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Physical Consequences of the Dirac Equation 61 where T replaces t by −t in the spinor argument, which only plays a role in non-stationary spinors. We may investigate the effect of S(T) on the state (3.25) ΨKλ,k(r) at the K point in graphene, K −i λeiφ−k −ik·r K′ S(T )Ψλ,k(r) = √2 1 e ∝ Ψλ,−k(r), where we have used exp(iφ−k) = exp[i(π − φk)] = − exp(iφk). One, thus, notices that the time reversal operation maps a state at the K point at wave vector k onto a state at K′ at wave vector −k.8 It is indeed a general statement that time reversal yields k → −k and that time-reversal symmetry in a crystal, therefore, leads to a band dispersion with ε−k = εk. 3.3 Physical Consequences of the Dirac Equa- tion Limits of the validity of the Dirac equation if interpreted as a one-particle quantum theory 3.3.1 Minimal length for the localisation of a relativis- tic particle 3.3.2 Velocity operator and “Zitterbewegung” 3.3.3 Klein tunneling and the absence of backscatter- ing 8Notice that comparison with Eq. (2.37) requires the interchange of the sublattice components of the spinor at the K′ point, as mentioned in Sec. 2.2.PDF Image | Physical Properties of Graphene
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