0.1 Klein-Gordon Equation 0.2 Dirac Equation

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0.5 NR Limit Nothing better shows the physics buried in the Dirac equation than taking the NR limit. To understand, we must add the coupling of the Dirac electron to the electromagnetic field (discussion below in Dirac Equation as a Field Equation). γ µ (−i∂ µ + eA µ ) ψ + mψ = 0 (8) or in Hamiltonian form H = −eA 0 ψ + γ 0 γ i (p i + eA i )ψ + mγ 0 ψ (9) Thus Hψ = Eψ in two-component form with E moved to left is ( m − eA 0 − E ⃗σ · (⃗p + eA) ⃗ ) ( ) χ ⃗σ · (⃗p + eA) ⃗ −m − eA 0 − E χ ′ = 0. Eliminate χ ′ in favor of χ, and A 0 = φ: ( (m − E − eφ) + ⃗σ · ⃗Π 1 m + E + eφ ) ⃗σ · ⃗Π χ = 0 (10) where we used the abbreviation ⃗ Π = ⃗p+e ⃗ A. To get to the NR limit, subtract the rest energy from E to get the NR energy W = E − m and write the denominator 1 2m + (W + eφ) ≈ 1 2m − 1 1 (W + eφ) + 4m2 8m (W + 3 eφ)2 + . . . First term in () in Eq.(10) is O(mv 2 ); second is O(mv 2 ) plus m times higher powers of v 2 . Using the commutator [p i + eA i , p j + eA j ] = −ieF ij and the identity σ i σ j = g ij + iɛ ijk σ k , and keep terms up to order mv 2 , this becomes ( −W − eφ + 1 Π 2m ⃗ 2 + e ) ⃗σ m 2 · ⃗B χ = 0. (11) This is the Pauli equation for a spinning NR particle. The last term is the interaction of the spin magnetic moment with the magnetic field. The magnetic moment is (e/m) S ⃗ rather than the orbital (e/2m) L, ⃗ indicating the g-factor of 2 put in by hand in pre-Dirac times and justified a postiori as the Thomas precession. 6
In higher order, the equivalence of DE to Pauli equation is approximate because the energy eigenvalue appears in the expansion of the second term in Eq.(10) so that is does not have the same form as the Pauli equation. Nonetheless, we can obtain an approximate Pauli equation from the next term in the expansion above as follows: − 1 4m 2 (⃗σ · ⃗Π)(W + eφ)(⃗σ · ⃗Π) = − 1 4m 2 (⃗σ · ⃗Π) 2 (W + eφ) − 1 4m 2 (⃗σ · ⃗Π)[eφ, ⃗σ · ⃗Π]. It is correct to order v 4 to replace W + eφ in the first term on the right by (⃗σ · ⃗Π) 2 according to Eq.(11). In the absence of a magnetic field ⃗ A = 0, the result is the order v 4 relativistic kinetic energy correction −p 4 /8m 3 . Taking the commutator in the second term and using the σ i σ j identity yields second = ie 4m (δ 2 ij + iɛ ijk σ k )Π i ∂ j φ in which the first term is known as the Darwin term and the second is the spin-orbit term which was added phonomenologically to the Pauli equation in the olden days. To appreciate the physical significance of these terms, let’s specialize to A ⃗ = 0, whence the Darwin term becomes Darwin = − ie 4m 2 (∇2 φ − ⃗ E · ⃗∇). This spin independent term represents a relativistic correction to the effective potential which has obscure experimental consequences. For the pure coulomb field, the first term is a delta function which shifts the energy of H- atom s-states. The significance of the spin-orbit term appears if we specialize to central potentials φ(r) where it becomes SO = − e 1 ∂φ ⃗σ · ⃗L. 4m 2 r ∂r COMMENT A more systematic and elegant way of approaching the NR approximation is the Foldy-Wouthuysen (FW) transformation. The idea is to make a unitary transformation of the states and operators, ψ = Uψ and O F W = UOU † with UU † = 1, such that the new Hamiltonian is block diagonal, decoupling the upper and lower components and giving separate two-component equations for each. For a free particle, this is accomplished by √ m + ω U = 2ω + γ i p √ i , 2ω(ω + m) 7
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 for boosts Σ 0i = − 1 2 ( 0
Page 9 and 10: is conserved. The operator P define
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

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