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Relativistic Wave Equations 47 mc2 −mc 2 k E Figure 3.1: Dispersion relation of a relativistic particle. The dispersion consists of two energy (positive and negative) branches, which are separated by twice the particle’s mass energy mc2. energy dispersion is obtained from the Lorentz-invariant norm of this vector, Pμ pμpμ ≡ E2/c2 − p2 = m2c2, given in terms of the mass of the particle, which is an invariant scalar with respect to the Lorentz transformations in space-time. A slightly more detailed discussion of these aspects, concerning the covariance of special relativity, may be found in Sec. 3.2.2. 3.1.1 Relativistic Schr ̈odinger/Klein-Gordon equation Evidently, it is more useful, in order to avoid complicated functions of differ- ential operators, to introduce the “quantum” substitution (3.2) not directly in Eq. (3.4), but in its square. This substitution yields the equation −h ̄2∂t2ψ(r, t) = m2c4 − h ̄2c2∇2 ψ(r, t), (3.5) which has been derived independently by Schr ̈odinger, Klein and Gordon in 1926/27 and which is called relativistic Schr ̈odinger or else Klein-Gordon equation. The latter name is more common, and we follow this convention here. The Klein-Gordon equation raises, however, more puzzling questions for the physicists of that time than it was supposed to solve. A first consequence of the Klein-Gordon equation is that it possesses solutions of negative energy,PDF Image | Physical Properties of Graphene
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