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Suppose we have a function $f\in L^2(\mathbb R^n)$ and we know the functions $x\mapsto|f(x)|$ and $p\mapsto|\hat f(p)|$, where $\hat f$ is the Fourier transform of $f$. Can we reconstruct the function $f$ itself, up to a phase factor? If $\lambda\in\mathbb C$ with $|\lambda|=1$, then $f$ and $\lambda f$ give exactly the same data, but I see no other obstructions. (Update: The answer is no. See this article. A related question is given in the next paragraph.)

Alternatively, one could consider the same problem for $f\in L^2(\mathbb T^n)$ on the torus $\mathbb T^n=\mathbb R^n/\mathbb Z^n$: Do the functions $\mathbb T^n\ni x\mapsto |f(x)|$ and $\mathbb Z^n\ni p\mapsto|\hat f(p)|$ determine $f$ uniquely? (Note: Since Qiaochu Yuan gave an answer to the Euclidean case, this is the only remaining question. I originally chose to leave this part out, but I suppose it's better to edit the question now than to ask a new one.)

If $f$ is a quantum mechanical wave function, position measurements give the function $x\mapsto|f(x)|^2$ and momentum measurements the function $p\mapsto|\hat f(p)|^2$. Assuming we can prepare a particle to the same state (same wave function) as many times as we wish, we can in principle carry out these measurements. From a physical point of view, the question is whether these measurements determine the wave function up to the nonphysical phase factor. Position and momentum are in a sense a good pair of observables, so I would expect the answer to be affirmative on physical grounds.

One could write $f(x)=\alpha(x)r(x)$ where $|\alpha(x)|\equiv1$ and $r(x)\in\mathbb R$ and similarly $\hat f(p)=\beta(p)s(p)$, and then try to use identities for Fourier transforms of products and find one of the unknown functions $\alpha$ and $\beta$ in terms of the known functions $r$ and $s$. (Actually, we can only possibly find $\alpha$ in the support of $r$, but that's enough.) I don't see an obvious way to solve the resulting equations, even with formal calculations.

If the problem becomes easier, an answer for a Schwartz function $f$ in dimension $n=1$ is ok. I would like to see a method for reconstructing $f$ from the given data, but a proof that the data does determine $f$ uniquely would also be great.

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    $\begingroup$ The answer appears to be no. Google led me to journals.cambridge.org/…. $\endgroup$ Dec 5, 2014 at 8:18
  • $\begingroup$ @QiaochuYuan, great, thanks! I had originally left out another case I was interested in (on the torus), but I included it now since that article settles the original question. $\endgroup$ Dec 5, 2014 at 8:50
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    $\begingroup$ Related question on Phys.SE: physics.stackexchange.com/q/81303/2451 $\endgroup$
    – Qmechanic
    Dec 5, 2014 at 9:53

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