Final answer:
The frequency of a harmonic oscillator in the third excited state with an energy of 0.2 eV is approximately 1.38 × 10^13 Hz, which is not among the given options.
Step-by-step explanation:
To find the frequency of vibration for a linear harmonic oscillator in the third excited state with an energy of 0.2 eV, we use the quantum mechanical formula for the energy levels of a harmonic oscillator:
En = (n + 1/2)hv
Where En is the energy of the nth quantum state, n is the quantum number (> 0 for the ground state), h is Planck's constant (4.135667696 × 10-15 eV·s), and v is the frequency of the oscillator.
For the third excited state (n=3), the energy is:
E3 = (3 + 1/2)hv = 0.2 eV
We can solve for the frequency (v):
v = E3 / (3.5h)
v = 0.2 eV / (3.5 × 4.135667696 × 10-15 eV·s)
v ≈ 1.38 × 1013 Hz
Therefore, the frequency of vibration of the oscillator is approximately 1.38 × 1013 Hz, none of the provided options (a. 0.2 Hz, b. 0.4 Hz, c. 0.6 Hz, d. 0.8 Hz) are correct.