Hi all In Paul Tipler's book on modern physics (with Ralph LLewelyn) I read an explanation for the formation and stability of a molecule, which is based on Pauli's exclusion principle. This principle was responsible for a term in the energy equation, which yields also (naturally) a term in the force equation. Is this quantum force an independent interaction or is it possible to decompose this exchange interaction in terms of gravitational, eletromagnetic, weak and strong nuclear force? Best wishes DaTario
The "exchange interaction" is due to requirement that the many-body wave-function of a system of electrons be total antisymmetric with respect to particle interchange. In terms of single-particle wave-function this means that the electron "fill up" single particle states and no more than one eloectron can be in the same state. People do not usually consider exchange to be another type of force since it is just due to the symmetry of the wavefunction. And no it can't be decomposed into gravitation, electromagnetic, etc.
If it cannot be thought of as combination of the four fundamental interactions, and if it is responsible (at least in part) to something important and relevant as the stability of molecules, I see no choice other than consider exchange interaction as one of the fundamentals. Best wishes DaTario
Perhaps it is not considered a fundamental force because it acts instantaneously across the entire system (i.e. it has an infinite speed) and because iif there were an associated boson mediating the force it would have to have inifnite energy. Otherwise I too wonder why it wouldn't be considered a force...
The book is somehow wrong or you had not understanded it. The exclusion principle is not a force, but an effect of the anti-simmetry of the fermion wavefunction. There's a term in the energy due to the anti-simmetry, but the force in the term is one of the four fundamental forces. The term is due the exclusion principle, but the force in question is one of the four basic interactions.
Tipler says the energy term associated with exclusion principle has the form U_{ex} = \frac{A}{r^n} where A and n are to be determined based on the experimental data of the specific elements which constitutes the molecule, and r is the distance from the nucleus. The term would be responsible for the repulsion, otherwise the coulomb attraction would imply the crashing of both nuclei. There seems to exist, therefore, a force term (which in the case of K Cl is of the form A/r^{10} according to a solved exercise) implied by quantum mechanics. It still seems plausible to ask if this force can or cannot be derived from any one of the four fundamental interactions. Learning from Casimir effect, we may think of this force as eletromagnetic, in nature, but originated from quantum fluctuations (the r{-10} dependence reminds me the casimir term) But no clue on how to estabilish this result. Best wishes DaTario P.S. in no place we can read in the book that coulomb repulsion between protons take part in this process.
The force is electromagnetic In fact it's the coulomb force. Many people thinks ferromagnetism is due to magnetic force. It's impossible once magnetic force can do no work and ferromagnetism can do work. In fact ferromagnetism is due to the coulomb force in the exchange energy term.
I think danime has it right: if you set the charge of the particle to zero, then there is no force, exchange or otherwise! So there you go. The force is EM. What about the centrifugal barrier in ordinary mechanics? You know there is a term in the effective potential that goes like L^2/(2mr^2) in three dimensions, and a similar term appears in the quantum Hamiltonian. What "fundamental" force do you associate with this term? It comes from the "centrifugal force" but we would never consider such a force "fundamental" - as a matter of fact, some people would claim that it isn't even a FORCE in the rigorous sense. It is strictly a consequence of the non-inertial nature of the reference frame. I think you can make an analogy here. The "exchange FORCE" is not a "force" in the rigorous sense, but is a consequence of the quantum-mechanical nature of matter's electromagnetic interactions, in the same way that the centrifugal "force" is a consequence of the reference frame.
Descartes said "In trascendental matters be transcendentally clear". There's no such thing as exchange force. There's a term in the hamiltonian due to the electromangnetic interaction that lowers the total energy when identical particles are in such state, which depends on the hamiltonian in question.
As an example we can cite the ferromagnetism. Once the electrons are fermions their total wave-function must be antisymmetric. When you construct the hamiltonian for say more than one electron interacting through coulomb force there are negative cross terms. If the terms are non null the total energy will be less so the nature prefers to make this terms non null. But the only way this terms can be non null is if the spins are parallel, once anti-parallel spin wave functions are orthogonal. This creates an effect contrary to the coulomb repulsion once the more the wave-functions overlap in the space the more effective is the exchange term. It's a brief description of the ferromagnetic effect. There other things as domains, anisotropies, etc... But that have nothing to do with what we are dealling.
I appreciate your mentioning the smilarities beween cantrifugal force and this exchange force. I would say that in the former case, the inertial contribution has a form which perfectly matches what we consider and expect of a force, while in the case of the exchange force, what we have in hands is a statement that poses restrictions on the probability of being at some specific portion of space. This geometrical and probabilistic nature of the exchange term is what puzzles me the most, for we are not used to force terms with this appearance. Besides, it seems that you both agree that exchange force is the electrical force plus quantum fluctuation effects. Note that you don't mention the quantum mystery of the magic numbers 2, 8, 18, that represent the electronic capacity of each quantum level in the atomic structure. The exclusion principle as I know it, is an "ad hoc" term in quantum theory. It does not appear as consequence of the Shroedinger equation (correct me if I am wrong...). Hoping we can see some light on this subject, DaTario
The similarities are only superficial. Centrifugal force is just an inertial effect and depends on the frame of reference. The exchange term do not depends on it. It's as real as the fact that exists solids, which is a quantum effect due to the exclusion principle too. No quantum fluctuations here. Just fermionic wave-functions plus electrical force. Correcting: Exclusion principle is not "ad hoc". It's a due to the fact that elementary particles are really identical, so exchanging identical particles cannot modify the probability of something to happen. Once the probability is the modulus of the wavefunction (a complex number), the exchanging may multiply the wavefunction by 1 or -1. When the factor is 1 we are talking about bosons, when -1 fermions. This alone produces the exclusion principle in fermions and the tendency to have many particles in the same state in bosons (ex: laser beams).
By the way this numbers are not magic. They are consequence of the exclusion principle and the solution of the schroedinger equation for the coulomb potential.
Many physicists call them "magic numbers." Not sure why you would say this. All force terms are geometrical and probabilistic. The electromagnetic force, for example, and the probability "sphere" of a single bound electron, look very similar. In fact, I would think you could recharacterize the magnitude of the force as the probability of a single quantized interaction (i.e. a photon exchange). The probability then decreases with the square of the distance. The quantum exchange "force" behaves similarly, with the exception that it doesn't depend on the distance. Again, the only substantive difference I see between the quantum exchange interaction and the four fundamental forces is that the former is non-local.
It means nothing. Every physicist knows where they come from. Another meaningless statement. It's not how it works... The exchange "force" depends of the distance once it depends on the overlapping of the wavefunctions. The substantive difference between the exchange interaction and the four fundamental forces is that the former are not a fundamental force, being an effect of the four fundamental forces
Now you're just being pedantic. Why don't you illuminate us then? And then when it is triggered, it acts instantaneously across the entire system, i.e., non-locally.
Sorry if I sounded pedantic. It's a matter of the first course in quantum mechanics in every university. It's not the probability that falls with square of the distance, but the potential. The probability has to do with the mean values whose calculate using operator acting on the wave-function, which has to do with state of the particle. Certainly, but it have never to do with what I said. You spoke about the time evolution of the state and I spoke about the spatial spread of the wave-function. Do you want to have some guiding in studying quantum mechanics?
Finally the topic was "Four or Five fundamental forces". By now four because even if you consider the exchange force as a real force It's not fundamental because it requires the electromagnetic interaction which is fundamental itself.
Let me try again. I tried previously to use the centrifugal barrier to give some intuition about the exchange force, but that didn't pan out. Let me try this one - the analogy is MUCH more appropriate: What about the "6-12 potential" that appears in many branches of physics? This is an interaction that exists between molecules in a diatomic gas, for example. In reality: this is nothing more than the "leading order" effects of the quantum mechanical system and the electromagnetic interactions. We don't consider the "7-13-force" to be anything special - it is nothing more than an EFFECTIVE (macroscopic) interaction between the molecules. In fact, the 6-12 potential can be derived from the fundamental Coulomb interaction between the molecules and their constituents, INCLUDING things like the "exchange force". And there are other examples as well. Infinitely many of them, in fact! In summary: the "exchange interaction" is NOT a special force - it is part of the quantum realization of the fundamental force (in this case, E&M). It's almost misleading to call it a force at all!