I Matrix Representation of the Angular Momentum Raising Operator

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Summary
Matrix Representation of the Angular Momentum Raising Operator. Calculating L(+).
In calculating the matrix elements for the raising operator L(+) with l = 1 and m = -1, 0, 1 each of my elements conforms to a diagonal shifted over one column with values [(2)^1/2]hbar on that diagonal, except for the element, L(+)|0,-1>, where I have a problem.

This should be value [(2)^1/2]hbar; however, I get L(+)|0,-1> = (0(0+1)-(-1)((-1)+1))^1/2|0,-1+1> = 0hbar|0,0>. This would be a 0 value not [(2)^1/2]hbar. Not sure where I making my mistake.
 

WWGD

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Please use ## ( beginning and end) for Latex rendeting to make post easier to read.
 
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L+ with l = 1 and m = -1, 0, 1

L+|0,-1> = (0(0+1)-(-1)((-1)+1))^1/2|0,-1+1> = 0ħ|0,0>. This would be a 0 value not √2ħ. Not sure why this wrong.
 
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L+ with l = 1 and m = -1, 0, 1

L+|0,-1> = √[0(0+1)-(-1)((-1)+1)]|0,-1+1> = 0ħ|0,0>. This would be a 0 value not √2ħ. Not sure why this wrong.
 

vanhees71

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In the following I use ##\hbar=1## and write ##|m \rangle## (we are in the subspace with ##l=1## anyway). Then the "raising operator" acts on these basis states by
$$\hat{L}_+|1 \rangle=0, \quad \hat{L}_+ |0 \rangle=\sqrt{2}|1 \rangle, \quad \hat{L}_{+} |-1 \rangle=\sqrt{2} |0 \rangle.$$
Here I used the general formula
$$|\hat{L}_+ |l,m \rangle=\sqrt{(l-m)(l+m+1)} |l,m+1 \rangle.$$
The matrix representation with the basis ##|1 \rangle##, ##0 \rangle##, ##|-1 \rangle## (in that order) thus reads
$$\hat{L}_+=\begin{pmatrix} 0 & \sqrt{2} & 0 \\ 0 & 0 & \sqrt{2} \\ 0 & 0 &0 \end{pmatrix} $$
 
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Applying the general formula

$$|\hat{L}_+ |l,m \rangle=\sqrt{(l-m)(l+m+1)} |l,m+1 \rangle.$$

$$|\hat{L}_+ |l,m \rangle=\sqrt{(0-(-1))(0+(-1)+1)} |l,m+1 \rangle.$$

$$|\hat{L}_+ |l,m \rangle=\sqrt{(+1)(0)} |0,0 \rangle.$$

$$|\hat{L}_+ |l,m \rangle=\sqrt{0} |0,0 \rangle.$$

I don't see how matrix element column 3 row 2 can be$$\sqrt{2} $$I still get 0. Thank you for the response.
 
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Got it! Thanks! Did not correctly apply l = 1 as the subspace and the m values (1,0,-1) as the basis states.
 

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