- #1
binbagsss
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I'm looking at how a l j1,j2,jm> state can be operated on by the ladder operators, changing the value of m, and how this state can be expressed as a linear combination of l j1, j2, m1, m2> states.
Where the following relationships must be obeyed (x2):
m=m1+m2
l j1-j2 l ≤ j ≤ j1+j2
So I'm looking at an example where we start with state j1=1/2 and j2=1/2.
From the above relations I know :
m1=m2=[itex]\pm[/itex]1/2
j=j1+j2=1
m=-1,0,0,1
I am in state l j1,j1,j,m> = l 1/2,1/2,1,1>.
I operate with L- to get:
l 1/2, 1/2, 1,0>
Now I want to express this in terms of l j1, j2, m1, m2> bases. So looking at the possible values of m1 and m2, I can see this can be produced by a linear combination of l j1, j2, m1, m2> = l 1/2,1/2,1/2,-1/2> , l 1/2,1/2,-1/2,1/2>
So I need to figure out the coefficients.
Now here is my question:
I have the relation: J[itex]\pm[/itex] l j1,j2,j,m> = (j(j+1)-m(m[itex]\pm[/itex]1))[itex]^{\frac{1}{2}}[/itex] l j1,j2,j,m[itex]\pm[/itex]1>, *
Which would give the coefficients as [itex]\sqrt{\frac{3}{4}}[/itex]
So I don't understand where the coefficients come from : ([itex]\sqrt{\frac{1}{2}}[/itex] is the solution).
I can see (i think ) that this makes sense from a normalization point of view. But then what happens to the coefficients attained from * ?
Many thanks for any assistance, greatly appreciated !
Where the following relationships must be obeyed (x2):
m=m1+m2
l j1-j2 l ≤ j ≤ j1+j2
So I'm looking at an example where we start with state j1=1/2 and j2=1/2.
From the above relations I know :
m1=m2=[itex]\pm[/itex]1/2
j=j1+j2=1
m=-1,0,0,1
I am in state l j1,j1,j,m> = l 1/2,1/2,1,1>.
I operate with L- to get:
l 1/2, 1/2, 1,0>
Now I want to express this in terms of l j1, j2, m1, m2> bases. So looking at the possible values of m1 and m2, I can see this can be produced by a linear combination of l j1, j2, m1, m2> = l 1/2,1/2,1/2,-1/2> , l 1/2,1/2,-1/2,1/2>
So I need to figure out the coefficients.
Now here is my question:
I have the relation: J[itex]\pm[/itex] l j1,j2,j,m> = (j(j+1)-m(m[itex]\pm[/itex]1))[itex]^{\frac{1}{2}}[/itex] l j1,j2,j,m[itex]\pm[/itex]1>, *
Which would give the coefficients as [itex]\sqrt{\frac{3}{4}}[/itex]
So I don't understand where the coefficients come from : ([itex]\sqrt{\frac{1}{2}}[/itex] is the solution).
I can see (i think ) that this makes sense from a normalization point of view. But then what happens to the coefficients attained from * ?
Many thanks for any assistance, greatly appreciated !