For a quantum optics experiment I come across the following problem:

To 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 {align}

sup_ { psi} J ( psi) \

text {with restriction} int _ { mathbf {R}} | psi ^ 2 (x) | dx = 1

end {align}

right.

end {equation}

where Tf is the Fourier transform

$ psi $ functions should be fluid enough for every purpose. Therefore I do not explicitly say in which room they live.

I tried it like this:

1) If I refer every term upwards *independently*I can get X + P with a $ psi ^ 2 (x) = Delta (X) $ and $ Tf ( psi ^ 2 (p)) = Delta (p) $ , From now on, the goal is to make the most of the relationship between the two $ psi $ and his Tf

2) In quantum physics, the problem can be described in the following form:

begin {equation}

J ( psi) = | int _ {- X} ^ {X} x | psi ^ 2 (x) | dx + int _ {- P} ^ {P} psi ^ * (x) frac { partially psi} { partial x} – psi (x) frac { partial psi ^ *} { partial x} dx |

end {equation}

I tried to distinguish J in terms of $ psi $ and $ psi ^ * $ to apply Lagrangian multipliers to the functional space of $ psi $ functions. **With X = P**, I found:

begin {equation}

exists lambda \ text {so that} \

2 i frac { partial psi} {x} + (x – lambda) psi = 0

end {equation}

what I solved with

begin {equation}

psi = C exp left ( frac {i (x ^ 2 – lambda x)} {2} right)

end {equation}

Could you please help me with the general problem? Would you know that literature treats it? It seems to me that this is related to the Pauli problem, but a little differently.

Thank you in advance.