A = (m, (x) + imz (x) 1. We consider the eigenvalue problem of the Dirac Hamiltonian AY = 24 with m; (2) m2(x)), 4(x)=(26) ia A is a 2 x 2 matrix operator and is a 2 x 1 column matrix of two complex functions. The eigenvalue problem is defined with real 2, on the interval x [0, b) and all m, (x), m2(x) and m3(x) are real. We also define the inner product (4,4g) = (x)45(x)dx. a) Show that the boundary conditions .(0) .(b) = el 200) 2(b) for 6, 6 R, make A truly self-adjoint with respect to the inner product. [20 marks] b) Find the eigenfunctions, and eigenvalues in the case that m = m2 = 0 while 8, = 0 and is an arbitrary real angle. [20 marks] = A = (m, (x) + imz (x) 1. We consider the eigenvalue problem of the Dirac Hamiltonian AY = 24 with m; (2) m2(x)), 4(x)=(26) ia A is a 2 x 2 matrix operator and is a 2 x 1 column matrix of two complex functions. The eigenvalue problem is defined with real 2, on the interval x [0, b) and all m, (x), m2(x) and m3(x) are real. We also define the inner product (4,4g) = (x)45(x)dx. a) Show that the boundary conditions .(0) .(b) = el 200) 2(b) for 6, 6 R, make A truly self-adjoint with respect to the inner product. [20 marks] b) Find the eigenfunctions, and eigenvalues in the case that m = m2 = 0 while 8, = 0 and is an arbitrary real angle. [20 marks] =