Propagator of the free particle (9 pt) we are specifically interested in the propagator for a free particle of mass m. a) (1 pt) write down the hamiltonian h for the free particle. what is the action of h on a wave function (x, t)? (explain) b) (2 pt) consider the wave packet described by the the following initial wave function : (c,0) = pipot/ne-32/(442) (2742)1/4" what are the mean position and the mean momentum of this wave packet? c) (3 pt) what are the uncertainty on the momentum and position for this wave packet? is there an uncertainty relation for these two quantities (explain why)? if yes, what is it, and is it respected ? d) (2 pt) consider the classical function w.) = x2. what is the quantum operator 2 associated with this classical function? is it true for this ope- rator that (12(x) =w(( for a gaussian wave packet (x) = ae-(x-a)/(44²) ? e) (1 pt) the time evolution of a wave function can be described by a propagator, \\(t)) = u(t) |_(0)) u(t) = e-it/h given a wave function at t = 0, i. e. v(, 0), and a propagator u(t), what is the wave function (x, t) at any later time t > 0? derive the general expression, in terms of the matrix elements utc) of the propagator and of the initial wave function y(x, 0).

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