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2002AugQualQUANT - Ben Sauerwine Practice for Qualifying...

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Ben Sauerwine Practice for Qualifying Exams Problem Source: CMU August 2002 Qualifying Exam This problem considers the hyperfine structure of the 1s level of hydrogen. Splitting of this level is responsible for the famous “21 cm” emission line in radio astronomy studies of interstellar hydrogen clouds. The hydrogen atom Hamiltonian can be expressed as: hf f W W H H + + = 0 where the non-relativistic Hamiltonian R e m P H 2 2 0 2 = v Recall that the 1s level is the hydrogen atom orbital ground state, an eigenstates of that is positive definite and spherically symmetric (with zero angular momentum), and varies like 0 H ( ) 0 , , a R e R φ θ ψ . The “fine structure” corrections are of relativistic origins and may be expanded as D SO mv f W W W W + + = with ( ) ( ) R c W S L R b W P a W D SO mv δ v v v = = 3 4 with a, b, c constants. The “hyperfine structure” corrections result from interaction of the electron with the nuclear magnetic moment. Expressed in terms of nuclear spin I v , electron spin , and angular momentum S v L v , ( ) ( ) ( ) ( ) [ ] ( ) ( ) R I S f I S n I n S R e I L d W hf v v v v v v v v v δ + + = ˆ ˆ 3 3 with d, e, and f constants and the unit vector in the direction of n ˆ R v .
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