Question

Hyperfine lifetime calculation: what is the spin eigenfunctions?

Answer 1

I'm trying to calculate the lifetime of the 21 cm line in hydrogen and have the following expression:

1τ=4α3ω3if|⟨af|x⃗ |ai⟩|21τ=4α3ωif3|⟨af|x→|ai⟩|2.

The initial state is ai=a|F=1,Fz=1⟩+b|F=1,Fz=0⟩+c|F=1,Fz=−1⟩ai=a|F=1,Fz=1⟩+b|F=1,Fz=0⟩+c|F=1,Fz=−1⟩ and the final state is af=|F=0,Fz=0⟩af=|F=0,Fz=0⟩.

My problem arises when calculating the matrix element |⟨af|x⃗ |ai⟩|2|⟨af|x→|ai⟩|2, what are the spin wave functions?

So far I've thought to useai=|1s⟩(a|↑↑⟩+b2√(|↑↓⟩+|↓↑⟩)+c|↓↓⟩)ai=|1s⟩(a|↑↑⟩+b2(|↑↓⟩+|↓↑⟩)+c|↓↓⟩) andai=|1s⟩12√(|↑↓⟩−|↓↑⟩)ai=|1s⟩12(|↑↓⟩−|↓↑⟩) but then the matrix element becomes zero.

Answer 2

$hello \: frnd$

to calculate lifetime of 21 cm line in hydrogen and have following expression:

1π=4a3w3ifl<afix[]al>l21π=4a3wif3l
<afix→lai>i2

the initial state is ai=alF=1,Fz=1>+BL
F=1,Fz=0>+CLF=1,Fz=1>ai=al
F=1,Fz=1>+blF=1,Fz=0>+clF=1,Fz=-1> and the final state is af=lF=0,Fz=0>AF=of=0,Fz=0>.

my problem arises when calculating matrix element.

$hope \: it \: helps$