Harmonic oscillator: An electron is confined in a harmonic oscillator potential well. What is the longest wavelength of light that the electron can absorb if the net force on the electron behaves as though it has a spring constant of 74 N/m? (m_el = 9.11 × 10^-31 kg, c = 3.00 × 10⁸ m/s, 1 eV = 1.60 × 10^-19 J, ħ = 1.055 × 10^-34 J ∙ s, h = 6.626 × 10^-34 J ∙ s) A) 210 nm B) 200 nm C) 220 nm D) 230 nm

Question

Quizplus · Accepted Answer

The longest wavelength of light that the electron can absorb is given by the equation:$$\lambda = \frac{hc}{E_n - E_0}$$where:* $\lambda$ is the wavelength of light* $h$ is Planck's constant* $c$ is the speed of light* $E_n$ is the energy of the electron in the excited state* $E_0$ is the energy of the electron in the ground stateThe energy of the electron in the ground state is given by:$$E_0 = \frac{1}{2}hf_0$$where $f_0$ is the frequency of the oscillator.The energy of the electron in the excited state is given by:$$E_n = \left(n + \frac{1}{2}ight)hf_0$$where $n$ is the principal quantum number.The longest wavelength of light that the electron can absorb is when $n = 1$. Therefore, the longest wavelength of light that the electron can absorb is given by:$$\lambda = \frac{hc}{\left(\frac{3}{2}ight)hf_0}$$$$\lambda = \frac{2hc}{3hf_0}$$$$\lambda = \frac{2c}{3f_0}$$The frequency of the oscillator is given by:$$f_0 = \sqrt{\frac{k}{m}}$$where $k$ is the spring constant and $m$ is the mass of the electron.Therefore, the longest wavelength of light that the electron can absorb is given by:$$\lambda = \frac{2c}{3\sqrt{\frac{k}{m}}}$$Plugging in the given values, we get:$$\lambda = \frac{2(3.00 	imes 10^8 	ext{ m/s})}{3\sqrt{\frac{74 	ext{ N/m}}{9.11 	imes 10^{-31} 	ext{ kg}}}}$$$$\lambda = 210 	ext{ nm}$$Therefore, the correct answer is A.

Harmonic Oscillator: an Electron Is Confined in a Harmonic Oscillator