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54 The Dirac Equation for Relativistic Fermions 3.2.1 Eigenstates of the 2D Dirac Hamiltonian In order to obtain the eigenstates, we make the plane-wave ansatz for a 2-spinor, 1 uλ ip·r Ψλ,p(r) = √ v e , (3.23) 2λ and one obtains from its substitution in Eq. (3.21) (cosβ−λ)uλ +sinβe−iφpvλ =0. A possible choice for the 2-spinor components, which respects the normali- sation, is which yields for the eigenstates uλ =p1+λcosβ and vλ =λp1−λcosβeiφp, λm m2+p2 Ψλ,p(r) = √2 q1 + √ 1 One notices that the ultra-relativistic limit (m → 0) yields the same states, 1 1 ip·r Ψλ,p(r) → √2 λeiφp e (3.25) as those of low-energy electrons in graphene (2.37) if we consider one Dirac point K or K′. These low-energy electrons may, therefore, indeed be viewed as massless 2D relativistic particles. In the opposite limit of large mass (m ≫ |p|), one obtains, as in the 3D case, two decoupled equations for the components, with Ψλ=+,p(r) → 10 eip·r, Ψλ=−,p(r) → 01 eip·r, apart from an unimportant global phase factor in the second line. In the case of small, but non-zero, values of the momentum, one obtains a small ad- mixture by the “small” components vλ=+ and uλ=−. In relativistic quantum mechanics, the components uλ=+ and vλ=− are sometimes also called “large” components. λq1 − √ m2 +p2 eip·r. λm eiφp (3.24)PDF Image | Physical Properties of Graphene
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