130_notes.dvi

12.4 Identical Particles

It is not possible to tell the difference between two electrons. Theyare identical in every way. Hence,
there is a clearsymmetry in nature under the interchange of any two electrons.

We define theinterchange operatorP 12. By our symmetry, this operator commutes withHso
we can have simultaneous eigenfunctions of energy and interchange.

If we interchange twice, we get back to the original state,

P 12 ψ(x 1 ,x 2 ) =ψ(x 2 ,x 1 )

P 12 P 12 ψ(x 1 ,x 2 ) =ψ(x 1 ,x 2 )

so thepossible eigenvalues of the interchange operator are just +1 and -1.

P 12 ψ±=±ψ

It turns out that both possibilities exist in nature. Some particles likethe electron, always have
the -1 quantum number. The are spin one-half particles and are calledfermions. The overall
wavefunction changes sign whenever we interchange any pair of fermions. Some particles, like the
photon, always have the +1 quantum number. They are integer spinparticles, calledbosons.

There is an important distinction between fermions and bosons whichwe can derive from the in-
terchange symmetry. If any two fermions are in the same state, the wave function must be zero in
order to be odd under interchange.

ψ=ui(x 1 )uj(x 2 )→ui(x 1 )uj(x 2 )−uj(x 1 )ui(x 2 )

(Usually we write a state likeui(x 1 )uj(x 2 ) when what we mean is the antisymmetrized version of
that stateui(x 1 )uj(x 2 )−uj(x 1 )ui(x 2 ).) Thus, no two fermions can be in the same state. This is
often called thePauli exclusion principle.

In fact, the interchange symmetry difference makes fermions behave like matter and bosons behave
like energy. The fact thatno two fermions can be in the same statemeans they take up space,
unlike bosons. It is also related to the fact that fermions can only becreated in conjunction with
anti-fermions. They must bemade in pairs. Bosons can be made singlyand are their own
anti-particle as can be seen from any light.

12.5 Sample Test Problems

1.*Calculate the Fermi energy forNparticles of massmin a 3D cubic “box” of sideL. Ignore

spin for this problem.

Answer

The energy levels are given in terms of three quantum numbers.

E=

π^2 ̄h^2

2 mL^2

(n^2 x+n^2 y+n^2 z)

The number of states with inside some (n^2 x+n^2 y+n^2 z)max(^18 of a a sphere innspace) is

N=

1

8

4

3

π(n^2 x+n^2 y+n^2 z)

(^32)
max