# What the heck is meant by Pauli force/effect ?

1. Mar 4, 2012

### nonequilibrium

What the heck is meant by "Pauli force/effect"?

I'm a last year physics undergrad and whenever I have a physics class given by an experimental physicist (e.g. solid state physics), they sometimes say things like "... and because of the Pauli force these two electrons are repelled ..." and whenever I enquire about what is meant, I get an answer like "it's due to the Pauli exclusion principle".

But the latter only states that two fermions cannot be in exactly the same state; it says nothing about a repulsive force that acts on two fermions close to each other. In essence the Pauli exclusion principle does nothing to prevent two fermions in being arbitrarily similar states, as long as the states are not exactly the same.

So what is the deal: is there an actual Pauli force additional to the Pauli exclusion principle (NB: let's not get into a semantic discussion about the word "force", call it what you will, I'm simply referring to the so-called repulsive effect of two fermions close to each other)? Or are my experimental physicis professors botching the concept of Pauli exclusion principle, making their arguments using it fallacious (i.e. there is only the Pauli exclusion principle, no repulsive effect)?

2. Mar 4, 2012

### Ilmrak

Re: What the heck is meant by "Pauli force/effect"?

You're perfectly right, your professor used the Pauli exclusion principle in a wrong way.
I think it's common in solid state physic to do the same error, I've read the same thing in the Ashcroft and Mermin book to justify the Lennard-Jones potential repulsive term.

Pauli principle states that the state of a system of fermions is anti-symmetric in the excange of two fermions.
It can be interpret with a effective force when, for example, you use the Hartree-Fock approximation. In doing this you find that the anti-symmetry of the wave function acts as an additiona term in the Hamiltonian, the "excange energy". This energy is though not necessarily positive, for example in the "jellium" model you find its effect is actractive. The specific form of this energy depends on both the interaction and the wavefunction you're using as an approximation.

In conclusion it's right to think about the Pauli principle as an effective energy in some approximation, but you can't know a priori if it's an actractive or a repulsive contribution.

Ilm

Last edited: Mar 4, 2012
3. Mar 4, 2012

### nonequilibrium

Re: What the heck is meant by "Pauli force/effect"?

I see, interesting... Do you know of a source that addresses this issue, i.e. that mentions the common fallacious reformulations and in what ways it is correct (apparently related to the Hartree-Fock approximation)?

4. Mar 5, 2012

### Ilmrak

Re: What the heck is meant by "Pauli force/effect"?

I don't know where to find a general treatment of this issue.

You can simply try to find something on the excange energy and then constate it can be both positive or negative, maybe checking this explicitly in different examples.

For a simple treatment of Hartree-Fock approximation you can read almost every solid state physic book (check it on physics forum), even Ashcroft, Mermin, Solid State Physics (I personally hated this book ).
You can certainly find some example here where anti-symmetry of the state produce a positive term in the energy of the system.
Only believe in what is demonstrated though

For a rigorous treatment (but not so easy) of Hartree-Fock approximation look for books on many-body quantum physics like Fetter, Walecka, Quantum Theory of Many-Particle Systems.
Here you can find even the very simple example of the jellium model (not using Hartree-Fock if I remember it right), where anti-symmetry of the state produce a negative term in the mean energy of the system.

edit: to avoid misunderstanding, to my knowledge treating anti-symmetry of fermions states as an effective potential is not strictly related to Hartree-Fock, which is only one of the possible approximations resulting in such a term in the Hamiltonian.

Ilm

Last edited: Mar 5, 2012
5. Mar 5, 2012

### Zarqon

Re: What the heck is meant by "Pauli force/effect"?

This question has in fact been asked a couple of times before on PF, for example by myself a year or two ago, see here: