# Pauli Exclusion principle.

Having looked into neutrinos and the process in which they were found I've started looking more in Wolfgang Pauli himself. I've read into this principle but there are a few things I would like to clear up. I have picked out the information I am interested in learning abou.

"The Pauli exclusion principle is a quantum mechanical principle formulated by Wolfgang Pauli in 1925. It states that no two identical fermions may occupy the same quantum state simultaneously. A more rigorous statement of this principle is that, for two identical fermions, the total wave function is anti-symmetric. For electrons in a single atom, it states that no two electrons can have the same four quantum numbers"

- http://en.wikipedia.org/wiki/Pauli_exclusion_principle

1) These first few questions I am interested more in what the words in bold actually mean. A fermion is a particle with a half integer spin. I know that, but as a fact and do not have any reason to believe that other than I have found it on the internet. What does it mean by half integer spin?

2) What is a wave function, and what would make it anti-symetric?

3) What are quantum numbers?

Thanks, any help would be great. I have looked on the wikipedia page but I sometimes get lost reading, as I am not working at a partciularly high level (AS Level) and this is not in the curriculum, but is just out of interest.

What does it mean by half integer spin?
The electron, for instance, carries quantum angular momentun of 1/2 h. h is Plancks constant which has the dimensions of angular momentum.

What is a wave function, and what would make it anti-symetric?
A wave function describes a given physical setup, and allows us to calculate the probabilities of various events, and to make predictions about how the set-up will evolve.

A mathematical function is anti-symmetric is the sign changes on exchanging labels. For instance

x - y is antisymmetric in x, y because swapping x and y changes the sign. x + y is symmetric.

The wave function of a collection of electrons is anti-symmetric under exchange of electrons ( I'm not 100% sure of this, but it's close).

What are quantum numbers?
They are the numbers that describe states in quantum mechanics. Two electrons can't have the same state, therefore they must have different quantum numbers.

1) These first few questions I am interested more in what the words in bold actually mean. A fermion is a particle with a half integer spin. I know that, but as a fact and do not have any reason to believe that other than I have found it on the internet. What does it mean by half integer spin?

2) What is a wave function, and what would make it anti-symetric?

3) What are quantum numbers?

Thanks, any help would be great. I have looked on the wikipedia page but I sometimes get lost reading, as I am not working at a partciularly high level (AS Level) and this is not in the curriculum, but is just out of interest.

Pauli exclusion principle
The Pauli exclusion principle can be written in following form: particles of half-integer spin have antisymmetric wavefunctions, and particles of integer spin have symmetric wavefunctions.
In other words the question arise (see Feynman lectures), why "particles with half-integral spin are Fermi particles whose amplitudes add with the minus sign". In his last lecture R. Feynman (Feynman, 1987) sketched an elementary argument for above question, using the topological behaviour of lepton wave function.
There is (Gottfried and Weisskopf, 1986; Gould, 1995) a remarkable property of lepton in three dimensional space: when a lepton is rotated 360 degrees (what means that the wave function phase shifts on 360 degrees), it returns to a state that looks the same geometrically, but that is topologically distinct with respect to its surroundings: a twist has been introduced. A second full rotation (a total of 720 degrees) brings the object back to its original state.
Feyman considered a belt; and the belt ends A and B in two positions 1 and 2 he used for demonstration of rotations of Dirac wave function.
To see this (see fig from R. Feynman paper), first grasp the two ends of a belt, one end in each hand; then interchange the position of your hands. So we have introduced a "twist", which is topologically equivalent to having rotated one end of the belt by 360 degrees.
Thus, when fermions are interchanged, one must keep track of this "implied rotation" and the phase shift, sign change, and destruction interference to which it gives rise. For example, if A(1)B(2) describes "electron 1 in state A and electron 2 in state B," then the state with electrons interchanged must be -A(2)B(1) and their superposition is A(1)B(2) - A(2)B(1)
It could say that according to R. Feynman, if particle field has the Moebius strip topology, it must obey the Pauli exclusion principle..

Feynman, R.P. (1987). The reason for Antiparticles//Elementary particles and the laws of
physics: the 1986 . Dirac Memorial Lectures/. Cambtidge University Press,
1987. – Pp. 1-59.
Gottfried, K. and Weisskopf, V.F. (1984). Concepts of Particle Physics. Oxford.
Gould, Roy R. (1995). Am. J. Phys., Vol 63, No. 2, February