Matter waves: How fast must a nonrelativistic electron move so its de Broglie wavelength is the same as the wavelength of a 3.4-eV photon? (m_el = 9.11 × 10^-31 kg, c = 3.00 × 10⁸ m/s, 1 eV = 1.60 × 10^-19 J) A) 2000 m/s B) 1900 m/s C) 1700 m/s D) 1600 m/s E) 1400 m/s

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Quizplus · Accepted Answer

The de Broglie wavelength of an electron is given by:$$\lambda = \frac{h}{p}$$where h is Planck's constant and p is the momentum of the electron. The momentum of a photon is given by:$$p = \frac{E}{c}$$where E is the energy of the photon and c is the speed of light. The energy of a photon is given by:$$E = hf$$where f is the frequency of the photon. The wavelength of a photon is given by:$$\lambda = \frac{c}{f}$$So, the de Broglie wavelength of an electron is the same as the wavelength of a photon if:$$\frac{h}{p} = \frac{c}{f}$$Substituting the given values, we get:$$\frac{6.63 	imes 10^{-34} 	ext{ J s}}{p} = \frac{3.00 	imes 10^8 	ext{ m/s}}{3.4 	ext{ eV} 	imes 1.60 	imes 10^{-19} 	ext{ J/eV}}$$Solving for p, we get:$$p = 3.32 	imes 10^{-24} 	ext{ kg m/s}$$The speed of the electron is then:$$v = \frac{p}{m} = \frac{3.32 	imes 10^{-24} 	ext{ kg m/s}}{9.11 	imes 10^{-31} 	ext{ kg}} = 3.64 	imes 10^6 	ext{ m/s}$$So, the correct answer is A) 2000 m/s.

Matter Waves: How Fast Must a Nonrelativistic Electron Move So