While undergoing a transition from the $n=1$ to the $n=2$ energy level, a harmonic oscillator absorbs a photon of wavelength $6.50$ $\mu$m. What is the wavelength of the absorbed photon when this oscillator undergoes a transition (a) from the $n=2$ to the $n=3$ energy level and (b) from the $n=1$ to the $n=3$ energy level?
(c) What is the value of $\sqrt{\left(k^{\prime }/m\right)}$, the angular oscillation frequency of the corresponding Newtonian oscillator?

Verified step by step guidance

1. Step 1: Understand the energy levels of a quantum harmonic oscillator. The energy levels are given by the formula E_n = (n + 1/2)ℏω, where n is the quantum number, ℏ is the reduced Planck's constant, and ω is the angular frequency of the oscillator. The energy difference between two levels determines the energy of the absorbed or emitted photon.
2. Step 2: Relate the energy of the photon to its wavelength using the equation E = hc/λ, where E is the energy of the photon, h is Planck's constant, c is the speed of light, and λ is the wavelength of the photon. For the given transition from n=1 to n=2, use the provided wavelength (6.50 μm) to calculate the energy difference ΔE_12 = hc/λ.
3. Step 3: Use the energy difference ΔE_12 to determine the angular frequency ω of the oscillator. From the energy difference formula ΔE_12 = ℏω, solve for ω = ΔE_12/ℏ. This angular frequency will be used for subsequent calculations.
4. Step 4: For part (a), calculate the energy difference for the transition from n=2 to n=3 using ΔE_23 = E_3 - E_2 = ℏω. Then, use the relation λ = hc/ΔE_23 to find the wavelength of the absorbed photon for this transition.
5. Step 5: For part (b), calculate the energy difference for the transition from n=1 to n=3 using ΔE_13 = E_3 - E_1 = 2ℏω. Then, use the relation λ = hc/ΔE_13 to find the wavelength of the absorbed photon for this transition. For part (c), the value of √(k'/m) is directly related to the angular frequency ω by the formula ω = √(k'/m). Use the previously calculated ω to solve for √(k'/m).

Key Concepts

Energy Levels in Quantum Mechanics

In quantum mechanics, energy levels refer to the discrete values of energy that a quantum system, such as an electron in an atom or a harmonic oscillator, can occupy. Transitions between these levels occur when the system absorbs or emits energy, typically in the form of photons. The difference in energy between levels determines the wavelength of the absorbed or emitted photon, following the relation E = hc/λ, where E is energy, h is Planck's constant, c is the speed of light, and λ is the wavelength.

Harmonic Oscillator

A harmonic oscillator is a system that experiences a restoring force proportional to its displacement from an equilibrium position, leading to oscillatory motion. In quantum mechanics, the harmonic oscillator model describes particles in a potential well, where energy levels are quantized. The energy levels are given by E_n = (n + 1/2)ħω, where n is the quantum number, ħ is the reduced Planck's constant, and ω is the angular frequency of the oscillator.

Angular Frequency and Spring Constant

The angular frequency (ω) of a harmonic oscillator is related to the spring constant (k') and mass (m) of the system by the formula ω = √(k'/m). This relationship indicates how quickly the oscillator oscillates; a larger spring constant or smaller mass results in a higher frequency. Understanding this concept is crucial for calculating the angular frequency of the oscillator in the given problem, which is essential for determining the system's behavior during transitions between energy levels.