Aparticle, initially (t -> 0) in the ground state of an infinite, 1d potential box with walls at r 0 and = a, is subjected at time t = 0 to a time-dependent perturbation v (r, t) et/7, with eo a small real number a) calculate to first order the probability of finding the particle in an excited state for t 0. consider all final states. are all possible transitions allowed? b) examine the time dependence of the problem: make a sketch and describe with words the behavior of the transition probability as a function of time induced by the perturbation c) calculate the probability that, after a sufficiently long time, the system will have made such a transition (i. e., analyze the limit t -> 00) d) how would the previous results you found in a)-c) change if the particle starts from the ground state in a 1d harmonic oscillator potential? be specific and justify your answers appropriately

Astudent throws a water balloon with speed v0 from a height h = 1.76 m at an angle θ = 21° above the horizontal toward a target on the ground. the target is located a horizontal distance d = 9.5 m from the student’s feet. assume that the balloon moves without air resistance. use a cartesian coordinate system with the origin at the balloon's initial position. (a) what is the position vector, rtarge t, that originates from the balloon's original position and terminates at the target? put this in terms of h and d, and represent it as a vector using i and j. (b) in terms of the variables in the problem, determine the time, t, after the launch it takes the balloon to reach the target. your answer should not include h. (c) create an expression for the balloon's vertical position as a function of time, y(t), in terms of t, vo, g, and θ. (d) determine the magnitude of the balloon's initial velocity, v0, in meters per second, by eliminating t from the previous two expressions.