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48 The Dirac Equation for Relativistic Fermions as may be seen from the plane-wave ansatz ψ(r, t) = u e−i(Et/ ̄h−k·r), which, substituted in Eq. (3.5), yields E2u = (m2c4 + h ̄2c2k2)u and, thus, the two solutions E± = ±pm2c4 + h ̄2c2k2, (3.6) instead of a single positive one. Both solutions are plotted in Fig. 3.1. Notice that negative-energy states are not problematic in a classical theory because of the gap 2mc2 between the two energy branches – negative and positive energy particles would simply not communicate. However, such states are a severe problem for a quantum theory, where the ground state, alias the lowest energy state, may not be correctly defined for an energy spectrum, which has no lower bound. A particle may, in principle, “fall” from a positive-energy state to one of negative energy by emitting, e.g., a photon the energy of which corresponds to the energy difference between the initial and the final states. We will discuss this problem in more detail in Sec. 3.3. Another problem of the Klein-Gordon equation (3.5) is related to its second-order time derivative. One would, in principle, need to specify not only ψ(r,t = t0), i.e. the wave function at some reference time t0, but also its time derivative. As a consequence, the probability density ρ(r, t), which appears in the continuity equation ∂tρ(r, t) + ∇ · j(r, t) = 0 , (3.7) is no longer necessarily positive, in contradiction to a physicists intuition of a probability density. In order to illustrate this point, we consider first the non-relativistic Schr ̈odinger equation (3.3). If we multiply Eq. (3.3) by ψ(r,t)∗ and substract the result from the complex conjugate of Eq. (3.3) times ψ(r, t), we obtain ih ̄ [ψ(r, t)∗∂tψ(r, t) or else ∂t[ψ(r,t)∗ψ(r,t)]+∇·ψ(r,t)∗ h ̄ ∇ψ(r,t)−ψ(r,t) h ̄ ∇ψ(r,t)∗=0. 2im 2im (3.8) + ψ(r, t)∂tψ(r, t)∗] h ̄ 2 = −2m ψ(r, t)∗∇2ψ(r, t) − ψ(r, t)∇2ψ(r, t)∗PDF Image | Physical Properties of Graphene
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