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Unformatted text preview: 1 3. A compact way of writing the Bohr energy level formula for hydrogen is E n = − α 2 mc 2 2 n 2 , where = ke 2 / c ≈ 1/137 is a dimensionless number, and m is the electron mass. Use the classical relations for a circular orbit: K = − 1 2 U , and U = − ke 2 / r for the electronproton system. a. Derive the formula v n = c 1 n for the electron’s speed in the n th orbit. b. Derive the formula r n = mc n 2 for the radius of the n th orbit. c. Show from these that the angular momentum is given by L = n . a. Since K + U = E and U = − 2 K we see that K = − E , so we set 1 2 mv n 2 = 2 mc 2 2 n 2 and Fnd the claimed result. b. Since K + U = E and K = − 1 2 U we see that E = 1 2 U . We set − 2 mc 2 2 n 2 = − 1 2 ke 2 / r n = − 1 2 c / r n , and Fnd the claimed result. c. Since L n = mv n r n , we Fnd the claimed result. Physics 54 Summer 2011 2...
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 Summer '09
 Thomas
 Physics, Electron, Photon, Light, Fundamental physics concepts

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