Calculating Probability of Excited State of Harmonic Oscillator

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Harmonic Oscillator

9. An ideal harmonic oscillator has energy levels separated by a constant 0.05 eV. That
is, the difference between the ground state and the first excited state of this oscillator is
an energy of 0.05 eV. This single oscillator is brought into contact with a large solid
composed of an enormous number of oscillators, and characterized by a temperature of
T=150 K. What is the probability that the single oscillator will be found in its first excited
state?
1. 0.021
2. 0.055
3. 0.098
4. 0.144
5. 0.234
6. 1.000


What relationship should I even start with? Thanks for the help!

 

[uber]

This is a simple Boltzmann factor problem.

Recall that the probability of an individual energy level is:

exp(-e/(kT))/sum(exp(-ne/(kT))

where

e is the separation between energy levels
k is the Stefan-Boltzmann constant (make sure you get its units right!)
T is the absolute temperature

in this case, we can neglect all terms n > 1 as they decay rapidly to zero. Also, recal that the energy level for the ground state is 0.

That should help you complete the problem.