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solp(1)

# solp(1) - Physics 742 Graduate Quantum Mechanics 2 Solution...

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Physics 742 – Graduate Quantum Mechanics 2 Solution Set P 1. [25] A hydrogen atom is in interstellar space in the 1s state, but not in the true ground state ( F = 0), but rather in the hyperfine excited state ( F =1), specifically in the state 1 2 , , , , 1,0, ,1,0 I F n l j F m φ = = . It is going to transition to the true ground state 1 2 , , , , 1,0, ,0,0 F F n l j F m φ = = via a magnetic dipole interaction. (a) [5] Write out the initial and final states in terms of the explicit spin states of the electron and proton , ± ± . Find all non-zero components of the matrix F I φ φ S , where S is the electron spin operator. The explicit form for these spin states can be found, for example, eq. (9.35). We therefore have ( ) ( ) 1 2 1 2 1,0 0,0 = + − + − + = + − − − + The spin operator acting on the initial state yields ( ) ( ) ( ) ( ) ( ) ( ) 1 1 1 2 2 8 1 1 1 2 2 8 1 1 1 1 2 2 2 8 10 , 10 , 10 00 . x x y y z z S S i i S σ σ σ = + − + − + = − − + + + = + − + − + = − − − + + = + − + − + = + − − − + = = = = = = = = Therefore, the only non-zero matrix element is S z , so 1 2 ˆ F I φ φ = S k = (b) [8] Show that the rate for this transition for a wave going in a specific direction with a definite polarization is given by ( ) ( ) 2

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solp(1) - Physics 742 Graduate Quantum Mechanics 2 Solution...

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