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EXAMPLE: Prove Eq. 14) and determine ρ and j for this value of ψ. Taking the derivatives of ψ with respect to time and space, we obtain ∂ψ/∂t = −iEψ and ∇2 ψ = −p2 ψ. Substituting these into Eq. 3), we find the nonrelativistic expression for the kinetic energy, Eq. 1). Then using the definitions of ρ and j in terms of ψ, we find ρ = |N |2 p j = |N |2 . 16) This is the nonrelativistic solution. However, Eq. 1) is not Lorentz covariant. 2. Derivation of the cross section 29 EXAMPLE: Show this. Write p2 = p2T + p2z to separate the transverse and longitudinal momenta.
62) Tf i = i d4 xφ∗f [ie(Aμ ∂μ + ∂μ Aμ )] φi . Since the partial derivative in the second term of Eq. 62) acts on both Aμ and φi , we can integrate by parts to get it to act on φ∗f instead. 63) ≡ −i d4 xjfμi Aμ , where we have multiplied the definition of the current in Eq. 19) by −e to obtain the electromagnetic current, called the current for ‘spinless’ electron scattering in Halzen and Martin [4]. EXAMPLE: If φi (x) = Ni exp(−ipi · x) and φf (x) = Nf exp(−ipf · x), calculate jfμi . We assume N is real.
Cross sections hadrons. This technique is referred to as the wave-optical model and provides a rather general description of scattering. Consider a plane wave, normalized to unity, ψi = exp(ikz), where k = 1// λ and λ = 2π/ λ is the de Broglie wavelength. The time depen dence has been omitted. Instead of considering a preferred direction, such as two particles moving toward each other, the plane wave is rep resented by a superposition of incoming and outgoing spherical waves. ) At a large distance r from the center, kr � 1, the spherical wave has the form exp(±ikr)/kr to conserve probability.