The Spin-Statistics Theorem

An electron is cnrrently considered to be a pointlike elementary particle with no substructure. High-energy electron-positron collision experiments show no evidence for a nonzero electron size and pnt an npper limit of 3 X 10 m on the radius of an electron [D. Bourilkov, Phys. 

The wave function specifying the state of an electron depends not only on the coordinates X, y, and z but also on the spin state of the electron. What effect does this have on the wave functions and energy levels of the hydrogen atom  

To a very good approximation, the Hamiltonian operator for a system of electrons does not involve the spin variables but is a function only of spatial coordinates and derivatives with respect to spatial coordinates. As a result, we can separate the stationary-state wave function of a single electron into a product of space and spin parts  

In quantum mechanics the uncertainty principle tells us that we cannot follow the exact path taken by a microscopic particle. If the microscopic particles of the system all have different masses or charges or spins, we can use one of these properties to distinguish the particles from one another. But if they are all identical, then the one way we had in classical mechanics of distinguishing them, namely by specifying their paths, is lost in quantum mechanics because of the uncertainty principle. Therefore, the wave function of a system of interacting identical particles must not distinguish among the particles. For example, in the perturbation treatment of the helium-atom excited states in Chapter 9, we saw that the function li(l )2i(2), which says that electron 1 is in the li orbital and electron 2 is in the 2s orbital, was not a correct zeroth-order wave function. 

We now derive the restrictions on the wave function due to the requirement of in-distinguishability of identical particles in quantum mechanics. The wave function of a systan of n identical microscopic particles depends on the space and spin variables of the particles. For particle 1, these variables are x, yi, Zi, Let stand for aU four of these variables. Thus = q, . Qn)- 

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The relation between the spin and statistics of particles is given by the spin-statistics theorem. 

Chapter 10 gives two further quantum-mechanical postulates that deal with spin and the spin-statistics theorem. 

Chapter 10 Electron Spin and the Spin-Statistics Theorem... 

Many proofs of varying validity have been offered for the spin-statistics theorem see I. Duck and E. C. G. Sudurshan, Pauli and the Spin-Statistics Theorem, World... 

Scientific, 1997 Am. J. Phys., 66, 284 (1998) Sudurshan and Duck, Pramana-J. Phys., 61, 645 (2003) (available at www.ias.ac.in/pramana/v61/p645/fulltext.pdf). Several experiments have confirmed the validity of the spin-statistics theorem to extremely high accuracy see G. M. Tino, Fortschr. Phys., 48, 537 (2000) (available at arxiv.org/ abs/quant-ph/9907028). 

The spin-statistics theorem has an important consequence for a system of identical fermions. The antisymmetry requiranent means that... 

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