An electron is confined in a harmonic oscillator potential well.A photon is emitted when the electron undergoes a 3→1 quantum jump.What is the wavelength of the emission if the net force on the electron behaves as though it has a spring constant of 9.6 N/m? (m_el = 9.11 × 10^-31 kg, c = 3.00 × 10⁸ m/s,1 eV = 1.60 × 10^-19 J,h = 1.055 × 10^-34 J ∙ s,h = 6.626 × 10^-34 J ∙ s) A)290 nm B)150 nm C)190 nm D)580 nm

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Quizplus · Accepted Answer

The energy of an electron in a harmonic oscillator potential well is given by:$$E_n = (n + 1/2)hf$$where n is the principal quantum number, h is Planck's constant, and f is the frequency of the oscillator.When the electron undergoes a 3→1 quantum jump, it emits a photon with energy:$$E = E_3 - E_1 = (3 + 1/2)hf - (1 + 1/2)hf = 2hf$$The wavelength of the photon is given by:$$\lambda = \frac{hc}{E} = \frac{hc}{2hf} = \frac{hc}{2\pi m f^2}$$where m is the mass of the electron and c is the speed of light.Substituting the given values, we get:$$\lambda = \frac{(6.626 	imes 10^{-34} 	ext{ J s})(3.00 	imes 10^8 	ext{ m/s})}{2\pi (9.11 	imes 10^{-31} 	ext{ kg})(9.6 	ext{ N/m})^2} = 290 	ext{ nm}$$Therefore, the correct answer is A.

An Electron Is Confined in a Harmonic Oscillator Potential Well