B.2.2. Anti-Symmetrized Momentum Eigenstates for Fermi-Dirac Particles

An anti-symmetrized state can be constructed as

| k 1 ,, k N ( ) = P ( ) P P| k 1 ,, k N  

= Sum of all N! signed permutations of the k i  's in | k 1 ,, k N                (B.25)

For example,

        P ( ) P P| k 1 , k 2 , k 3 =| k 1 , k 2 , k 3 +| k 2 , k 3 , k 1 +| k 3 , k 1 , k 2

                                | k 2 , k 1 , k 3 | k 3 , k 2 , k 1 | k 1 , k 3 , k 2                                   (B.26)

Now, if a momentum k α  appears twice, permutation between only them will give rise to 2 identical terms with opposite sign.  Hence, the sum vanishes identically.

For example, setting k 1 = k 2  in (B.26) gives

        P ( ) P P| k 1 , k 1 , k 3 =| k 1 , k 1 , k 3 +| k 1 , k 3 , k 1 +| k 3 , k 1 , k 1

                                | k 1 , k 1 , k 3 | k 3 , k 1 , k 1 | k 1 , k 3 , k 1   

                        =0

The orthonormality of the 1-particle states implies

        k a , k b ,, k l | k a' , k b' ,, k l' = δ aa' δ bb' δ ll'

Thus, for a given permutation P( k 1 ,, k N ) , we have

        P( k 1 ,, k N ) | k 1 ,, k N ( ) =1

so that

        k 1 ,, k N | k 1 ,, k N ( ) =N!

An orthonormal set of anti-symmetrized states is therefore

          | k 1 ,, k N ( A ) = 1 N! | k 1 ,, k N ( )

= 1 N! P ( ) P P| k 1 ,, k N                     (B.25)

with

        k 1 ,, k N | k 1 ,, k N ( A ) =1                                            (B.27)

Now, for a fixed k 1 ,, k N , all the N!  distinct permutations P( k 1 ,, k N )  give rise to only 1 distinct anti-symmetrized state, i.e., | k 1 ,, k N ( A ) = ( ) P P | k 1 ,, k N ( A ) .  Therefore, in a sum k 1 ,, k N | k 1 k N ( A ) , each distinct anti-symmetrized state will appear N!  times.  Since the completeness relation involves a sum with each distinct orthonormal state counted once, we have

        1 N! k 1 k N | k 1 k N ( A ) k 1 k N |   = 1 ˆ ( A )                                (B.28)

The wave function r 1 ,, r N | k 1 ,, k N ( A )  can be written as

        r 1 ,, r N | k 1 ,, k N ( A ) = 1 N det| r 1 | k 1 r 1 | k N r i | k j r N | k 1 r N | k N |             (B.29)

which is called the Slater determinant.