The Dispersion Relation for Free Relativistic Electron Waves Is

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The Dispersion Relation for Free Relativistic Electron Waves Is

Atomic and Molecular physics/Lecture notes presented by Dr. Fouad Attia Majeed/Third year students/College of education (Ibn Hayyan)/Department of Physics/University of Babylon. 3.5 Phase and group velocities We know matter waves (de Broglie waves) h h p mv where v is the velocity of a particle of the mass m. For particles and according to relativity 2 E mc and since E h then mc2 mc2 hv v h If we define u as the phase velocity and is given by h mc2 c2 u mv h v Thus, the velocity of the matter waves is larger than speed of light in vacuum, u > c, unless v > c. This result seems disturbing because it appears that the matter waves would propagate faster than the speed of light and would not be able to keep up with particles whose motion they govern. However, the phase velocity of a wave is the velocity of the wave front, not its amplitude. The maximum of the amplitude of a given wave can propagate at different velocity, called group velocity. At this velocity the energy (information) is transmitted. Usually, vg = u, but in the case of dispersion, u(), the group velocity vg < u. Thus, Dr. Fouad Attia Majeed 22 Atomic and Molecular physics/Lecture notes presented by Dr. Fouad Attia Majeed/Third year students/College of education (Ibn Hayyan)/Department of Physics/University of Babylon. the matter wave should be dispersive to match the requirement of vg < u when u > c. The group velocity is given by d v g d where is the angular frequency and is the wave number. Phase Velocity (u) Group Velocity (v ) g 2 since 2 and then we get 2 u u 2 Dr. Fouad Attia Majeed 23 Atomic and Molecular physics/Lecture notes presented by Dr. Fouad Attia Majeed/Third year students/College of education (Ibn Hayyan)/Department of Physics/University of Babylon. Question: The dispersion relation for free relativistic electron waves is 2 2 2 2 c mc (a) Calculate expressions for the phase velocity u and group velocity vg of these waves and show that their product is constant, independent of . (b) From the result (a), what can you conclude about vg if u > c? Solution: (a) From the definition of the phase velocity 2 2 2 mc u c h We see that the phase velocity u > c. From the definition of the group velocity d 1 2c2 v g 2 d 2 c2 2 mc2 h c2 c c 1 mc h2 1 mc h2 Thus, the group velocity vg < c. However, the product 2 mc2 c 2 2 uvg c c h 1 mc h2 is constant independent of . (b) We see from (a) that in general for dispersive waves for which u > c, the group velocity vg < c. Only when u = c, the group velocity vg = c . Dr. Fouad Attia Majeed 24 .

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