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Chapter 7 Relativistic Quantum Mechanics

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Chapter 7 Relativistic Quantum Mechanics Chapter 7 Relativistic Quantum Mechanics In the previous chapters we have investigated the Schr¨odinger equation, which is based on the non-relativistic energy-momentum relation. We now want to reconcile the principles of quantum mechanics with special relativity. Schr¨odinger actually first considered a relativistic equation for de Broglie’s matter waves, but was deterred by discovering some apparently unphysical prop- erties like the existence of plane waves with unbounded negative energies E = p2c2 + m2c4: − Classically a particle on the positive branch of the square root will keep E mc2 forever ≥p but quantum mechanically interactions can induce a jump to the negative branch releasing δE 2mc2. Schr¨odinger hence arrived at his famous equation in the non-relativistic context. ≥ Two years later Paul A.M. Dirac found a linearization of the relativistic energy–momentum relation, which explained the gyromagnetic ratio g = 2 of the electron as well as the fine structure of hydrogen. While his equation missed its original task of eliminating negative energy solutions, it was too successful to be wrong so that Dirac went on, inspired by Pauli’s exclusion principle, to solve the problem of unbounded negative energies by inventing the particle-hole duality that is nowadays familiar from semiconductors. In the relativistic context the holes are called anti-particles. While Dirac first tried to identify the anti-particle of the electron with the proton (the only other particle known at that time) this did not work for several reasons and he concluded that there must exist a positively charged particle with the same mass as the electron. The positron was then discovered in cosmic rays in 1932. Dirac thus made the first prediction of a new particle on theoretical grounds and, more generally, showed the existence of antimatter as an implication of the consistency of quantum mechanics with special relativity. This initiated the long development of the modern quantum field theory of elementary particles and interactions. After a brief discussion of the problem with negative energy solutions we now construct the Dirac equation and analyze its nonrelativistic limit. The issue of Lorentz transformations and 127 CHAPTER 7. RELATIVISTIC QUANTUM MECHANICS 128 further symmetries will be taken up in the chapter on symmetries and transformation groups. The Klein-Gordon-equation. If we start with the relativistic energy-momentum relation E2 = m2c4 + c2~p 2. (7.1) and use the correspondence rule E i~∂ and ~p ~ ~ we obtain → t → i ∇ (i~∂ )2 ψ(~x, t)=(m2c4 c2~2∆)ψ(~x, t), (7.2) t − which can be written as m2c2 + ψ(~x, t) = 0 (7.3) ~2 2 in terms of the d’Alembert operator := 1 ∂ ∆, briefly called “box.” While Schr¨odinger c2 ∂t2 − already knew this relativistic wave equation, it was later named after Klein and Gordon who first published it. An immediate problem for the use of (7.3) as an equation for quantum mechanical wave functions is the existence of negative energy solutions i i Et+ ~p~x 2 2 4 2 ψ (~x, t)= e− ~ ~ with E = mc + p c mc (7.4) E − ≤ − p so that the energy is unbounded from below. Once interactions are turned on electrons could thus emit an infinite amount of energy, which is clearly unphysical. If we try to avoid the negative energy solutions of the Klein-Gordon equation (7.3) by using an expansion of the positive square root, 2 2 4 6 2 p 2 p p p 2 E = mc 1+ 2 2 = mc + 3 2 + 5 4 . mc , (7.5) r m c 2m − 8m c 16m c − ≥ the Hamiltonian becomes an infinite series with derivatives of arbitrary order and we loose locality. More concretely, it can be shown that localized wavepackets cannot be constructed without contributions from plane waves with negative energies [Itzykson,Zuber]. 7.1 The Dirac-equation Dirac tried to avoid the problem with the negative energy solutions by linearization of the equation for the energy. We get an idea for how this could work by recalling the equation σiσj = 2 2 ½ δij ½ + iεijkσk for the Pauli matrices σi, which implies (~σ~v) = σiviσjvj = ~v . For massless particles we thus obtain the relativistic energy-momentum relation from a linear equation 2 2 2 2 2 ½ Eψ = c~σ~pψ E ½ = c (~p~σ) = c p . (7.6) ± ⇒ CHAPTER 7. RELATIVISTIC QUANTUM MECHANICS 129 Upon quantization the suggested Hamiltonian H = c~p~σ becomes a 2 2 matrix of momentum ± × operators because 0 1 0 i 1 0 p3 p1 ip2 ~ ∂3 ∂1 i∂2 ~σ = , − , ~p~σ = piσi = − = i − . (7.7) 1 0 i 0 0 1 ⇒ p1 + ip2 p3 ∂1 + i∂2 ∂3 − − − Equation (7.6) indeed shows up as the massless case of the Dirac equation. It is called Weyl equation and it describes the massless neutrinos of the standard model of particle physics. In 1928 Dirac made the following ansatz for a linear relation between energy and momenta 2 E ½ = cp α + mc β (7.8) · i i with four dimensionless Hermitian matrices αi and β. Taking squares and demanding the relativistic energy-momentum relation (7.1) we find 2 2 4 2 2 3 2 2 4 ½ E ½ =(m c + c pipi) = c αiαjpipj + mc pi(αiβ + βαi)+ β m c , (7.9) which is equivalent to the matrix equations 2 ½ β = ½, α , β = 0, α , α = 2δ , (7.10) { i } { i j} ij where A, B = AB + BA denotes the anticommutator. Assuming the existence of a solution { } we arrive at the free Dirac equation ~ i~∂ ψ = Hψ, H = cα ∂ + βmc2 (7.11) t i i i with H = H†. The coupling to electromagnetic fields is achieved as in the non-relativistic case with the replacement ~p ~ ~ e A~ and E i~∂ V . With V = eφ this leads to the → i ∇ − c → t − interacting Dirac equation ∂ ~ e i~ eφ ψ = cα ∂ A ψ + βmc2ψ, (7.12) ∂t − i i i − c i for charged particles in an electromagnetic field, where we still need to find matrices αi and β representing the Dirac algebra (7.10). While α = σ would satisfy α , α = 2δ ½ it is impossible to find a further 2 2 matrix i i { i j} ij × β solving (7.10). A simple argument shows that the dimension of the Dirac matrices has to be 2 even: Since β = ½ and α β = βα cyclicity of the trace tr(Mβ) = tr(βM) implies i − i tr α = tr α β2 = tr(α β)β = tr β(α β)= tr β(βα )= tr α (7.13) i i i i − i − i and hence tr α = 0 (similarly tr β = tr(βα )α = tr α (βα ) = tr α (α β) = tr β shows i 1 1 1 1 − 1 1 − 2 2 that the trace of β also has to vanish). On the other hand, αi = β = ½ implies that all CHAPTER 7. RELATIVISTIC QUANTUM MECHANICS 130 eigenvalues of α and β must be 1 so that tr α = tr β = 0 entails that half of the eigenvalues i ± i are +1 and the other half are 1. This is only possible for even-dimensional matrices. Dirac − hence tried an ansatz with 4 4 matrices and found the solution × 1 0 σi ½ 0 αi = αi† = αi− αi = , β = , 1 , (7.14) σi 0 0 ½ β = β = β − † − where we used a block notation with 2 2 matrices ½, 0 and σi as matrix entries. In full gory 0001× 0 0 0 i 0 0 1 0 10 0 0 000101 00 0 i −01 00 0 0 11 001 0 0 1 − detail the Dirac matrices read α1 = B0100C, α2 = B0 i 0 0C, α3 = B1 0 0 0 C, β = B0 0 1 0 C, but one B C B − C B C B − C @1000A @i 0 0 0A @0 10 0 A @00 0 1A − − should never use these explicit expressions and rather work with the defining equations (7.10), or possibly with the block notation (7.14) if a splitting of the 4-component spinor wave function T ψ = (ψ1,ψ2,ψ3,ψ4) into two 2-component spinors is unavoidable like in the non-relativistic limit (see below). It can be shown that (7.14) is the unique irreducible unitary representation of 1 1 the Dirac algebra (7.10), up to unitary equivalence α Uα U − , β UβU − with UU † = ½. i → i → Relativistic spin. In order to derive the spin operator for relativistic electrons we observe that the orbital angular momentum L~ = X~ P~ is not conserved for the free Dirac Hamiltonian × [H,L ] = [c~αP~ + βmc2,ε X P ]= ~ cε α P = 0 (7.15) i ijk j k i ijk j k 6 ~ ~ ~ ~ because [Pl, Xj]= i δlj. A conserved operator J = L + S can be constructed by observing that [H,ε α α ]= ε [c~αP~ , α α ]= ε cP ( α , α α α α , α ) (7.16) ijk j k ijk j k ijk l { l j} k − j{ l k} = ε cP (2δ α 2δ α ) = 4cε P α ijk l lj k − lk j ijk j k because [A, BC]= ABC + BAC BAC BCA = A, B C B A, C . The spin operator − − { } − { } ~ ~ σi 0 Si = εijkαjαk = (7.17) 4i 2 0 σi therefore yields a conserved total angular momentum J~ = L~ + S~. 1 Lorentz covariant form of the Dirac equation. Multiplication with the matrix c β from the left recasts the relation E eφ = c~α(~p e A~)+ βmc2 into − − c β( 1 E e φ) β~α(~p e A~) mc ψ = 0. (7.18) c − c − − c − The standard combination of coordinates, energy-momenta and gauge potentials into 4-vectors xµ =(ct, ~x), ∂ =( 1 ∂ , ~ ), pµ =( 1 E, ~p),Aµ =(φ, A~), (7.19) µ c t ∇ c which entails the relativistic version p P = i~∂ of the quantum mechanical correspondence µ → µ µ (2.3), suggests the combination of β and βα into a 4-vector γµ of 4 4 matrices as i × 1 0 0 0 µ µ ν µν µν 00 1 0 01 ½ − γ =(β, β~α) γ ,γ = 2g with g = B0 0 1 0C.
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