Sunday, March 13, 2011

Mastering Physics: Classical and Quantum Harmonic Oscillators

Consider a harmonic oscillator with mass m=0.100\; \rm kg and k=50\;\rm N/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 \hbar=1.055\times10^{-34}\; \rm J\cdot s.

Part A
Let this oscillator have the same energy as a mass on a spring, with the same k and m, released from rest at a displacement of 5.00 \;\rm cm from equilibrium. What is the quantum number n of the state of the harmonic oscillator?
Express the quantum number to three significant figures.

  n  = 2.650×1031

Part B
What is the separation DeltaE between energy levels in this harmonic oscillator?
Express your answer in joules to three significant figures.

  DeltaE  = 2.360×10−33
  \rm J
This energy is far smaller than you could possibly measure in an experiment with a mass on a spring. Just as for a classical harmonic oscillator, in experiments this huge quantum oscillator would appear as though its energy could take any value.

Part 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.
Express your answer in meters to three significant figures.

  \rm m
Since the diameter of an atomic nucleus is on the order of 10^{-15}\; \rm 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 seem to be able to take all values.
It is interesting to see that quantum mechanics reduces to classical mechanics on the scales of energy and size for which classical mechanics has been successful. However, to truly understand how the strange quantum world gives rise to the classical world of everyday experience requires the principle of decoherence, which describes how quantum states reduce to classical ones through the interactions of large systems with their environment.


  1. Yet again you have saved my life. haha. I did not understand Part C at all. Thank you!

  2. You're welcome! That's my goal.