Matrix Representation of the Angular Momentum Raising Operator

[hnicholls]

TL;DR 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]

Please use ## ( beginning and end) for Latex rendeting to make post easier to read.

 

[hnicholls]

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.

 

[hnicholls]

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]

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
$$\hat{L}_+=\begin{pmatrix} 0 & \sqrt{2} & 0 \\ 0 & 0 & \sqrt{2} \\ 0 & 0 &0 \end{pmatrix} $$

 

[hnicholls]

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.

 

[PeterDonis]

Applying the general formula

You calculated this wrong. These states all have $l = 1$, not $l = 0$. Try the formula with $l = 1$.

 

[hnicholls]

Got it! Thanks! Did not correctly apply l = 1 as the subspace and the m values (1,0,-1) as the basis states.