Dirac Equation. Consider the motion of an electron in the absence of an electromagnetic field. In classical relativity, electron energy, , is related to electron momentum, , according to the well-known formula. (1112) where is the electron rest mass. The quantum mechanical equivalent of this expression is the wave equation. The Dirac Equation The Hydrogen Atom Why do we need the Dirac Equation? The mathematical Formalism Klein-Gordon equation Dirac equation Dirac equation Ansatz In order to find a wave equation that describes a free electron Dirac made the ansatz H D = α.p+βm. α ≡ (α x,α y,α z) and βare hermitian operators only working on the spin 5.4 The Dirac Equation The problems with the Klein-Gordon equation led Dirac to search for an alternative relativistic wave equation in 1928, in which the time and space derivatives are first order. The Dirac equation can be thought of in terms of a "square root" of the Klein-Gordon equation. In covariant form it is written: iγ0 ∂ ∂t So that the equation obeys by the 4-component spinor uα(p;λ) describes a particle which is fion-shellfl i.e. p2 =m2. The Dirac equation for the wave-function of a relativistic moving spin-1 2 particle is obtained by making the replacing pµ by the operator i∂µ giving iγµ∂µ m β α Ψβ(x) = 0; which has solution Ψα(x) = e ipxuα(p Because the Dirac equation implies the Klein Gordon equation, we need to satisfy p 2+ m = 0. Since mis real, this implies that p corresponds to a future pointing object. In particular, we can go to a reference frame where p~= 0, and p0 = m= E. In this reference frame, the Dirac equation takes the following form 0 B B @ m 0 E 0 0 m 0 E E 0 m 0 0 Abstract. In this chapter, the solutions of the Dirac equation for a fermion in an external electromagnetic field are presented for the cases of a pure magnetic field of arbitrary strength, of a strong magnetic field when fermions occupy the ground Landau level, and of a crossed field. The density matrix of the plasma electron in a magnetic The symmetry properties of Maxwell's equations are derived from the symmetries of the massless Dirac equation, and the conservation laws for the electromagnetic field are obtained from those for In particle physics, the Dirac equation is a relativistic wave equation derived by British physicist Paul Dirac in 1928. In its free form, or including electromagnetic interactions, it describes all spin-1 ⁄ 2 massive particles such as electrons and quarks for which parity is a symmetry.It is consistent with both the principles of quantum mechanics and the theory of special relativity, and to act upon. We introduce the Dirac spinor field ↵(x), an object with four complex components labelled by ↵ =1,2,3,4. Under Lorentz transformations, we have ↵(x) ! S[⇤]↵ (⇤ 1x)(4.22) where ⇤=exp 1 2 ⌦ ⇢M ⇢ (4.23) S[⇤] = exp 1 2 ⌦ ⇢ S ⇢ (4.24) Although the basis of generators M⇢ and S⇢ are di↵erent, we use the The Dirac Equation Our goal is to find the analog of the Schrödinger equation for relativistic spin one-half particles, however, we should note that even in the Schrödinger equation, the interaction of the field with spin was rather ad hoc. There was no explanation of the gyromagnetic ratio of 2. One can incorporate spin into the non-relativistic equation by using the Schrödinger-Pauli 3. Dirac equations. - We must now direct our attention to a more correct theory of electric phenomena. The Dirac