For the purpose of a quantum optics experiment, I come to the following problem :
Let $X,P \in \mathbf{R}^2$
\begin{equation} J(\psi) = | \int_{-X}^{X} x |\psi^2(x) | dx + \int_{-P}^{P} p |Tf(\psi)^2(p) | dp | \end{equation}
Find:
\begin{equation} \left\{ \begin{aligned} \sup_{ \psi } J(\psi) \\ \text{with constraint} \int_{\mathbf{R}} |\psi^2(x) | dx = 1 \end{aligned} \right. \end{equation}
where Tf is the Fourier transform
$\psi$ functions are supposed to be smooth enough for any purpose. That's why I don't explicitly state in what space they live.
I tried in this way :
1) If i upper bound each term independantly, I can obtain X+P with a $\psi^2(x) = \delta(X) $ and $Tf(\psi^2(p)) = \delta(p)$ . From now on, the aim is to make the best of the relation between $\psi$ and its Tf
2) In quantum physics, the problem can be rewritten in the following form :
\begin{equation} J(\psi) = | \int_{-X}^{X} x |\psi^2(x) | dx + \int_{-P}^{P} -i \psi^*(x) \frac{\partial \psi}{\partial x} dx | \end{equation}
I tried to differentiate J with respect to $\psi$ and $\psi^*$ in order to apply Lagrange multipliers on the functional space of $\psi$ functions.
With X = P and the constraint changed
\begin{equation} \left\{ \begin{aligned} \sup_{ \psi } J(\psi) \\ \text{with constraint} \int_{-X}^X |\psi^2(x) | dx = 1 \end{aligned} \right. \end{equation}
I found:
\begin{equation} \exists \lambda \text{ such that } \\ i \frac{\partial \psi}{\partial x} + (x - \lambda) \psi = 0 \end{equation} which I solved with \begin{equation} \psi = C \exp \left( i(x^2 - \lambda x) \right) \end{equation}
Could you please help me with the general problem ? Would you be aware of any litterature treating it ? It seems to me this is linked to the Pauli Problem, yet slightly different.
Thanks in advance.
