Problem Set 06 Dispersion Relation of a De Broglie Wave 4 of 22> Part A Consider

Question

Problem Set 06 Dispersion Relation of a De Broglie Wave 4 of 22> Part A Consider a nonrelativistic free particle of mass m moving with velocity u. Find a dispersion relation ? k Express your answer in terms of mass m, wave number k, and Planck's constant h. for the de Broglie wave of this particle. In 1924, the French physicist Louis de Broglie resolved the long-standing problem of wave-particle duality in nature by postulating that a free particle has a characteristic wavelength determined by its momentum; that is, View Available Hint(s) 2m where h is Planck's constant. Hence, if a particle is a wave, it must have a characteristic dispersion relation. A dispersion relation is simply a representation of a wave's frequency as a function of wavelength, i.e., f(A). However, most often, scientists use angular frequencyw 2mf and wave number VCorrect k instead of frequency and wavelength. By knowing the dispersion relation(k), one can find the phase and group velocities of a Part B Ugroup Use the dispersion relation to find the group velocity group and phase velocity vphase Express your answer in terms of velocity v only. Separate your answers with a comma. Uphase- Ugroup. Vphase

Explanation / Answer

B)

Phase velocity, v_ph = w/k = hk/m

Group velocity, v_g = 2hk/m

V_ph = h 2pi/(2pi m lambda) = v

V_g = h 2pi/ (pi m lambda) = v/2

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