Consider a harmonic oscillator with mass m=0.100kg and k=50N/m. You may have worked similar problems before, as a mass on a spring using classical mechanics, but this time you will use the solution to the Schrödinger equation for the harmonic oscillator. Keep in mind that this system would be enormous by quantum standards, and in practice you would never expect to use quantum mechanics to describe a mass on a spring. Nonetheless, it is interesting to see what quantum mechanics predicts here. Throughout this problem, use ℏ=1.055×10−34J⋅s. a) Let this oscillator have the same energy as a mass on a spring, with the same k and k, released from rest at a displacement of 5.00cm from equilibrium. What is the quantum number n of the state of the harmonic oscillator?
b) What is the separation ∆E between energy levels in this harmonic oscillator?
c) Nodes are the points where the wave function (and hence the probability of finding the particle) is zero. What is the separation between nodes of the wave function for the mass on a spring described in this problem? Assume that all of the nodes occur in the classically allowed region. Since the diameter of an atomic nucleus is on the order of 10-15 m, the separation that you've calculated is far too small to be measureable in any experiment. Just as for a classical harmonic oscillator, the position of this mass would appear to be able to take all values.