Sunday, March 13, 2011

Mastering Physics: Applying the Harmonic Oscillator

The quantum harmonic oscillator is very important in analyzing the spectra of diatomic molecules. It can also be extended to the analysis of polyatomic molecules. In this problem, you will look at how the energy levels of the harmonic oscillator relate to the spectrum of carbon monoxide.


Part A
There is a strong line in the infrared spectrum of carbon monoxide with a wavelength of 4.61 \;\rm \mu m. What is the energy E of a photon in this line?
Express your answer in joules to three significant figures.
ANSWER:

  E = 4.310×10−20
 \rm J


Part B
It can be shown that this line corresponds to a transition between adjacent energy levels in a harmonic oscillator. If this is true, what is the angular frequency omega of the oscillator? Use \hbar=1.055\times10^{-34}\; \rm J \cdot s.
Express your answer in inverse seconds to three significant figures.
ANSWER:

  omega = 4.09×1014
 \rm s^{-1}


Part C
Find the value of k, the effective spring constant. Use 16.0 and 12.0 atomic mass units for the masses of oxygen and carbon, respectively. (1\;{\rm amu}=1.66\times 10^{-27}\;\rm kg.)
Express your answer in newtons per meter to two significant figures.
ANSWER:

  k = 1900
 \rm N/m
This value is around an order of magnitude smaller than the spring constant for the springs in an average car suspension. Also, this is similar to the effective spring constant for a large trampoline.

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