that the quantum model of the atom takes the electron to being a wavefunction having a probability density. This is because London argues that with the advent of quantum mechanics, it was seen that the resulting wavefunctions did not allow for the previous theories of permanent multipole moments (induction and orientation effects) to hold out due to the newly calculated electron clouds. I guess one could argue that since it is a fluctuating polarization that it could be accounted in the fluctuation of the position of the electron within its cloud but then the electron clouds would impose restrictions on the fluctuations (for example, a 3D isotropic fluctuating polarization works for an s orbital but the p, d and f orbitals are not isotropic yet the London force is valid for molecules and atoms where the valence orbitals are not isotropic).

Milonni's Quantum Vacuum text provides a nice and succinct answer. He shows that the fluctuating polarization that London assumes in his model is due to the coupling of the electron in the atom with the fluctuation fields of the vacuum. Since the creation and annihilation operators for QED are harmonic oscillators, the coupled fluctuating polarization in the atom falls into being a harmonic oscillator naturally. Of course this still involves our quantum vacuum which Jaffe has argued does not need be introduced at all to derive the Casimir Force (take for example Schwinger's Source Theory though I hate to think of working out the Casimir Force using it). I do not think I have come across a physical explaination of the fluctuation without using the vacuum in some manner (I know that Casimir and Polder's original paper assume that the atoms are interacting with the fluctuating field). Perhaps Schwinger talks of it though I can't recall what he might of said.

But regardless of which, I don't think that with quantum mechanics you can fully explain the source of the fluctuation in the polarization. It seems to require QED to get that far however quantum mechanics does allow for a framework to assume the fluctuation which is what London did.

#### cgk

I've been away from the forums so I haven't seen these discussions so I apologize if this has already been stated.

London did not specify where the fluctuations came from. [...] I guess one could argue that since it is a fluctuating polarization that it could be accounted in the fluctuation of the position of the electron within its cloud [...]
That's precisely what happens: The r^6 force is caused by the instantaneous ("dynamic") correlation of the motion between different electrons. That requires no vacuum, relativity or quantum electrodynamics. QED is only what you need when you want to calculate the short-range special relativistic correction (Casimir-Polder-Force correction to the London force) as explained by alxm.

[...]but then the electron clouds would impose restrictions on the fluctuations (for example, a 3D isotropic fluctuating polarization works for an s orbital but the p, d and f orbitals are not isotropic yet the London force is valid for molecules and atoms where the valence orbitals are not isotropic).
That also happens. The force is not strictly r^6, but also contains (weak) higher contributions. These are caused by non-isotropic contributions (i.e., higher multipole cross-polarization), back-polarization effects, and the fact that molecules have finite spatial extend. In practice people often express dispersive forces as a combination of C6, C8 and C10 coefficients, for r^6, r^8 and r^10 contributions, respectively.

Milonni's Quantum Vacuum text provides a nice and succinct answer. He shows that the fluctuating polarization that London assumes in his model is due to the coupling of the electron in the atom with the fluctuation fields of the vacuum. [...]

But regardless of which, I don't think that with quantum mechanics you can fully explain the source of the fluctuation in the polarization. It seems to require QED to get that far however quantum mechanics does allow for a framework to assume the fluctuation which is what London did.
It is perfectly possible to calculate the dispersion coefficients without any reference to any vacuum or QED. And this is not some obscure variant, but the standard way of calculating such forces for real molecules. There are three standard ways of doing that:
(i) Calculate the potential energy curve of the molecules using a highly correlated wave function, then fit to r^6/r^8 asymptotic behavior. This is the most straight-forward approach. Note that there is *NO* special handling of dispersion at all: Dispersion is not put into the ansatz, but arises automatically because it is part of the Schroediger equation.
(ii) Calculate the C6 coefficients directly via the dynamic polarizability, which can be obtained by linear response theories of correlated wave functions.
(iii) Calculate the C6 coefficients directly via a molecular interaction-specialized method such as DFT-SAPT or CCSD-SAPT (symmetry adapted perturbation theory'').

None of those approaches contain any QED, any fluctuation, or any relativity at all, and all of them can be used to obtain accurate C6 coefficients, directly from the non-relativistic Schroedinger equation.

#### Q-reeus

I'd like to throw something into the mix here which may be naive, or a relevant argument in favor of one or the other viewpoint. The energetics of dispersion forces between dielectric molecules seems without any real problem - there is afaik no jumps in atomic/molecular energy levels etc. involved and forces are purely conservative. What has bugged me for quite a while is the situation for imperfectly conducting bodies. Suppose a thin film of such is in isolation in a vacuum, initially at absolute zero. Owing to the 'continuous' band structure in a metal, conduction electrons can be excited into higher energy levels of almost arbitrarily small value. Supposing ZPF fluctuations are at work here. Particularly for the lower frequency modes, the ZP fields should be driving stochastic but collective currents in the film on time scales perhaps many orders of magnitude larger than the relaxation time in the metal. How can this not lead to ohmic heating in the film - hence it acting as a stochastic 'antenna' 'powered' by the vacuum owing to what seems fundamentally nonreciprocal and irreversible behavior? In other words, the imperfectly conducting metal is on this view a one-way 'catalyst' for converting ZPF into real photons! Hard to imagine this has not been considered by the fraternity involved, but then is there a simple explanation for why this cannot happen? Merely invoking conservation of energy is not imo a satisfactory answer.

#### alxm

London did not specify where the fluctuations came from.
http://www.springerlink.com/content/rx775241231700w0/" Right in the abstract of his first paper on the subject:
Hinzu tritt als dritter Bestandteil der Wechselwirkung die kurzperiodische gegenseitige Störung der inneren Elektronenbewegung der Moleküle, welcher Beitrag bei den ein-fachsten nichtpolaren und auch noch bei schwach polaren Molekülen den Haupt-bestandteil der Molekularattraktion darstellt. Er macht insbesondere die früher unvermeidliche Annahme einer Quadrupolstruktur der Edelgase entbehrlich.
He correctly attributed the dispersion force to "fast periodic mutual perturbations of the inner electron motions", which would today be called correlation.

That is, I do not think you can just simply attribute it to the fact that the quantum model of the atom takes the electron to being a wavefunction having a probability density.
As cgk already pointed out, all you need to correctly get the London force between atoms is the Coulomb potential and the Schrödinger equation.
It does not require anything more or anything less than what's already required to describe the atom.

This is because London argues that with the advent of quantum mechanics, it was seen that the resulting wavefunctions did not allow for the previous theories of permanent multipole moments (induction and orientation effects) to hold out due to the newly calculated electron clouds.
Where did he argue that?.
But regardless of which, I don't think that with quantum mechanics you can fully explain the source of the fluctuation in the polarization. It seems to require QED to get that far however quantum mechanics does allow for a framework to assume the fluctuation which is what London did.
Well that's a ridiculous assertion given that London forces are calculated on a daily basis these days,
using completely ordinary molecular Hamiltonians with nothing more and nothing less than the Coulomb potential between electrons.

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#### Gavroy

okay, thanks for your precise explanation of the Van der Waals forces

i have a document attached, where they calculate the van der waals forces with quantum mechanical methods between 2 hydrogen atoms.

they use the spherical symmetry of the 1s orbital in order to calculate this force.

however, i don't understand why they can just do it so, as the potential is definately not spherical symmetric at all and therefore it should, imho, matter in which direction the orbital is directed?

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