Pauli Exclusion Principle

Commonly stated as the common sense notion that two things can not be in exactly the same place at exactly the same time. In physics it is given the specific meaning that no two identical fermions can have the same quantum numbers at any given time. This accounts for the fact that electrons in atoms stack up in progressively higher energy levels, which in turn accounts for the fact that chemistry is interesting.

If two identical particles are forced to try to have the same quantum numbers they will respond with a force called the degeneracy pressure, or the Pauli Repulsion. There is a type of star, called a degenerate white dwarf, that is held up entirely by this force.

To understand the Pauli exclusion principle you just need to know how states are calculated in quantum mechanics.

A state is simply the way something is. Every unique arrangement of a system (say an atom) is considered to be an individual state. States are labeled by the quantum numbers associated with that arrangement of the system. Say you have a Hydrogen atom with the electron in it's lowest energy. You hit the electron with a photon, it gains energy, so the principle quantum number(the quantum number that describes the energy of the electron) increases. Before and after the photon hits are two different states whoose quantum numbers have different values.

Now something else you need to know about the quantum numbers is that they are arguments to a function. The function is called the Wavefunction of the system and it describes the system completely. Lets call the quantum numbers n,s,l,m then f(n,s,l,m) = state of the system where f is the wavefunction.

Often we want to know how likely a certain set of quantum numbers is to appear in a system, i.e. how likely the system is to be in a certain state f. If i have an ultraviolet photon and I chuck it at my hydrogen atom how likely am I to ionise the atom, for example.

Quantum mechanics describes a formalism for accurately calculating the probabilities that any state of a system exists. As input you feed in the wavefunction with the appropriate quantum numbers and as output you get a probability. The machine that converts the quantum numbers to a probability is an integral or more properly an overlap integral. I can't format integrals in HTML, but let's call the funny s shaped part I. The integration is over all of space so we will call the domain dx the integral looks like

I f(n,s,l,m)f^*(n,s,l,m) dx .

The * indicates that we take the complex conjugate of the wavefunction when calculating the probability and the reason we do this is to ensure that the result will be positive definite⁺ (the function can sometimes be complex).

You don't really have to worry about that to understand the Pauli exclusion principle. The exclusion principle comes about because the wavefunction for an electron is always anti-symmetric. This is the definition of a Fermion. When you have many electrons and you you try to write down the probability of two of them being in exactly the same place then you have to calculate the overlap integral but the function f is strictly anti-symmetric. Now from basic calculus you should learn that the integral of an anti-symmetric wavefunction is 0^!. The probability of two electrons being in the same sate is 0. It can never happen, it is not even that it is very unlikely, it is actually impossible. The magic of the Pauli exclusion principle comes about because of the anti-symmetry of the electron's wavefunction.

By contrast an object which has a symmetric wavefunction will quite like to be in the same state as another similar object. Particles with symmetric wavefunctions are called Bosons. Bose noticed this behavior and wrote a letter to Einstein about it. Einstein got the work published (Bose was pretty unknown at the time) and this work is the basis behind Bose-Einstein condensates.

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+ This was stumbled upon in the 1926 by Max Born when people didn't really know what was going on

! The positive part of the function exactly cancels the negative part of the function when you calculate the area under the function.

The Pauli Exclusion principle states that the quantum states of all fermions must be orthogonal. This follows from the assertion that the wavefunction of the universe is antisymmetric under the exchange of any two identical fermions.

Another way of saying the same thing is, "If you were to take two identical fermions in the universe and switch them instantaneously, then the sign of the wavefunction of the universe would be reversed but otherwise the wavefunction would remain the same."

So, let's see what this implies. Suppose two identical fermions are in the same quantum state. Since the particles are in the same exact state, swapping them should have no effect upon the wavefunction of the universe. However, swapping always reverses the sign of the wavefunction of the universe! That's a change!

The solution is that the wavefunction of this situation is zero. If you reverse the sign of zero, you get zero back. Thus you can swap them, and get the same thing back, at the same time as swapping them and reversing sign. And the wavefunction which is zero is the wavefunction for a situation which is not occurring. Therefore, it cannot be that two identical fermions are in the same state.... because if it was true, it wouldn't be!

Now, we just considered the case where the fermions were in identical states. The thing about quantum mechanics is that real states are composed of an infinite number of components. Fortunately, we are free to choose how to divvy up these components, and consider them separately, so long as we make them all orthogonal. So, for any two fermions' wavefunctions, we split off their 'shared' component, and make it zero. This renders the full wavefunctions orthogonal (actually, if you want to be super-correct, you antisymmetrize them so that swapping back and forth actually looks like a multiplication by -1; but the only observable effects of this are sufficiently complicated that I won't do that here).

There is one last gap here: to connect this orthogonality to the whole idea of not overlapping. Orthogonality implies that two fermions' wavefunctions can't have the same sign of value in the same places without cancelling exactly elsewhere. If we try to put two fermions into the same place, we will have to put some of both of them into a second location where they cancel. But now we have two fermions and two locations. That's OK. Let's add a third fermion. Well, the math gets trickier, but it ends up being that we have to add a third location... hopefully, you see where this is going. In reality, you won't use discrete locations, but integrate over regions - but the same idea applies: in order for the wavefunctions to be orthogonal, they must not overlap, except in ways that do not make the heuristic statement that they are 'not in the same place' obscenely inaccurate.

This heuristic only holds if you understand (or at least accept... :) ) that two electrons are considered in different 'places' even if the only difference in their state is that they have opposite quantum spin. That is why you can fit two electrons into one S orbital, for example.
To tie this in: the way one expresses 'opposite spin' is via the spinor of a particle... and spinors that refer to opposite directions of spin will be orthogonal to each other. No, spinors are not the axial ω vectors like in classical mechanics. ω vectors contain direction and magnitude of ω. Spinors contain direction, phase, and magnitude... of the wavefunction, not the frequency.

 

Now, it may seem that all of this is saying that a mathematical principle is capable of exerting a force upon an object (i.e. degeneracy forces that hold up atoms and the degenerate matter of white dwarves and neutron stars). It's not a force like the other forces of nature, though. The fermion does not exert forces on the particles it's excluding with to maintain that exclusion; it just can't go there. The other particle will end up pushed on, though, because antisymmetrization means that it's also getting pushed on by whever force is pushing on the first electron. So what's holding up white dwarves and neutron stars? The kinetic part of the energy of a particle is proportional to the curvature of its wavefunction. If you try to pack a fermion into a tight space, it needs to go from being low density to very very high density back down to low density in a very short distance. This requires high curvature, and thus high energy. Looked at from a k-space perspective, you always have unlimited room at shorter wavelengths, but pushing particles into those short wavelength (and thus high energy) states requires a lot of energy.

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siren: Fermions' wavefunctions need not by antisymmetric (odd) functions. Indeed, the very concept of fermions' wavefunctions being "antisymmetric" makes no sense as a universal principle, since it implies that there is a preferred center for the universe for fermions (and thus, pretty much, the entire universe) to be antisymmetric around. Also, this would require all fermions to exist equally on one half of the universe as the other. While this is a conceivable rule, it certainly does not lead to the results we are talking about.

The flaw is that when you calculate the probability associated with a wavefunction, you first multiply it by its complex conjugate (essentially, square it). This removes all negative portions, so there cannot be any cancellation of left versus right, because it's all positive. All cancellation must occur as addition of wavefunctions to each other, before this step. What you said about integrating odd functions is not relevant (though it is true).