Q:

Fourier transform and expectation values in QM? 

In Quantum mechanics every wave funtion in position space is of the form
$$\psi(x,t)=\frac{1}{\sqrt{2\pi \hbar}}\phi(p,t)e^{ip \cdot x/\hbar}dp,$$ and the corresponding wave function in momentum space is
$$\phi(p,t)=\frac{1}{\sqrt{2\pi \hbar}}\psi(x,t)e^{-ip \cdot x/\hbar}dx.$$
Both functions are probability amplitudes and due to Parsevals theorem:
$$\int_{-\infty}^{\infty}|\psi(x,t)|^2dx=\int_{-\infty}^{\infty}|\phi(p,t)|^2dp=1.$$
Average values in position space are:
$$\langle x_j \rangle = \int_{-\infty}^{\infty}x_j|\psi(x,t)|^2dx$$
and due to the transformation of derivative
$$\langle p_k \rangle = \int_{-\infty}^{\infty}p_k|\phi(p,t)|^2dp=\int_{-\infty}^{\infty}-i \hbar \partial_k\psi(x,t)\overline{\psi(x,t)}dx.$$
This is all very clear, but how can we, for example, obtain the following average value:
$$\langle x_j \rangle \langle p_k \rangle = \int_{-\infty}^{\infty}-x_j \cdot i \hbar \partial_k\psi(x,t)\overline{\psi(x,t)}dx.$$
Above result is a consequence of the well-known Born's rule, but surely(?)  we should be able to find it by direct calculation.  It would be boring if we can calculate expectation values for $x$ and $p$, but not for their compositions without postulating the Born's rule. 

A:

The average of any observable $A$ is given by $\langle \Psi|A|\Psi\rangle$, which can be written in configuration space as
$$\langle  A\rangle =\int_{\mathbb{R}^3} \bar \Psi(\vec r,t)\left(A\Psi(\vec r,t)\right)\,d^3\vec r$$
If $A=x_jp_k$, which in configuration space is $A=x_j\left(-i\hbar\frac{\partial}{\partial x_k}\right)$, then we have
$$\langle x_jp_k\rangle =\int_{\mathbb{R}^3} \bar \Psi(\vec r,t)x_j(-i\hbar )\frac{\partial \Psi(\vec r,t)}{\partial x_k}\,d^3\vec r$$

