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The Pauli exclusion principle asserts that the total electron wave function
is
antisymmetric (equivalently, only two electrons, with opposite spins, can
occupy the same orbital). But suppose I have two separate hydrogen atoms
and
let them interact to form a hydrogen molecule. At what point does the wave
function become antisymmetric? The permutation group is discrete, so
becoming
antisymmetric is an all-or-none process, inconsistent with
"intermediate
excited states" and the like while the molecule is forming.
It seems to be necessary to suppose that the electron wave function is antisymmetric even before the two atoms interact, but no chemist would believe this. For one thing, it flies in the face of any notion of electron localization. In any problem, THE SYSTEM has to be large enough to encompass everything you want to model. This means that, if one wants to consider the collision of two hydrogen atoms, one has to include both atoms in the wavefunction for THE SYSTEM. This wavefunction begins, then, as either symmetric or antisymmetric with respect to electron interchange--which simply says that electrons can have "spin up" or "spin down" only in reference to something else, whether that is an external magnetic field (as in ESR) or another electron. A single electron, without reference to anything external to itself, can have neither "spin up" nor "spin down." Or, to preserve the classical, localized-electron picture (beloved of chemists, including myself): Two hydrogen atoms coming together need not form a bond. There are two possible states in which electrons have parallel spins when the atoms come together (uu or dd), and two in which the electrons have opposite spins (ud or du). Only collisions in which the electrons happen to have opposite spins can result in bond formation; this leads to a naive prediction that exactly 50% of H atom collisions will result in bond formation. I have no idea whether this prediction is true or not; I suspect it is not, because the localized electron picture presumes that electrons are distinguishable from one another. Assuming that electrons are distinguishable leads to the conclusion that a two-electron wave function can NEVER be antisymmetric to electron interchange! A note on the Pauli exclusion principleThe Pauli exclusion principle is even more general than you have stated: for any number of interacting identical leptons, the total wavefunction must be antisymmetric; for interacting identical bosons, the total wavefunction must be symmetric. This means that, considering a pair of electrons such as in a molecule of dihydrogen (H2), exchanging the electrons must give an overal wavefunction that is minus-one (-1) times the initial wavefunction.Suppose that we have two electrons (1 and 2), each of which may have sets of properties a or b. The electrons can have the same properties, either a(1)a(2) or b(1)b(2), or they may have different properties, a(1)b(2) or a(2)b(1). It is intuitively obvious that the combinations in which both electrons have identical properties are symmetric to electron interchange. We'll call them ya = a(1)a(2) But it gets worse. As it stands, NONE of the possible combinations is antisymmetric with respect to exchange of electrons! Exchanging electrons in the "different properties" situations results in a totally different function, not the additive inverse! That is to say, |a(1)b(2)| ¹ |a(2)b(1)| unless the two combinations are equivalent (the common-sense interpretation, but this assumes that the a and b operators commute), in which casea(1)b(2) = a(2)b(1) and interchange of electrons is symmetric no matter what you do!The indistinguishability of electrons comes to our rescue here. Electrons don't come with little tags we can label "1" or "2," and so we have to take linear combinations of the two possible microstates in which the electrons have different properties. That is, we can't just use a(1)b(2) or a(2)b(1); we have to use
y1 = a(1)b(2) + a(2)b(1)
My sources are McQuarrie's Quantum Chemistry, and remembered discussions in my graduate-school quantum chemistry course.
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