Question

Let this oscillator have the same energy as a mass on a spring, with the same k and m, released from rest at a displacement of 5.00cm from equilibrium. what is the quantum number n of the state of the harmonic oscillator? express the quantum number to three significant figures.

Answer 1

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.

Answer 2

There are a lot of same examples that you may have worked before, where the mass on a spring uses a classics when it comes to mechanics. So in this system, always put in your mind that there is an enormous quantum standard that one can use in the equation. It should be 2.10x10 raise to a negative sixth. J.