PDF Publication Title:
Text from PDF Page: 050
46 The Dirac Equation for Relativistic Fermions 3.1 Relativistic Wave Equations In order to obtain a relativistic wave equation, one may proceed in the same manner as in the derivation of the Schr ̈odinger equation for non-relativistic particles from classical Hamiltonian mechanics. Remember that the classical Hamiltonian of a particle in a potential V (r) is a function p2 H(p, r) = 2m + V (r), (3.1) defined in phase space, the space spanned by the particle’s position r and its canonical momentum p. In quantum mechanics, the position and the momentum no longer commute, and one reinterprets Eq. (3.1) in terms of operators acting on vectors of a Hilbert space, when one replaces H → i h ̄ ∂ t a n d p → − i h ̄ ∇ , ( 3 . 2 ) where ∇ is the gradient with respect to the position r, and ∂t is the time derivative. This procedure yields the well-known Schr ̈odinger equation ih ̄∂tψ(r, t) = − h ̄ ∇2 + V (r) ψ(r, t) (3.3) 2m of basic quantum mechanics, where the wave function ψ(r,t) is the spatial representation of the Hilbert vector. One may indeed apply this procedure to a free relativistic particle of mass m the energy dispersion of which is given by E=pm2c4+p2c2 , (3.4) in terms of the velocity of light c. Whereas the taylor expansion for |p| ≪ mc yields the non-relativistic dispersion mc2 + p2/2m, apart from the mass energy mc2, the ultra-relativistic limit of vanishing mass yields E = c|p|. This indicates that we need to treat, in relativistic (quantum) mechanics the energy E and the momentum p on the same footing, in contrast to the non-relativistic limit, where the energy is proportional to the square of the momentum. In relativistic mechanics, one treats indeed the energy, due to the Lorentz invariance, as the time-component of a D + 1 dimensional momentum vector, pμ ≡ (E/c, p), or its ”covariant” vector pμ = (E/c, −p), to account for the particular metric of the D+1-dimensional space-time. ThePDF 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 | RSS | AMP |