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Relativistic Wave Equations 49 Comparison with the continuity equation (3.7) yields the probability density ρ(r, t) = ψ(r, t)∗ψ(r, t), (3.9) which corresponds to the probability of finding the quantum particle at the position r at time t, and the probability current density j(r, t) = h ̄ [ψ(r, t)∗∇ψ(r, t) − ψ(r, t)∇ψ(r, t)∗] . (3.10) 2im If we proceed in the same manner for a quantum particle described in terms of the Klein-Gordon equation (3.5), one obtains the equation ψ(r, t)∗∂t2ψ(r, t) − ψ(r, t)∂t2ψ(r, t)∗ − c2 ψ(r, t)∗∇2ψ(r, t) − ψ(r, t)∇2ψ(r, t)∗ = 0, and the corresponding probability and probability current densities are, thus, after multiplication of the above equation by ih ̄/2mc2, and ρ(r, t) = ih ̄ [ψ(r, t)∗∂tψ(r, t) − ψ(r, t)∂tψ(r, t)∗] 2mc2 j(r, t) = h ̄ [ψ(r, t)∗∇ψ(r, t) − ψ(r, t)∇ψ(r, t)∗] , 2im (3.11) (3.12) respectively. Whereas Eq. (3.12) coincides with the probability current den- sity (3.10) of the Schr ̈odinger case, the probability density (3.11) is clearly different from Eq. (3.9). In contrast to our intuition and non-relativistic quantum mechanics, this probability density may become negative, and it is, therefore, no longer possible to interpret it as the probability of finding a (relativistic) quantum particle at the position r at time t. 3.1.2 Dirac equation The above-mentioned problems of the Klein-Gordon equation led Dirac, in 1928, to propose an alternative relativistic wave equation [27]. Whereas its solutions yield positive probability densities, which are more straight-forward to interpret physically, it did not solve the problem of negative energy states. The latter have finally found a compelling interpretation by Dirac in terms of anti-particles, as we will discuss later in this chapter.PDF Image | Physical Properties of Graphene
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