I'm having a little trouble with something that's 'easy to check' according to the script I'm using. I consider the kinetic energy operator
$$\hat T = \frac{p^2}{2m} = -\frac{\hbar^2}{2m} \frac{d^2}{dx^2} $$
and the wave function in momentum space, which is obtained using the Fourier transform
$$\tilde \psi(k) = \mathcal{F}[\psi(x)] = \frac{1}{\sqrt{2 \pi}} \int_{-\infty}^{+\infty} \psi(x) e^{-ikx} dx \ .$$
According to the script it's easy to see that
$$\hat T \tilde \psi(k) = \frac{\hbar^2 k^2}{2m} \tilde \psi(k)\ ,$$
i.e. that $\tilde \psi(k)$ is an eigenstate of the kinetic energy operator. I don't manage to replicate that result. In particular, using integration by parts a few times I get
$$\begin{align} \hat T \tilde \psi(k) &= -\frac{\hbar^2}{2m} \frac{1}{\sqrt{2 \pi}} \int_{-\infty}^{+\infty} \frac{d^2}{dx^2} ( \psi(x) e^{-ikx} ) dx \\ &= … \\ &= -\frac{\hbar^2}{2m} \tilde \psi(k) (-k^2 + 2k^2 - k^2) \\ &= 0 \end{align}$$
I see that $\mathcal{F}[\hat T \psi(x)] = \frac{\hbar^2 k^2}{2m} \mathcal{F}[\psi(x)]$ but $\mathcal{F}[\hat T \psi(x)] \neq \hat T \tilde \psi(k)$, right? So I'm confused ...
Can anybody show me where I go wrong?