We wish to write the states of total angular momentum
in terms of the
We will do this by operating with the
operator and setting the coefficients
so that we have eigenstates.
We choose to write the the quantum number
This is really just the defintion of the dummy variable
(Other choices would have been possible.)
The z component of the total angular momentum is just the sum of the z components from
the orbital and the spin.
There are only two product states which have the right
If the spin is up we need
and if the spin is down,
We will find the coefficients
will be an eigenstate of
So operate on the right hand side with
And operate on the left hand side.
Since the two terms are orthogonal, we can equate the coefficients for each term,
giving us two equations.
terms on the LHS and
terms on the RHS, we get two equations.
Now we just cross multiply so we have one equation with a common factor of
While this equation looks like a mess to solve, if we notice the similarity between the
LHS and RHS, we can solve it if
If we look a little more carefully at the LHS, we can see that another solution
(which just interchanges the two terms in parentheses) is to replace
These are now simple to solve
So these are (again) the two possible values for
We now need to go ahead and find
into our first equation,
we get the ratio between
We will normalize the wave function by setting
So lets get the squares.
So we have completed the calculation of the coefficients.
We will make use of these in the hydrogen atom, particularly for
the anomalous Zeeman effect.
Writing this in the
notation of matrix elements or Clebsch-Gordan coefficients of the form,