PDF Publication Title:
Text from PDF Page: 057
The 2D Dirac Equation 53 Continuity equation We finally notice that the Dirac equation yields positive probability densities, which were Dirac’s main motivation to propose an alternative to the Klein- Gordon equation, as mentioned above. In order to obtain the continuity equation (3.7), we proceed in the same manner as for the Schro ̈dinger or the Klein-Gordon equation, i.e. we multiply the differential equation (3.13) from the left by the spinor ψ†(r, t) and subtract the Hermitian conjugate of Eq. (3.13), multiplied from the right by ψ(r,t). This yields the continuity equation ∂tψ†(r,t)ψ(r,t)+c∇·ψ†(r,t)αψ(r,t) , and one thus obtains the positive probability density ρ(r, t) = ψ†(r, t)ψ(r, t) and for the probability current density j(r, t) = cψ†(r, t)αψ(r, t). 3.2 The 2D Dirac Equation After this short excursion to the 3D Dirac equation, we now concentrate on the 2D equation (3.16), H2D =p·σ+mσz, which interests us in the case of graphene, where we choose a unit system with c = h ̄ = 1, for the remainder of this section. We may rewrite the Hamiltonian as H2D =ǫ cosβ sinβe−iφp , (3.21) sin βeiφp − cos β where we have defined ǫ = pm2 +p2, cosβ = m/ǫ, sinβ = |p|/ǫ, and φp = arctan(py/px), as in Eq. (2.38) for the wave vector q, which is replaced by the momentum p. The eigenvalues of the unitary and Hermitian matrix on the r.h.s. of Eq. (3.21) are λ = ± and correspond to the positve and negative energy states, respectively, and we have Eλ = λǫ = λpm2 + p2, (3.22) in agreement with Eq. (3.6).PDF Image | Physical Properties of Graphene
PDF Search Title:
Physical Properties of GrapheneOriginal File Name Searched:
CoursGraphene2008.pdfDIY PDF Search: Google It | Yahoo | Bing
Salgenx Redox Flow Battery Technology: Power up your energy storage game with Salgenx Salt Water Battery. With its advanced technology, the flow battery provides reliable, scalable, and sustainable energy storage for utility-scale projects. Upgrade to a Salgenx flow battery today and take control of your energy future.
CONTACT TEL: 608-238-6001 Email: greg@infinityturbine.com (Standard Web Page)