Configuration Space

What is the position of the exclusion principle on the new quantum mechanics?

Heisenberg showed that wave mechanics leads to qualitatively different conclusions for particles of the same kind (for instance for electrons) than for particles of different kinds.

It is impossibile to distinguish one of several like particles from the other.

As a result, the wave functions describing an ensemble of a given number of like particles in the configuration space are sharply separated into different classes of symmetry.

These classes can never be transformed into each other by external perturbations.

“Configuration space” includes the spin degree of freedom.

This is described in the wave function of a single particle by an index with only a finite number of possible values.

For electrons, this number is equal to 2.

The configuration space of N electrons has therefore 3 N space dimensions and N indices of « two-valuedness ».

There are different classes of symmetry.

The most important ones are the symmetrical class.

Its wave function does not change its value when:

• the space and spin coordinates of 2 particles are permuted and
• the antisymmetrical class, in which for such a permutation the wave function changes its sign.

3 different hypotheses are logically possible concerning the actual ensemble of several like particles in Nature.

1. This ensemble is a mixture of all symmetry classes.

This is never realized in Nature.

2. Only the symmetrical class occurs.

3. Only the antisymmetrical class occurs.

Only Assumption 3 is in accordance with the exclusion principle, since an antisymmetrical function containing 2 particles in the same state is identically zero.

Assumption 3 is therefore the correct and general wave mechanical formulation of the exclusion principle.

It is this possibility which actually holds for electrons.

Instead of it there was for electrons still an exclusion: not of particular states any longer, but of whole classes of states, namely the exclusion of all classes different from the antisymmetrical one.

The impression that the shadow of some incompleteness fell here on the bright light of success of the new quantum mechanics seems to me unavoidable.

Heisenberg was also able explain the existence of the 2 non-combining spectra of helium which I mentioned.

There is a rigorous separation of the wave functions into symmetry classes with respect to space-coordinates and spin indices together.

Exchange Energy

Aside from this, there is an approximate separation into symmetry classes with respect to space coordinates alone.

The latter holds only so long as an interaction between the spin and the orbital motion of the electron can be neglected.

In this way, the para- and ortho-helium spectra could be interpreted as belonging to the class of symmetrical and antisymmetrical wave functions respectively in the space coordinates alone.

The energy difference between corresponding levels of the 2 classes has nothing to do with magnetic interactions.

Instead, it is of a new type of much larger order of magnitude called “exchange energy”.

Of more fundamental significance is the connection of the symmetry classes with general problems of the statistical theory of heat.

This theory means that a system’s entropy is (apart from a constant factor) given by the logarithm of the number of quantum states of the whole system on a so-called energy shell.

One might first expect that this number should be equal to the corresponding volume of the multidimensional phase space divided by hf

• h is Planck’s constant
• f is the number of degrees of freedom of the whole system.

However, it turned out that for a system of N like particles, one had still to divide this quotient by N! in order to get a value for the entropy in accordance with the usual postulate of homogeneity.

This homogeneity says that the entropy must be proportional to the mass for a given inner state of the substance.

In this way, a qualitative distinction between like and unlike particles was already preconceived in the general statistical mechanics.

It is a distinction which Gibbs tried to express with his concepts of a generic and a specific phase.

In the light of the result of wave mechanics concerning the symmetry classes, this division by N! can easily be interpreted by accepting either Assumption 2 or 3, according to both of which only one class of symmetry occurs in Nature.

The density of quantum states of the whole system then really becomes smaller by a factor N! compared with the density which had to be expected according to an assumption of the type I admitting all symmetry classes.

In an ideal gas, the interaction energy between molecules can be neglected.

Its deviations from the ordinary equation of state must be expected.

This is because only one class of symmetry is possible as soon as the mean De Broglie wavelength of a gas molecule becomes of an order of magnitude comparable with the average distance between 2 molecules, that is, for small temperatures and large densities.

• For the antisymmetrical class, the statistical consequences have been derived by Fermi and Dirac
• For the symmetrical class, the same had been done before the discovery of the new quantum mechanics by Einstein and Bose.

The former case could be applied to the electrons in a metal.

• It could be used to interpretat magnetic and other properties of metals.