The force exerted by the walls of an infinite square well on a wave packet: Ehrenfest theorem, revivals and fractional revivals

The temporal evolution of a wave packet psi(x, 1) confined to an infinite square well (occupying the region 0 less than or equal to x less than or equal to L), its manifested by fractional revivals of the probability density P(x, t) equivalent to \psi(x,t)\(2) and the time-dependence of the expectation values of position, momentum and force (X(t) equivalent to <x>(t), Pi(t) equivalent to <(p) over cap >(t) and Phi(t) equivalent to <(F) over cap >(t), respectively) are studied, using wave packets for which Pi(0) = 0. In the region accessible to the wave packet (0 < x < L), (p) over cap and the Hamiltonian (H) over cap commute; still, (p) over cap is, in general, not a constant of the motion, and the three expectation values are all time-dependent, except when the initial packet has a definite parity, that is psi(x, 0) = +/-psi(L - x, 0), which implies X(t) = L/2 and Pi(t) = 0 = Phi(t). For it packet with no definite parity, Y(t) equivalent to X(t) - L/2 stays, if the packet is sufficiently narrow at t = 0, close to zero nearly all the time, but executes large excursions in the neighbourhood of instants at which the probability density experiences a mirror revival or a revival. This behaviour, with no classical counterpart. can be understood by paying attention to the presence of infinite potential barriers (at x = 0 and x = L) and taking account of the net force felt by an initially off-centred wave packet. It is shown that though plots (against time) of expectation values catch it glimpse of the revival structure of a wave packet, such plots fail to reveal some fractional revivals.