Final answer:
The quantum number (n) of a state of a harmonic oscillator can be determined using the formula n = (E - 1) / hν. The quantum number for the given harmonic oscillator with the same energy as a mass on a spring is approximately 54.4.
Step-by-step explanation:
The quantum number (n) of a state of a harmonic oscillator can be determined using the formula:
n = (E - 1) / hν
where E is the energy of the oscillator state, h is Planck's constant, and ν is the frequency of the oscillator.
In this case, the energy of the oscillator is the same as that of a mass on a spring. We can calculate the energy of the system by using the equation:
E = 0.5kA² = 0.5(0.030 kg)(4π² Hz)(0.050 m)² = 0.03 J
Substituting the values into the formula for n, we get:
n = (0.03 J - 1) / (6.626 × 10⁻³⁴ J·s)(4π² Hz) ≈ 54.4
Rounding to three significant figures, the quantum number (n) of the state of the harmonic oscillator is 54.4.