hw10 - momentum and its z-component quantum numbers,...

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HOMEWORK # 10 Physics 6572 Friday, 11/21/08; due MONDAY 12/1/08 1. Study carefully section 5 . 3( d ), Electric Quadrupole HyperFne Splitting of a Hydrogenic Atom, and reproduce the results (5 . 127) and (5 . 128). 2. Contributions to the magnetic moment of an atom (or nucleus) arise from the orbital motion of the charged particles and the intrinsic spins in the system. Generally, the magnetic moment of the operator of an atom may be assumed to have the structure m = e ¯ h 2 mc ( g L L + g S S ). (1) Since both L and S are vector operators with respect to the total angular momentum J = L + S , m is also a vector operator. An external magnetic Feld B is in the +z-direction. Use the Frst order perturbation the- ory and the Wigner-Eckart theorem to Fnd the energy level splitting in the basis | nlsjm a , where n , l , s , j , and m . are the principle, orbital angular momentum, spin, and total angular
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Unformatted text preview: momentum and its z-component quantum numbers, respectively. 3. A hydrogen atom in an excited state decays to a lower energy state by emitting a photon via electric dipole or magnetic dipole transitions. The electric dipole transition depends on the matrix element v d = A n ′ l ′ m ′ | vx | nlm a . (2) a) Use the Wigner-Eckart theorem to write the matrix elements of v d as a product of a Clebsch-Gordan coe±cient and a reduced matrix element. b) ²ind the reduced matrix element in terms of the radial wave functions R nl ( r ) for the initial and Fnal states deFned by A v r | nlm a = R nl ( r ) Y lm ( θ, φ ). (3) c) What are the selection rules for the matrix elements of v d ?...
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This note was uploaded on 03/27/2011 for the course PHYS 6572 at Cornell.

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