The position $\hat{q}$ and momentum $\hat{p}$ has $[\hat{q},\hat{p}]=i$.
And we set there eigenstates as $|s\rangle\_q$ and $|s\rangle\_p$ with eigenvalue s.


In the paper [Phy Rev A. 79, 062318 (2009)], the eigenstates of $\hat{q}$ and $\hat{p}$ can be transfered as
\begin{equation}
\begin{array}{c}|s\rangle\_{p}=\frac{1}{\sqrt{2 \pi}} \int\_{-\infty}^{\infty} d r e^{i r s}|r\rangle\_{q}=F|s\rangle\_{q} \\ |s\rangle\_{q}=\frac{1}{\sqrt{2 \pi}} \int\_{-\infty}^{\infty} d r e^{-i r s}|r\rangle\_{p}=F^{\dagger}|s\rangle\_{p}\end{array}
\end{equation}
Here $F$ is the Fourier transformation and has the form
\begin{equation}
F=e^{i \pi\left(\hat{q}^{2}+\hat{p}^{2}\right) / 4}
\end{equation}
Also $F$ is a special case for the rotation operator
\begin{equation}
R(\theta)=e^{i \theta\left(\hat{q}^{2}+\hat{p}^{2}\right) / 2}
\end{equation}
It rotates a state counter-clockwise in phase space by an angle $\theta$.


My question is how to prove
\begin{equation}
|s\rangle\_{p}=F|s\rangle\_{q} 
\end{equation}
Also, how to calculate the state rotation
\begin{equation}
R(\theta)|s\rangle\_{q} 
\end{equation}
Many thanks.



How to calculate Fourier transformation of eigenstates in CV quantum information
This is easiest to understand in terms of the annihilation operator $a$, related to $q$ and $p$ by $a=(q+ip)/\sqrt 2$. The operator $R(\theta)$ is then given by
$$
R(\theta)=e^{i \theta(q^2+p^{2}) / 2}=e^{i\theta/2}e^{i\theta a^\dagger a}.$$
Application of the commutation relation $[a,a^\dagger]=1$ shows that
$$R^\dagger(\theta)aR(\theta)=e^{i\theta}a.$$
Hence $R(-\pi/2)$ transforms $a$ into $-ia$, which means that $q=(a^\dagger+a)/\sqrt 2$ is transformed into
$$R^\dagger(-\pi/2)qR(-\pi/2)=(ia^\dagger-ia)/\sqrt 2=p.$$
The corresponding eigenstates are transformed as $|s\rangle\_p=R(\pi/2)|s\rangle\_q$.


A general $R(\theta)$ transforms $q$ into $q\cos\theta-p\sin\theta$ and it transforms $p$ into $q\sin\theta+p\cos\theta$.


