At time t = 0, a particle of mass m in a 2-dimensional harmonic oscillator potential V(x, y) (x + y), = mw 2 where w is a constant, is described by the wave function 2 p(x, y) = $20 (x, y) 02(x, y), 3 where the eigenfunctions Pn,n,(x, y) have energy Enny = hw (1 + nx + ny). Write down the full time-dependent wave function (x, y, t). Is the asso- ciated probability distribution independent of time? Would this change if a small perturbing potential was applied that was i) a function of radius only or ii) a function of y only? [4 marks]