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 IF nl jFm φ == . It is going to transition to the true ground state 1 2 1 , ,0 FF 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 FI 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 ( ) 11 1 2 28 1 2 1 1 22 10 , 10 , 10 00 . xx yy zz S Si i S σ = +−+−+ = −−+++ =+ + + = + + = +−−−+ = = Therefore, the only non-zero matrix element is S z , so 1 2 ˆ φφ = Sk = (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 2 4 FI F I EE m πα δ ω
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This note was uploaded on 02/15/2011 for the course PHYS 742 taught by Professor Unknow during the Spring '04 term at Harvard.

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

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