Current in a quantum antenna

Abstract

H a , a =1,2,..., N are identical copies of a one particle Hamiltonian acting in different Hilbert spaces and V ab , a < b are identical copies of the interaction Hamiltonian between two particles We try a wave function | ψ t > = ⊗ k - 1 N | ψ k t > https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9781003163626/7a1b5c8e-5b02-4966-b7fd-03fb766469c4/content/math1428.tif"/> Substituting this into the Schrodinger equation gives Σ k ⁢ ψ 1 t ⊗ … ⊗ ( i d ψ k t / d t ) ⊗ … ⊗ ψ n t = = Σ k ⁢ ψ 1 t ⊗ ... ⊗ H k ψ k t ⊗ ... ⊗ ψ n t + + ⁢ Σ a < b ψ 1 t ⊗ ... ⊗ V a b ψ a t ⁢ ... ⊗ ψ b t ⊗ ... ⊗ ψ n t https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9781003163626/7a1b5c8e-5b02-4966-b7fd-03fb766469c4/content/math1429.tif"/> Taking the inner product on both sides with ψ 1 t ⨶.. ψ kt ⨶...⨶ ψ nt where a hat above a symbol means omission of that symbol gives the approximate equations i d | ψ k t > / d t = H k | ψ k t > + Σ m ≠ k < I ⊗ ψ m t | V k m | ψ k t ⊗ ψ m t > https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9781003163626/7a1b5c8e-5b02-4966-b7fd-03fb766469c4/content/math1430.tif"/> 256 In terms of kernels, we can express this as i d ψ k t ( r ) / d t = H k ψ k t ( r ) + ( Σ m ≠ k ∫ V ( r , r ′ ) | ψ m t ( r ′ ) | 2 d 3 r ′ ) ψ k t ( r ) https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9781003163626/7a1b5c8e-5b02-4966-b7fd-03fb766469c4/content/math1431.tif"/> This equation however does not take into account the Pauli exclusion principle. For taking that into account, we must try an antisymmetrized wave function | ψ t > = Σ σ ∈ S n ⁢ | ψ σ 1 , t > ⊗ ... ⊗ | ψ σ n , t > https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9781003163626/7a1b5c8e-5b02-4966-b7fd-03fb766469c4/content/math1432.tif"/> Substituting and taking the inner products after assuming orthonormality of the component wave functions gives us i d | ψ k t > / d t = H k | ψ k t > + Σ m ≠ k < I ⊗ ψ m t | V | ψ k t ⊗ ψ m t > − Σ m ≠ k < ψ m t ⊗ I | V | ψ k t ⊗ ψ m t > https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9781003163626/7a1b5c8e-5b02-4966-b7fd-03fb766469c4/content/math1433.tif"/> which in coordinate form can be expressed as i d ψ k t ( r ) / d t = H k ψ k t ( r ) + Σ m ≠ k ⁢ ( ∫ V ( r , r ′ ) | ψ m t ( r ′ ) | 2 d 3 r ′ ) ψ k t ( r ) − Σ m ≠ k ⁢ ( ∫ V ( r , r ′ ) ψ ¯ m t ( r ′ ) ψ k t ( r ′ ) d 3 r ′ ) ψ m t ( r ) = 0 https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9781003163626/7a1b5c8e-5b02-4966-b7fd-03fb766469c4/content/math1434.tif"/> These equations are special cases of nonlinear Schrodinger equations of the form i d ψ k t ( r ) / d t = ∇ 2 ψ k t ( r ) / 2 m + V 0 ( r ) ψ k t ( r ) + Σ m ⁢ V k m ⁢ ( r , ψ 1 t , ... , ψ N t ) ψ m t ( r ) , k = 1 , 2 , ... , N https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9781003163626/7a1b5c8e-5b02-4966-b7fd-03fb766469c4/content/math1435.tif"/> The charge density is ρ ( t , r ) = − e ⁢ Σ k ⁢ | ψ k t ( r ) | 2 https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9781003163626/7a1b5c8e-5b02-4966-b7fd-03fb766469c4/content/math1436.tif"/> and its rate of increase is given by ρ t ( r ) = - e ⁢ Σ k ( ψ ¯ k t , t ( r ) ⁢ . ⁢ ψ k t ( r ) + ψ ¯ k t ( r ) ⁢ . ⁢ ψ k t , t ( r ) = ( i e / 2 m ) ⁢ Σ k [ ψ k t ( r ) ∇ 2 ψ ¯ k t ( r ) - ψ ¯ k t ( r ) ∇ 2 ψ k t ( r ) ] = d i v ( ( i e / 2 m ) ⁢ Σ k ⁢ [ ψ k t ( r ) ∇ ψ ¯ k t ( r ) - ψ ¯ k t ( r ) ∇ ψ k t ( r ) ] ) = - d i v J ( t , r ) https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9781003163626/7a1b5c8e-5b02-4966-b7fd-03fb766469c4/content/math1437.tif"/> 257 where the current density J ( t,r ) is given by J ( t , r ) = ( − i e / 2 m ) Σ k [ ψ k t ( r ) ∇ ψ ¯ k t ( r ) − ψ ¯ k t ( r ) ∇ ψ k t ( r ) ] https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9781003163626/7a1b5c8e-5b02-4966-b7fd-03fb766469c4/content/math1438.tif"/> It should be noted that this expression for the current density is the same as that derived using the linear Schrodinger equations but the wave functions ψ kt are calculated by solving the nonlinear Schrodinger equation. It should be noted that in deriving the above formula for the current density, we have made used of the relation V ¯ k m ( r , ψ 1 , … , ψ N ) = V m k ( r , ψ 1 , … , ψ N ) https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9781003163626/7a1b5c8e-5b02-4966-b7fd-03fb766469c4/content/math1439.tif"/> which is readily verified.