Expectation value of position operator for angular momentum states I have an exercice where it is asked to prove that this matrix element:
$$\langle{J=2,M=1}|\hat{x}^2\frac{\hat{\partial}}{\partial x}|{J=2,M=1}\rangle$$
is 0.
The problem is that I don't know how space and momentum operator act to angular momentum states. Any idea about how to face this problem?
 A: If ${\cal P}$ is the unitary and self-adjoint parity operator, it is well known that ${\cal P} |j, m\rangle = (-1)^j|j,m\rangle$ whereas
$${\cal P} X_k = -X_k {\cal P} \quad \mbox{and} \quad {\cal P} P_k = -P_k {\cal P}$$ for $k=1,2,3$ with $X_k$ position operator with respect to the $k$-th axis and $P_k$ the analogous momentum operator.
The text of the exercise is actually using a bit improper formalism as $|j,m\rangle$ only specifies the part of the vector in $L^2(\mathbb S^2, d\Omega)$ but nothing is said referring to the radial part: a further quantum number $n$ regarding the radial variable, $|n, j, m\rangle$, should be added, but it does not matter as it does not invalidate the above-written identity  ${\cal P} |n, j, m\rangle = (-1)^j|n, j,m\rangle$
The operator appearing in you bra-ket is $O= X_1^2 P_1$ up to numerical factors (and subtleties with domains I will ignore here) so that,
$${\cal P} O = -O{\cal P}\:.$$
Finally, with that $O$,
$$\langle j, m| O|j, m\rangle= (-1)^j \langle j, m| O {\cal P}|j, m\rangle = (-1)^j (-1)\langle j, m|  {\cal P} O|j, m\rangle
= (-1)^j (-1) (-1)^j\langle j, m|  O|j, m\rangle = -\langle j, m| O|j, m\rangle\:.$$
Summing up
$$\langle j, m| O|j, m\rangle= -\langle j, m| O|j, m\rangle$$
so that $$\langle j, m| O|j, m\rangle= 0$$ as wanted.
Observe that if explicitly writing the radial quantum number, we would obtain a more precise identity
$$\langle n, j, m| O|n', j', m'\rangle =0$$
along the same argument provided $j+j'$ is even. I stress that we may have $n \neq n'$ and $m \neq m'$ above.
As a final remark, it is not correct to say that this is the expectation value of $O$ since $O$ is not self-adjoint (nor Hermitian).
A: You need first to convert your angular momentum states into spherical harmonics, and then from spherical harmonics to Cartesian coordinates.  The problem does not mention any radial part so I assume it will be some generic $f(r)$.  
Alternatively, you can convert $\partial/\partial x$ and $x^2$ to spherical coordinates and then have those act on the appropriate spherical harmonics. 
In both cases you need to convert 
$$
\vert \ell,m\rangle \rightarrow f(r)Y_{\ell,m}(\theta,\phi)\, .
$$
