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CoompmlexpWleKBxAWpprKoxBimaAtiopnspinrGorxapimhenaetEiloecntrosn-iHnoleGWraavepguhidesnien MEaglnectictrFoielnd -Hole Waveguides in Magnetic Field 83 potential e δ(x) 0 vF <−ih ̄∇+cA,σ ̄ >G(x)+(U(x)−E)G(x)= 0 δ(x) , (2) G11(x) G12(x) G(x) = G21(x) G22(x) . These are to be used in derivation of electronic Gaussian beam asymptotic expansion. Ray asymptotic solution is the principal part of the method summation of Gaussian beams. 2.1 Ray asymptotic solutions Consider the axial gauge of the magnetic field A = B/2(−x2, x1, 0). The WKB ray asymptotic solution to the Dirac system U(x)−E vF[h ̄(−i∂x1 −∂x2)−iαx1 − αx2 ]u 0 F x1 x2 is to be sought in the form 22 αx1αx22 = (3) v[h ̄(−i∂+∂)+i −2]U(x)−E v 0 +∞ +∞ h ̄ h ̄ ψ = u = e i S ( x ) ∑ ( h ̄ ) j u j = e i S ( x ) ∑ ( h ̄ ) j ψ j ( x ) , ( 4 ) v j=0ivj j=0i where α = Be. Substituting this series into the Dirac system, and equating to zero c corresponding coefficients of successive degrees of the small parameter h ̄ , we obtain recurrent system of equations which determines the unknown S(x) and ψj(x), namely, (H − EI)ψ0 = 0, (H − EI)ψj = −Rψj−1, U(x) v (S −αx2 −i(S +αx1)) 0 ∂ −i∂ H= F x1 2 x2 2 , R=v x1 x2 , vF(Sx1 − αx2 +i(Sx2 + αx1 )) U(x) F ∂x1 +i∂x2 0 22 where I is the identity matrix. The hamiltonian H has two eigenvalues and eigenvectors where and where Sx1 shall use hα = U(x)±vF Heα=hαvα, α=1,2, (p1 − αx2 )2 +(p2 + αx1 )2 = U(x)±vFp, 22 1111 v1 = √2 p1−αx2 +i(p2+αx1 ) , v2 = √2 −p1−αx2 +i(p2+αx1 ) , 22 22 pp = p1 and Sx2 = p2. However, for the sake of simplicity instead of vα, below we 1 1 1 1 p −αx2 +i(p +αx1) e1=√ iθ , e2=√ iθ , eiθ=vF 1 2 2 2 , 2 e 2 −e E−U(x)PDF Image | GRAPHENE SYNTHESIS CHARACTERIZATION PROPERTIES
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