Harmonic oscillator (9 pt) a) (6 pt) calculate the expectation value of the following operators with respect to the state in) of simple harmonic oscillator( i. e. (nlî |xn), (n|p|n), (a) î (b) x (c) p. (d) ń 2 (e) 2 (f) (g) (h) 3 4 5 (i) î6 b) (3 pt) use the result of (a) to calculate the energy en for a particle in the eigenstaten) of the harmonic oscillator, whose hamiltonian is, à = "* +3mw? 82 – a83 – bx+4 +385

Asquare 1m x 1m plate is shown below. all four of its edges are kept at 0 co. at t=0the temperature distribution is given by: y xtsins n75)=.a) using separation of variables determine the temperature distribution t(x,y,t)along the plate at any time t.b) discretize the heat conduction equation +=22222 txtusing a second-order centered difference scheme for the two second derivatives and a first-order forward euler scheme for the first derivative. using von neumann stability analysis, determine the stability criterion for the proposed discretization scheme in terms of , hand t. solve for the maximum allowable time step tmax