1. If, as in non-relativistic quantum mechanics, the momentum operator has representation p = -i? x , then then show that the Hamiltonian H = x 3 p+px 3 is formally self-adjoint. 2. Solve Laplaces equation f xx (x, y) + f yy (x, y) = 0 subject to the boundary conditions f(x, 0) = sin(px), f(x, 1) = sin(px)e -p , f(0, y) = 0, f(1, y) = 0 and show that the solution is f(x, y) =sin(px)e -py 3. Show that f(x, y) = x 4 – x 2 y 2 + y 4 is a harmonic function.
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