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Tight-Binding Model for Electrons on the Honeycomb Lattice 23 bound state of Hla is described by the above-mentioned atomic wavefunction φ(a)(r − Rj). 2.1.1 Bloch’s theorem Another ingredient, apart from the atomic wavefunction, of the trial wave- function is a symmetry consideration – the trial wavefunction must respect the discrete translation symmetry of the lattice. This is the essence of Bloch’s theorem. In quantum mechanics, a translation by a lattice vector Ri may be described by the operator T =eipˆ·Ri, (2.3) Ri h ̄ in terms of the momentum operator pˆ, which needs to be appropriately defined for the crystal and which we may call quasi-momentum operator, as will become explicit below. The symmetry operator, because it describes a symmetry operation under which the physical problem is left invariant, commutes with the full Hamiltonian (2.2), [TRi,H] = 0. The eigenstates of H are, therefore, necessarily also eigenstates of TRi , for any lattice vector Ri, and the momentum p, which is the eigenvalue of the momentum operator pˆ, is a good quantum number. Because of the relation (1.5) between the basis vectors of the direct and the reciprocal lattices, this momentum is only defined modulo a reciprocal lattice vector Gj = m∗j a∗1 + n∗j a∗2, where m∗j and n∗j are arbitrary integers. Indeed, if we had chosen the momentum operator pˆ′ = pˆ + h ̄Gj instead of pˆ in the definition (2.3) of the discrete translation operator, we would have simply multiplied it with a factor of exp(iGj ·Ri) = exp(i2πn) = 1 because of the integer value n = m∗jmi +n∗jni. We, thus, need to identify, as pointed out in the last chapter, identify all momenta which differ by a reciprocal lattice vector, and it is more convenient to speak of a quasi-momentum p = h ̄k, which is restricted to the first BZ. The trial wavefunction, constructed from the atomic orbital wavefunc- tions φ(a)(r − Rj), fulfils the above-mentioned requirements, i.e. it is an eigenstate of the trans- lation operator (2.3).2 That the wavefunction (2.4) is indeed an eigenstate 2The sum PRj is a short notation for Pmj,nj and runs over all 2D lattice vectors Rj = mja1 + nja2, in the thermodynamic limit of an infinite lattice. X ψk(r) = eik·Rj φ(a)(r − Rj) (2.4) RjPDF Image | Physical Properties of Graphene
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