Problem 1;

Consider a free real scalar field o(T ), where ty = x, y, z for / = 1, 2, 3 and #4 = ict, satisfying the Klein-Gordon equation. (a) Write down the Lagrangian density for the system. (b) Using Eulerfe equations of motion, verify that $ does satisfy the Klein-Gordon equation. (c) Derive the Hamiltonian density for the system. Write down Hamil- tonte equations and show that they are consistent with the equation derived in (b)(a) Write down the Dirac equation in Hamiltonian form for a free par- ticle, and give explicit forms for the Dirac matrices. (b) Show that the Hamiltonian H commutes with the operator o . P where P is the momentum operator and o is the Pauli spin operator in the space of four component spinors. (c) Find plane wave solutions of the Dirac equation in the representation in which o- P is diagonal. Here P is the eigenvalue of the momentum operator.Consider an idealized (point charge) A1 atom (Z = 13,11 = 27). If a negative lepton or meson is captured by this atom it rapidly oaeeadee down to the lower n states which are inside the electron shells. In the case of p-capture: (a) Compute the energy E1 for the p in the n = 1 orbit; estimate also a mean radius. Neglect relativistic effects and nuclear motion. (b) Now compute a correction to E to take into account the nuclear motion. (c) Find a perturbation term to the Hamiltonian due to relativistic kinematics, ignoring spin. Estimate the resulting correction to E1. (d) Define a nuclear radius. How does this radius for Al compare to the mean radius for the n = 1 orbit from (a)? Discuss qualitatively what happens to the / when the atomic wave function overlaps the nucleus substantially. What happens to a 7- under the circumstances? Information that may be relevant: My = 105 MeV/c', SPIN(u) = 1/2, M. = 140 MeV/c2, SPIN(7) = 0