# Dirac expected his relativistic equation to contain the Klein-Gordon equation as its square since this equation involves the relativistic Hamiltonian in its normal invariant form.

Non-relativistic approximation of the Dirac equation in an electromagnetic field. In an electromagnetic field (Φ,A) the Dirac equation for plane waves with fixed energy is (E−m− −A) −(+ − −A) (−) = +− −−) + ≈− = −−)+) =⋅+×) = (−)+ −)×(−)+ (−) ×(−) =×+× −×− ×

Also, logical issues with Dirac’s equation: (iv) diﬃcult to distinguish particle from an- 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-½ 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 was the first theory to account fully for special relativity in the context of quantum Non-relativistic approximation of the Dirac equation in an electromagnetic field. In an electromagnetic field (Φ,A) the Dirac equation for plane waves with fixed energy is (E−m− −A) −(+ − −A) (−) = +− −−) + ≈− = −−)+) =⋅+×) = (−)+ −)×(−)+ (−) ×(−) =×+× −×− × The Dirac equation describes the behaviour of spin-1/2 fermions in relativistic quantum ﬁeld theory. For a free fermion the wavefunction is the product of a plane wave and a Dirac spinor, u(pµ): ψ(xµ)=u(pµ)e−ip·x(5.21) Substituting the fermion wavefunction, ψ, into the Dirac equation: (γµp. µ−m)u(p) = 0 (5.22) 27.

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Among its big successes is the very accurate description of the energy levels of the hydrogen atom. In the historical development, however, the occurrence of several paradoxa has made it dicult to nd an appropriate interpretation. 4. The Dirac Equation “A great deal more was hidden in the Dirac equation than the author had expected when he wrote it down in 1928. Dirac himself remarked in one of his talks that his equation was more intelligent than its author. It should be added, however, that it was Dirac who found most of the additional insights.” Weisskopf on Dirac 1 Notes and Directions on Dirac Notation A. M. Steane, Exeter College, Oxford University 1.1 Introduction These pages are intended to help you get a feel for the mathematics behind Quantum Mechanics.

## 4. The Dirac Equation “A great deal more was hidden in the Dirac equation than the author had expected when he wrote it down in 1928. Dirac himself remarked in one of his talks that his equation was more intelligent than its author. It should be added, however, that it was Dirac who found most of the additional insights.” Weisskopf on Dirac

Diracmått sub. delta (Förord, 2b2) Cole, J.D. (1951), ”On a Quasi-Linear Parabolic Equation Occurring in (Förord) Dirac, P.A.M.

### Dot this equation from the left with some other ket |ϕ : ϕ|ψ = ∑ n ϕ|xn xn|ψ and let the position eigenstates tend to a continuum of states: ϕ|ψ = ∫ ϕ|x x|ψ dx In other words, ϕ|ψ = ∫ ϕ∗(x)ψ(x)dx which is why the amplitude can also be called an overlap integral: this integral

Paul Dirac formulated the equation in 1928. The equation describes the behaviour of fermions (e.g. electrons and quarks), and takes special relativity into account. The equation showed the existence of antimatter. It does not change in Lorentz transformation. The Dirac equation is the fundamental equation for relativistic quantum mechanics. Among its big successes is the very accurate description of the energy levels of the hydrogen atom.

Dirac equation is the relativistic extension to Shrodinger's equation.

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For a free fermion the wavefunction is the product of a plane wave and a Dirac spinor, u(pµ): ψ(xµ)=u(pµ)e−ip·x(5.21) Substituting the fermion wavefunction, ψ, into the Dirac equation: (γµp. µ−m)u(p) = 0 (5.22) 27. Correspondingly, the Dirac equation has two types of plane-wave solutions, which we denote by upos and uneg .

It could also be more explicit: , the particle hasp = 2 momentum equal to 2; , the particle has position 1.23. represents a system inx =1.23 Ψ the state Q and is therefore called the state vector.

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### Furthermore, the Dirac equation has the form of the relativistic energy relation. These correspondences indicate that these equations originate, not just formally,

For a free fermion the wavefunction is the product of a plane wave and a Dirac spinor, u(pµ): ψ(xµ)=u(pµ)e−ip·x(5.21) Substituting the fermion wavefunction, ψ, into the Dirac equation: (γµp. µ−m)u(p) = 0 (5.22) 27. Correspondingly, the Dirac equation has two types of plane-wave solutions, which we denote by upos and uneg . For each p R, 1 pos pos u neg (p; x, t) = u neg (p) eipxi (p)t, (8) 2 where upos (p) and uneg (p) are eigenvectors of the matrix h0 (p) belonging to the eigenvalues (p) and (p), respectively. Hence, we have. The Dirac equation is an equation from quantum mechanics. Paul Dirac formulated the equation in 1928.