# If the Pauli Exclusion Principle is not a force, how does it result in pressure, or change the momentum of particles? [duplicate]

This question, and ones like it, seem to defy satisfactory explanation for non-physicists. PEP is referred to as Pauli "repulsion", or Pauli "pressure", while it is simultaneously denied to be a force, or result in a force. Comparisons are made to ideal gases. Apologies have been offered for technical jargon being mistaken for the lay person's notion of "force". Copious math Is done to describe acceptable and forbidden behavior of particles. Yet none of the four fundamental interactions, nor indeed a fifth force, is said to be responsible for keeping electron clouds from overlapping too much (solidity of matter questions), or neutrons from collapsing further in a neutron star.

Ideal gas behavior is straightforward and intuitive enough. It's easy to see that many small fast moving particles exert an average force on their container and vice versa. But at the particle level (can we assume hydrogen gas in a balloon?), the same question arises: what force, during the impulse time of an H atom collision with the container wall, changes the particle's momentum? Again, PEP would appear to be the cause, because this is really no different than the solid object question. But, PEP is not a force, so how does a particle undergo a change in momentum without a force?

## marked as duplicate by John Rennie, Cosmas Zachos, Qmechanic♦Jan 31 '18 at 23:25

• Possible duplicate of Is the electromagnetic force responsible for contact forces? – John Rennie Jan 31 '18 at 9:05
• It is arising from whichever of the four fundamental forces is confining the system. The force is confining a system of particles in a dynamic state through constantly changing their motion state, and the PEP is a constraint that governs the constrained particle dynamics and thus sets the steady state amount of confinement that arises from a given confining pressure. See this question here. In your ideal gas idea, it is the electromagnetic force that kicks the gas particles back into the container and confines them. The pressure arises .... – WetSavannaAnimal Jan 31 '18 at 10:28
• ... from their momentum which is needfully changed by the EM force. – WetSavannaAnimal Jan 31 '18 at 10:29
• Define "PEP", and do so in the title, please. I can't figure out what it means, and I'm sure there are others in the same situation. It's not an acronym in common use. – garyp Jan 31 '18 at 14:33
• – Stéphane Rollandin Jan 31 '18 at 14:37

Here's a slightly different approach that might help.

Consider two isolated particles interacting by an arbitrary central force, with angular momentum $L \neq 0$ with respect to each other. As long as the force remains finite, it's impossible for the particles to ever hit each other: whenever they get closer, they will necessarily speed up by conservation of angular momentum, creating a 'centrifugal potential barrier' $$V(r) = \frac{L^2}{2mr^2}.$$ In a reference frame that rotates with the particles, it looks like a new force has appeared, which gets stronger the closer the particles get, preventing them from occupying the same position.

This apparent force has nothing to do with any of the four fundamental forces and doesn't depend at all on what the force between the particles is -- it could be strong, electromagnetic, gravitational, or something else entirely. But the centrifugal force is not a new force; it just follows from the kinematics of the situation and the conservation laws.

Abstractly, you can think of the possible configuration space of the particles as being restricted to states with angular momentum $L$, and this restricted space contains no states with the particles on top of each other. Similarly, in the Pauli exclusion principle the state space is restricted to antisymmetric wavefunctions, which contain no configurations with the particles in the same state.

• oh, I really like this comparison. I guess, for the sake of precision, you should say 'interacting by an arbitrary central force,' since of course this would fail for a non-angular momentum conserving force (like a dipole-dipole interaction). – Rococo Feb 4 '18 at 18:22

All the Pauli exclusion principle does is ensure that two fermions cannot occupy the same quantum state. This means that ideal fermions at high density have large momenta, on average, whatever their temperature.

The implied pressure is calculated in exactly the same way, using exactly the same kinetic theory, as for a classical ideal gas - assuming point-like particles undergoing elastic collisions. What is responsible for those elastic collisions is swept under the carpet when you make the ideal gas (NB degenerate gases can be ideal too) approximation. If you relax the ideal assumption, then the details matter - e.g. coulomb interactions, Thomas-Fermi corrections, exchange interactions, inverse beta decay etc.

A good example is that ideal neutron degeneracy pressure does not support (all observed) neutron stars; it is the residual strong force between closely packed neutrons that is mostly responsible and the details do matter.

• Sorry for strong force is attractive - how does this help in resisting gravity, another attractive force? – SuperCiocia Jan 31 '18 at 23:19
• @SuperCiocia that is incorrect. The residual strong (nuclear) force becomes strongly repulsive at close separations. Which is why the deuteron has a finite size for instance. – Rob Jeffries Jan 31 '18 at 23:57
• – Rob Jeffries Feb 1 '18 at 0:03
• But the repulsive term in the nuclear form, isn’t it Pauli expulsion, like for Lennard Jones potentials? – SuperCiocia Feb 1 '18 at 0:12
• @SuperCiocia In what way does the PEP apply to a proton and a neutron in a deuteron? Perhaps at the quark level. But then one is admitting that the nucleons are not point-like and not ideal. – Rob Jeffries Feb 1 '18 at 0:23