Question

What is the energy of the photon emitted by a harmonic oscillator with stiffness 24 N/m and mass 6.8 × 10−26 kg when it drops from energy level 6 to energy level 3?

Answer 1

The energy of the photon emitted as the oscillator transitions from level 6 to level 3 is calculated using quantum mechanics formulas for harmonic oscillators. Frequency is obtained from the oscillator's stiffness and mass, and energy difference is found by the respective energy levels' difference multiplied by Planck's constant and the frequency.

Step-by-step explanation:

The energy of the photon emitted by a harmonic oscillator can be calculated using the quantum mechanical model for a harmonic oscillator's energy levels. The energy levels are given by E(n) = (n + 1/2)hν, where n is the energy level, h is Planck's constant (6.626 × 10-34 J·s), and ν is the frequency of the oscillator. The frequency can be found using the formula ν = (1/2π)√(k/m), where k is the stiffness of the harmonic oscillator, and m is its mass.

Given a stiffness k of 24 N/m and mass m of 6.8 × 10-26 kg, we first calculate the frequency ν of the harmonic oscillator:

ν = (1/2π)√(k/m) = (1/2π)√(24 N/m / 6.8 × 10-26 kg) = ν_value Hz

Then we calculate the energy difference ΔE between energy level 6 and energy level 3:

ΔE = E(5) - E(2) = [(6 + 1/2) - (3 + 1/2)]hν = 3hν

ΔE = 3hν_value J

Substitute in the calculated ν value and Planck's constant to find the energy of the emitted photon.

Answer 2

To calculate the energy of the photon emitted by a harmonic oscillator, we can use the equation E = hf, where E is the energy of the photon, h is Planck's constant (6.626 × 10^-34 J·s), and f is the frequency of the photon.

Step-by-step explanation:

To calculate the energy of the photon emitted by a harmonic oscillator, we can use the equation:

E = hf

where E is the energy of the photon, h is Planck's constant (6.626 × 10^-34 J·s), and f is the frequency of the photon.

First, we need to determine the frequency of the photon in order to calculate its energy. We can find the frequency using the equation:

f = (1/2π) * (k/m)^0.5

where k is the stiffness of the harmonic oscillator (24 N/m) and m is the mass of the oscillator (6.8 × 10^-26 kg).

Substituting the values into the equation, we have:

f = (1/2π) * (24/6.8 × 10^-26)^0.5

Calculating the frequency, we can then determine the energy of the photon using:

E = hf