B.2.3. Partition Functions and Expectation Values

The sum in the definition Tr A ˆ n n| A ˆ |n  is meant to run over every distinct, or independent, basis state exactly once.

Thus, for a system of N identical bosons, we have, [see B.2.1]

        Tr( A N ˆ )= k 1 ,, k N ( 1 N! α n α )    ( S ) k 1 ,, k N | A N ˆ | k 1 ,, k N ( S )

where A N ˆ  is invariant under particle exchanges.  Using

        | k 1 ,, k N ( S ) = 1 N! α n α ! | k 1 ,, k N ( + )          (B.22)

| k 1 ,, k N ( + ) = P P| k 1 ,, k N                         (B.31)

we have

        Tr( A N ˆ )= 1 ( N! ) 2 k 1 ,, k N    ( + ) k 1 ,, k N | A N ˆ | k 1 ,, k N ( + )

Now, for each given set of k 1 ,, k N ,

        P( k 1 ,, k N ) | A N ˆ | k 1 ,, k N ( + ) = k 1 ,, k N | A N ˆ | k 1 ,, k N ( + )            (B.30a)

Since there are N! permutations in (B.31), we have

        Tr( A N ˆ )= 1 N! k 1 ,, k N k 1 ,, k N | A N ˆ | k 1 ,, k N ( + )         (B.30)

 

Similarly, for a system of N identical fermions, we have, [see B.2.1]

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

where A N ˆ  is invariant under particle exchanges.  Using

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

| k 1 ,, k N ( ) = P ( ) P P| k 1 ,, k N                 (B.33)

we have

        Tr( A N ˆ )= 1 ( N! ) 2 k 1 ,, k N    ( ) k 1 ,, k N | A N ˆ | k 1 ,, k N ( )

Now, for each given set of k 1 ,, k N ,

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

Since there are N! permutations in (B.33), we have

        Tr( A N ˆ )= 1 N! k 1 ,, k N k 1 ,, k N | A N ˆ | k 1 ,, k N ( )         (B.32)

 

In summary, we have

        Tr( A N ˆ )= 1 N! k 1 ,, k N k 1 ,, k N | A N ˆ | k 1 ,, k N ( ± ) for bosons fermions

        Tr( A N ˆ )= k 1 ,, k N k 1 ,, k N | A N ˆ | k 1 ,, k N           for distinguishable particles.

Consider the special case where A N ˆ  is a sum of 1-particle operators, i.e.,

        A N ˆ = i=1 N A 1 ( i ) ˆ                   (B.34)

where A 1 ( i ) ˆ  is an 1-particle operator for particle i.

For the case of bosons, we have

        A ˆ =Tr( ρ N ˆ A N ˆ )

                = k 1 ,, k N ( 1 N! α n α )    ( S ) k 1 ,, k N | ρ N ˆ i=1 N A 1 ( i ) ˆ | k 1 ,, k N ( S )

                = 1 ( N! ) 2 k 1 ,, k N    ( + ) k 1 ,, k N | ρ N ˆ i=1 N A 1 ( i ) ˆ | k 1 ,, k N ( + )

Using the completeness relation

        1 ( N! ) 2 k 1 ,, k N | k 1 ,, k N ( + )    ( + ) k 1 ,, k N | = 1 ˆ ( S )

we have

A N ˆ = 1 ( N! ) 4 k 1 ,, k N k 1 ,, k N    ( + ) k 1 ,, k N | ρ N ˆ | k 1 ,, k N ( + )  

                ×    ( + ) k 1 ,, k N | i=1 N A 1 ( i ) ˆ | k 1 ,, k N ( + )

= N ( N! ) 2 k 1 ,, k N k 1 ,, k N k 1 ,, k N | ρ N ˆ | k 1 ,, k N ( + )        

                × k 1 ,, k N | A 1 ( 1 ) ˆ | k 1 ,, k N ( + )

where we've used (B.30a) and

           ( + ) k 1 ,, k N | A 1 ( i ) ˆ | k 1 ,, k N ( + ) =    ( + ) k 1 ,, k N | A 1 ( 1 ) ˆ | k 1 ,, k N ( + )

Now,

k 1 ,, k N | A 1 ( 1 ) ˆ | k 1 ,, k N ( + ) ={ k 1 | A 1 ( 1 ) ˆ | k a 0         if P( k 1 ,, k N )=( k a , k 2 ,, k N ) otherwise

Summing over k 1 ,, k N  gives

        k 1 ,, k N k 1 ,, k N | A 1 ( 1 ) ˆ | k 1 ,, k N ( + ) =N! k a k 1 | A 1 ( 1 ) ˆ | k a

where the factor N! is the number of times P( k a , k 2 ,, k N )  can appear in the sum.

Thus

A N ˆ = 1 ( N1 )! k a k 1 ,, k N k a , k 2 ,, k N | ρ N ˆ | k 1 , k 2 ,, k N ( + ) k 1 | A 1 ( 1 ) ˆ | k a

        = 1 ( N1 )! k a k 1 ,, k N k a , k 2 ,, k N | ρ N ˆ | k 1 , k 2 ,, k N ( + ) k 1 | A 1 ( 1 ) ˆ | k a (B.35)

Defining a reduced 1-body probability density by

        k a | ρ ( 1 ) ˆ | k b = 1 ( N1 )! k 2 ,, k N k a , k 2 ,, k N | ρ N ˆ | k b , k 2 ,, k N ( + )                 (B.37)

we can write (B.35) as

        A N ˆ = k a k 1 k a | ρ ( 1 ) ˆ | k 1 k 1 | A 1 ( 1 ) ˆ | k a   =Tr( ρ ( 1 ) ˆ A 1 ˆ )                    (B.36)

It is left as an exercise to show that (B.36) also applies to fermions provided

        k a | ρ ( 1 ) ˆ | k b = 1 ( N1 )! k 2 ,, k N k a , k 2 ,, k N | ρ N ˆ | k b , k 2 ,, k N ( )                 (B.38)