INT An electron is confined in a harmonic potential well that has a spring constant of 12.0 N/m. What is the longest wavelength of light that the electron can absorb?

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Understand the problem: The electron is confined in a harmonic potential well, which means its energy levels are quantized. The energy difference between these levels determines the wavelength of light the electron can absorb. The longest wavelength corresponds to the smallest energy difference, which is between the first two energy levels.

Recall the formula for the energy levels of a quantum harmonic oscillator: \( E_n = \left(n + \frac{1}{2}\right) \hbar \omega \), where \( n \) is the quantum number (\( n = 0, 1, 2, \dots \)), \( \hbar \) is the reduced Planck's constant, and \( \omega \) is the angular frequency of the oscillator.

Determine the angular frequency \( \omega \) using the relationship \( \omega = \sqrt{\frac{k}{m}} \), where \( k \) is the spring constant (12.0 N/m) and \( m \) is the mass of the electron (\( 9.11 \times 10^{-31} \, \text{kg} \)).

Calculate the energy difference between the first two levels: \( \Delta E = E_1 - E_0 = \hbar \omega \). This energy corresponds to the photon energy of the absorbed light, \( E = h f \), where \( f \) is the frequency of the light.

Relate the frequency \( f \) to the wavelength \( \lambda \) using the equation \( \lambda = \frac{c}{f} \), where \( c \) is the speed of light. Substitute \( f = \frac{\Delta E}{h} \) to express \( \lambda \) in terms of \( \Delta E \). This gives \( \lambda = \frac{h c}{\Delta E} \). Use this formula to find the longest wavelength of light the electron can absorb.

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Key Concepts

Here are the essential concepts you must grasp in order to answer the question correctly.

Harmonic Potential Well

A harmonic potential well is a model used in quantum mechanics where a particle, such as an electron, is subjected to a restoring force proportional to its displacement from an equilibrium position. This system is characterized by a parabolic potential energy function, leading to quantized energy levels. The behavior of particles in such wells can be described using the principles of quantum mechanics, particularly the Schrödinger equation.

Energy Levels and Quantum States

In a harmonic potential well, the electron occupies discrete energy levels, which are determined by the well's parameters, such as the spring constant. The energy levels are quantized, meaning the electron can only exist in specific states with defined energies. The difference in energy between these levels corresponds to the energy of photons that can be absorbed or emitted, which is crucial for understanding the interaction between light and matter.

Wavelength and Energy Relationship

The wavelength of light is inversely related to its energy, as described by the equation E = hc/λ, where E is energy, h is Planck's constant, c is the speed of light, and λ is the wavelength. To find the longest wavelength of light that the electron can absorb, one must calculate the energy difference between the lowest energy state and the first excited state in the harmonic potential well. This relationship allows us to determine the maximum wavelength corresponding to the energy transition.

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