Optical potential so far, we assumed that the potential v(x) is real. since the schrödinger equation is already complex, we can ask what happens if we add an imaginary term to it-this is called an 'optical potential'. in class, you derived that the total probability always satisfies p(t)0. evaluate this derivative again, but without assuming that v(x) is real. you will see that it doesn't vanish anymore. now assume that the imaginary part of v(x) is a constant -vo. find a differential equation for p(t). assuming that p(o) 1, solve it to find p(t) in terms of vo and note: we will not use this in the rest of this class, but you will see it again in nuclear physics, where it's used to model particles that can be absorbed or decay. can you see why?

In a given chemical reaction, the energy of the products is greater than the energy of the reactants. which statement is true for this reaction?

Suppose water is leaking from a tank through a circular hole of area ah at its bottom. when water leaks through a hole, friction and contraction of the stream near the hole reduce the volume of water leaving the tank per second to cah 2gh , where c (0 < c < 1) is an empirical constant. a tank in the form of a right-circular cone standing on end, vertex down, is leaking water through a circular hole in its bottom. (assume the removed apex of the cone is of negligible height and volume.) (a) suppose the tank is 20 feet high and has radius 8 feet and the circular hole has radius 2 inches. the differential equation governing the height h in feet of water leaking from a tank after t seconds is dh dt = − 5 6h3/2 . if the height of the water is initially 8 feet, how long will it take the tank to empty? (round your answer to two decimal places.)

Gas cloud 1 is likely to form a star. gas cloud 2 is not. based on this information, match the given conditions with each cloud