This preview shows pages 1–2. Sign up to view the full content.
This preview has intentionally blurred sections. Sign up to view the full version.View Full Document
Unformatted text preview: 1 3. A compact way of writing the Bohr energy level formula for hydrogen is E n = 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 electron-proton system. a. Derive the formula v n = c 1 n for the electrons 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...
View Full Document