Concise Physical Chemistry

c17 JWBS043-Rogers September 13, 2010 11:28 Printer Name: Yet to Come

278 THE VARIATIONAL METHOD: ATOMS

for the hydrogen atom, one for each dimension; but by using four quantum numbers,

we imply something beyond the solution in 3-space.

If helium is in the excited state, the electrons are in different spatial orbitals; but

in the ground state, both electrons have the same 1sorbital description. A distin-

guishing fourth quantum number called the spin quantum numbers=±^12 must be

introduced to give two distinct 1sorbitals in helium, one designated 1sαand the other

designated 1sβ.

17.5 SPIN

If we ionize the electrons and examine their spins, we may find that the first electron

ionized (the outer electron) has spinαimplying that the inner electron (2) has spinβ

and that the atomic orbital 1 sis

 1 s= 1 s(1)α+ 1 s(2)β

On the other hand, we may find that the first ionized electron has spinβleading to

 1 s= 1 s(1)β+ 1 s(2)α

These two results haveexactlythe same probability. We are in a logical dilemma

that results from the Hartree independent orbital hypothesis. It cannot be said that an

electron is in either theαspin orbital or theβspin orbital, only that they are both in

an orbital that is alinear combinationof equally weighted space-spin basis functions.

The two plausible basis functions

1 s(1)α 1 s(2)β and 1 s(1)β 1 s(2)α

can be combined in two possible ways:

1 s(1)α 1 s(2)β+ 1 s(1)β 1 s(2)α

and

1 s(1)α 1 s(2)β− 1 s(1)β 1 s(2)α

Only the lastantisymmetricallinear combination is acceptable for the 1selectrons in

helium.

17.6 BOSONS AND FERMIONS

There are only two kinds of elementary particles in the universe,bosonsandfermions.

Bosons are symmetrical under exchange and fermions are antisymmetrical under