- #1

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## Homework Statement

Let [itex]|0,0\rangle [/itex] be the simultaneous eigenstate of [itex]\mathbf{J}^2[/itex] and [itex]J_z[/itex] with eigenvalues 0 and 0. Find

[tex]

J_x|0,0\rangle \quad\quad J_y |0,0\rangle \quad\quad [J_x,J_y]|0,0\rangle

[/tex]

**2. The attempt at a solution**

It seemed reasonable to write [itex]J_x[/itex] and [itex]J_y[/itex] in terms of ladder operators

[tex]

J_{+}=J_x + iJ_y

[/tex]

[tex]

J_{-}=J_x -i J_y

[/tex]

and then to have them operate on the states (for the last one I would just use the x,y,z commutation relations). But I was looking at the normalization constant out front

[tex]

J_{+}|j,m\rangle = \hbar\sqrt{(j+m+1)(j-m)} |j,m+1\rangle

[/tex]

[tex]

J_{-}|j,m\rangle = \hbar \sqrt{(j-m+1)(j+m)}|j,m-1\rangle

[/tex]

but given the initial state, these seem to be zero... Please tell me I'm doing this wrong, zero is such a unsatisfactory answer...

Thanks,