Question 41.8: In Which State Is the Oscillator? A simple harmonic oscillat......
In Which State Is the Oscillator?
A simple harmonic oscillator in its ground state has a frequency of 3.450 MHz. Now its energy is found to be 2.400 × 10^{-26} J. What state is it in?
INTERPRET and ANTICIPATE
The frequency of the ground state is the frequency in the excited state. So we can use the given frequency and the energy of the excited state to find which state the oscillator is in.
SOLVE
Solve Equation 41.26 for n.
Substitute values. The answer must be an integer, so in this case we round up to the nearest whole number.
\begin{aligned}& n=\frac{\left(2.400 \times 10^{-26} J \right)}{\left(6.626 \times 10^{-34} J \cdot s \right)\left(3.450 \times 10^6 Hz \right)}+\frac{1}{2} \\& n=10.9988=11\end{aligned}CHECK and THINK
To check, let’s find the difference between adjacent levels: \Delta E=h f=\left(6.626 \times 10^{-34} J \cdot s \right) \left(3.450 \times 10^6 Hz \right)=2.286 \times 10^{-27} \quad J. So the system’s energy is about 2.400 \times 10^{-26} J / 2.286 \times 10^{-27} J =10.5 times greater than the difference between adjacent energy levels. The ground-state energy (n =1) is 0.5 hf, and the system is 10 levels above the ground state, so n = 1 + 10 = 11, as we found.