0.1 Klein-Gordon Equation 0.2 Dirac Equation

More documents

Recommendations

Info

$' \^' I$

in which ω is the operator + √ ⃗p 2 + m 2 which commutes with ⃗p. It is easily verified that UU † = 1, and a lengthy calculation shows that the FW Hamiltonian reduces to ( ) ω 0 H F W = . 0 −ω Thus the upper components represent the positive energy solutions and the lower components represent the negative energy solutions. Although the momentum operator is not altered by the FW transformation, since U commutes with ⃗p, the position operator is modified and becomes a non-local operator in a way that is difficult to calculate in this context. This is an indication of the fact that it is not possible to form a state that is localized to a single point using positive energy solutions only. At best, the paritcle can be localized to within a Compton wavelength m −1 . It is possible to do a FW transformation for particle subject to potentials, but it must be done sucessively for each higher order in v 2 . Omit it here. 0.6 H Atom 0.6.1 Hamiltonian The relativistic treatment of the H-atom is a major success of the <strong>Dirac</strong> theory. More generally, we treat a particle in a central field in the absence of a magnetic field. In particular, we neglect the (weak) magnetic field due to the nuclear magnetic moment. With the <strong>Dirac</strong> equation γ 0 (−i∂ 0 + eA 0 )ψ + γ i ∂ i ψ + mψ = 0 as a starting point, we rewrite in the form of a Schrödinger equation i∂ t ψ = Hψ with the Hamiltonian H = γ 0 γ i p i − α r + γ0 m ( ) m − α ⃗σ · ⃗p = r ⃗σ · ⃗p −m − α (12) r where α = e 2 /4π is the fine structure constant, value approximately 1/137. −α/r can be replaced by any central potential −eφ(r) or V (r). 0.6.2 Conserved Quantities As noted earlier, the total angular momentum ⃗ J is conserved. Therefore we can have simultaneous eigenstates of J 2 , J z , and H. in addition, the parity 8
is conserved. The operator P defined by P ψ(⃗r) = ψ(−⃗r) does not commute with the Hamiltonian because it changes the sign of the momentum operator. However P = γ 0 P does commute. Note that P commutes with the spin operator S i = i 4 ɛ ijkγ j γ k , so it does not change its value. This is proper because the spin, like the orbital angular momentum, should be a good axial vector and not change sign under space reflection. According to NRQM, operators J 2 , L 2 , and J z have eigenstates, generally denoted by Ylj m. j can be either l + 1 2 or l − 1 2 . We introduce the quantity ω with values ±1 such that j = l + 1 2ω. The parity P of the eigenstate is (−1) j− ω 2 so that states with different values of ω have different parity. We denote the eigenstates by Yωj m and they satisfy J 2 Y m ωj = j(j + 1)Y m ωj J z Y m ωj = mY m ωj L 2 Y m ωj = (j − ω 2 )(j − ω 2 + 1)Ym ωj PY m ωj = (−1) j− ω 2 Y m ωj Don’t need to know their explicit forms. 0.6.3 Eigenstates Look for solutions that are simultaneous eigenstates of H, P, and angular momentum. Since P = γ 0 P looks like ( ) P 0 P = , 0 −P the parity of the upper components is opposite to the parity of the lower components. Our eigenstate must look like ψ = 1 ( ) f(r)Y m ωj r ig(r)Y−ωj m , (13) where the i and 1/r are put in for convenience. The following identities ease the calculation along. Try to prove them, but proofs will be provided in class. ˆr is the unit radial vector. i.) ⃗σ · ⃗p = −i⃗σ · ˆr∂ r + 9 i⃗σ · ˆr ⃗σ · r ⃗L
Page 1 and 2: [revised Oct 15, 2010] 0.1 Klein-Go
Page 3 and 4: where ψ 0 is a constant 4-componen
Page 5 and 6: and for boosts Σ 0i = − 1 2 ( 0
Page 7: In higher order, the equivalence of
Page 11 and 12: and a similar series for G with coe
Page 13 and 14: expressed in terms of the fields. A
Page 15: This Lagrangian leads to solutions

0.1 Klein-Gordon Equation 0.2 Dirac Equation

Create successful ePaper yourself

Delete template?

Save as template?