(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 En = (1/2 + n)hw, n = 0,1,2,3... (plug in the function, differentiate, and confirm it is a solution).
(ii) Determine the (quantum) ground state energy for a simple harmonic oscillator consisting of a 100 g mass attached to a spring with a force constant of 10 N/m. What classical amplitude of oscillation A does this energy correspond to? (Recall the total energy of a classical simple harmonic oscillator is E = 1/2kA^2).
(iii) Now, suppose the mass from (ii) oscillates with amplitude A = 1.0 mm. What quantum excited state would this correspond to? Does the corresponding probability density make sense in comparison to what you would observe in the lab? Why or why not?
(iv) Use the time-independent Schroedinger equation with the SHO potential energy to show that the wave function and its second position derivative should be of the opposite sign when E > U and of the same sign when E < U. Show for the ground state SHO wave function that the classical turning points separate these regions.