B.3.e. Field Operators

The field operators are defined as

        ψ ˆ ( x )= α=0 φ α ( x )    a α   = α=0 x|α    a α                   (B.75)

        ψ + ˆ ( x )= α=0 φ α ( x )    a α +   = α=0 α|x    a α +                         (B.76)

so that

        [ ψ ˆ ( x ), ψ ˆ ( x ) ] = α=0 α'=0 x|α x | α    [ a α , a α' ] =0                 (a)

Taking the adjoint of (a) gives

        [ ψ + ˆ ( x ), ψ + ˆ ( x ) ] =0                              (b)

Also

        [ ψ ˆ ( x ), ψ + ˆ ( x ) ] = α=0 α'=0 x|α α | x    [ a α , a α' + ]   = α=0 α'=0 x|α α | x    δ αα'  

= α=0 x|αα| x    =x| x

=δ( x x )   = δ σσ' δ( r r )                              (B.79)

 

Now, eq(53,72) can be written as

        O ˆ = α=0 α'=0 α| O ˆ ( 1 ) ( x,p )| α    a α + a α'

                = dx    d x    α=0 α'=0 α|xx| O ˆ ( 1 ) ( x,p )| x    x | α a α + a α'

where x=( r,σ )  and dx = σ d 3 r .  Using (B.75-6), we have

        O ˆ = dx    d x    ψ + ˆ ( x )x| O ˆ ( 1 ) ( x,p )| x    ψ ˆ ( x )

Since

        x| O ˆ ( 1 ) ( x,p )| x =δ( x x ) O ( 1 ) ( x, i )

we have

        O ˆ = dx      ψ + ˆ ( x ) O ( 1 ) ( x, i )   ψ ˆ ( x )                  (B.74)

where we have removed the ^ over O ( 1 )  since it is not an operator in the n-representation.

 

Similarly, (B58,73) can be written as

        O ˆ = 1 2 α=0 β=0 α'=0 β'=0 αβ | O ˆ ( 2 ) ( x 1 , x 2 )| α β    a α + a β + a β' a α'

                = 1 2 dx    d x      ψ + ˆ ( x ) ψ + ˆ ( x ) O ( 2 ) ( x, x )   ψ ˆ ( x ) ψ ˆ ( x )                (B.77)

Examples

Number operator:

N ˆ ( 1 ) =1                     N ˆ ( N ) = i=1 N 1 =N

N ˆ = α=0 α'=0 α| α a α + a α'   = α=0 a α + a α   = dx      ψ + ˆ ( x )   ψ ˆ ( x )          (B.80)

The number density operator is defined as

N ˆ = dx    n ˆ ( x )

so that by comparison with (B.80) gives

n ˆ ( x )=   ψ + ˆ ( x )   ψ ˆ ( x )          (B.81a)

Derivation of (B.31) is similar to that for (B.82) below and will not be repeated here.

Spin operator:

S ˆ ( 1 ) =σ                      S ˆ ( N ) = i=1 N σ j

S ˆ = α=0 α'=0 α|σ| α a α + a α'  

Writing α=( k,σ )  where s is the z-component of the spin and k is a set of non-spin quantum numbers, we have

        S ˆ = k σ  σ' σ|σ| σ    a kσ + a kσ'

Setting

        ψ σ ˆ ( r )= k r|k    a kσ                ψ σ + ˆ ( r )= k k|r    a kσ +

we have

        S ˆ = σ  σ' d 3 r      ψ σ + ˆ ( x )σ|σ| σ    ψ σ' ˆ ( x )

                = σ  σ' kk' d 3 r     k|rr| k σ|σ| σ    a kσ + a k'σ'

                = σσ' kk' δ kk'   σ|σ| σ    a kσ + a k'σ'   = σσ' k σ|σ| σ    a kσ + a kσ'          (B.82a)

The spin density operator is defined by

        S ˆ = d 3 r    s ˆ ( x )

so that in comparison with (B.82a) gives

        s ˆ ( r )= σ  σ' kk' k|r r| k   σ|σ| σ    a kσ + a k'σ'                               (B.82b)

Assuming plane-wave states confined in volume V so that k are the wavevectors, we have

r|k= 1 V exp( i  kr )

and

        s ˆ ( r )= 1 V σ  σ' kk' exp[ i( k k )r ]   σ|σ| σ    a kσ + a k'σ'         (B.82)

Kinetic energy:

        T ( 1 ) ( x, i )  = p 2 2m = 2 2m 2

        T ˆ = 2 2m dx      ψ + ˆ ( x ) 2    ψ ˆ ( x )

                = σ 2 2m d 3 r      ψ σ + ˆ ( r ) 2    ψ σ ˆ ( r )                     (B.83)

For plane wave basis, we have

        T ˆ = σ  k 2 k 2 2m    a kσ + a kσ                                                (B.83a)