Answer Happy

Background The potential energy of a harmonic oscillator is given by VCR) A solution for the quantum mechanical harmonic oscillator (not the solution) differential equation is W-(x) = N,H,(a1/2x)e-ax? /2 where 1/2 a = and N, = 2*M131729)" v!) The energy expression for this wavefunction is 1/2 E, = 1 *(*)" (v +3) + ) where v = 0,1,2,.. V Part 1 1. Write a function for (x). We are only plotting for qualitative purposes so many constants can be considered to be 1 for simplicity. In this case make a = k = n = u = 1. The Hermite polynomials are a built-in function in NumPy (Google how to use it). 2. Plot the classical harmonic oscillator equation and the wavefunction for v = 0 on top of each other. Consider V(0) = 0. Change the vertical displacement of the wavefunction to be equal to the value of the first energy level given by the equation above. (So 4, (x) + E)

Part 2 3. Change the quantum number v to 1, and replot your graph with the two functions. Part 3 4. Re-plot the above working functions but this time plot the probability distribution of the wavefunction Part 4 We are going to compare the limit of the quantum probability distribution as it approaches the classical distribution. The classical distribution is given by 1 where A, = V2v + 1 5. Plot the classical distribution on top of the quantum mechanical distribution for v = 3,10, and 30 (three separate plots). Make sure to plot in a range that shows both functions. 6. Does it look like the quantum mechanical distribution is approaching the classical one? Comment or explain. 7. You should notice that the quantum distribution lays outside the classical region somewhat. Does the area outside the classical region increase or decrease with increasing v? How come?