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M2L8e.txt
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M2L8e.txt
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#
# File: content-mit-8-421-2x-subtitles/M2L8e.txt
#
# Captions for 8.421x module
#
# This file has 30 caption lines.
#
# Do not add or delete any lines.
#
#----------------------------------------
So this is hydrogen, and this h is now the Hamiltonian.
So the hyperfine coupling Hamiltonian, which is I
dot J, by using the expression for the total angular
momentum, I plus J. And then we square it.
When you evaluate the square, you get on the right hand side
an expression for I dot J. So I dot
J is nothing else than 1/2 F square, minus I square,
minus J square.
And therefore, for hydrogen, one where I, J, and S are all 1/2--
the proton has spin 1/2, the electron
has spin 1/2, that's it.
You have only two values or the resultant total angular
momentum.
1/2 and 1/2 can add up to 1 or 0.
And now the hyperfine splitting is into and F equals 1,
and F equals 0 state.
And one thing to remember is if you inspect the above formula
wit the quantum numbers, you find immediately
that compared to the degenerate line
without hyperfine splitting-- so what comes out
of the Dirac equation, the splitting is 1/4 and 3/4
of the hyperfine constant.
Since F equals 1 has a multiplicity of three,
two MF quantum numbers plus minus 1 and 0, the rule is
that the center of mass, of this level does not change,
due to hyperfine splitting.
So the center of mass of hyperfine states
is not changing.
So we introduce a level splitting, but no overall shift.