QM: Expectation value of raising and lowering operator

[barefeet]

Homework Statement

Using
$$J^2 \mid j,m_z \rangle = h^2 j(j+1) \mid j,m_z \rangle$$
$$J_z \mid j,m_z \rangle = hm_z \mid j,m_z \rangle$$

Derive that :
$$\langle j,m_z \mid J_-J_+ \mid j,m_z \rangle = h^2[ j(j+1) - m_z(m_z+1)]$$

Homework Equations

$$J_- = J_x - iJ_y$$
$$J_+ = J_x + iJ_y$$

The Attempt at a Solution

$$J_-J_+ = (J_x- iJ_y)(J_x + iJ_y) = J_x^2 + J_y^2 = J^2 - J_z^2$$

$$\langle j,m_z \mid J_-J_+ \mid j,m_z \rangle = \langle j,m_z \mid J^2 - J_z^2 \mid j,m_z \rangle = h^2[ j(j+1) - m_z^2]$$

Apparently I am missing a term here but I don't know where it should come from. I thought this should be true
$$J_z^2 \mid j,m_z \rangle = J_zJ_z \mid j,m_z \rangle = h^2m_z^2 \mid j,m_z \rangle$$
(Note: h is hbar everywhere )

 

[Orodruin]

You are assuming that $J_x$ commutes with $J_y$ when computing $J_-J_+$. This is not the case.