(i) Write down the time independent Schrödinger equation for the harmonic oscillator potential. Then confirm that the ground state wave function (n= 0) is a solution with the energy 퐸(=)*++푛.ℏ휔,푛=0,1,2,...(plug in the function, differentiate, and confirm it is a solution).

Atank has a shape of a cone with a radius at the top of 2 m and a height of 5 m. the tank also has a 1 m spout at the top of the tank. the tank is filled with water up to a height of 2 m. find the work needed to pump all the water out the top of the spout. (use 9.8 m/s2 for g and the fact that the density of water is 1000 kg/m3.)