Are wave functions uniquely determined by their position and momentum distributions?

Abstract

The problem of determining a square integrable function from both its modulus and the modulus of its Fourier transform is studied. It is shown that for a large class of real functions the function is uniquely determined from this data. We also construct fundamental subsets of functions that are not uniquely determined. In quantum mechanical language, bound states are uniquely determined by their position and momentum distributions but, in general, scattering states are not.

Phase retrieval techniques for radar ambiguity problems

“…Reisenbach [17] published the examples of Pauli partners found by Bargman, but only in 1978 Vogt [22] and Corbett and Hurst [7], [6] first started a systematic study of the subject and showed that there are infinitely many Pauli unique functions as well as infinitely many Pauli partners. To exhibit Pauli partners, they used a method based on the relation CF = F CZ.…”

“…However this is not the case as shown by Bargman (cf. [17]) and later by Corbett and Hurst [7] and Vogt [22]. We will give new examples, and show in particular that a function may have an infinity of Pauli partners.…”

“…the Paley-Wiener theorem and the Hadamard factorisation theorem, which will be used again in section 3. We will conclude this section with Pauli's problem and results by Corbett and Hurst [7], [6] and Vogt [22], and show how to build functions that have infinitely many Pauli partners. We will also list some connected questions which have probably not attracted the interest they deserve.…”

Complementary Observables in Quantum Mechanics

“…Example 5.2.1 shows that if the spectrum of the energy is purely discrete, then the position-energy and momentum-energy pairs are complementary, too. If the energy operator H is of the form H = 1 2m P 2 + V (Q), with V (Q) bounded and positive it is known that the probability distributions Q ρ , P ρ , H ρ do not suffice to determine the state ρ[39,41]. Not knowing the general answer to the posed question, werecall that if Q θ = U θ QU θ , with U θ = e iθH ,is a quadrature observable, then, not only the pair Q and P = Q π 2 , but, in fact, any pair (Q, Q θ ), θ / ∈ {0, π}, is complementary [42].…”

Phase retrieval in quantum mechanics

“…The importance of this problem for foundations of quantum mechanics was, as early as 1933, recognized by Pauli [15],who remarked on the mathematical [16].It has also been very thoroughly studied in the context of the informational completeness of quantum mechanics [17], being closely related to the problems of an optimal determination of the past of the system and an optimal forecasting of its future behavior [18,19]. In this field, efforts have mostly concentrated on finding counterexamples when no unique reconstruction is possible [20]. It is known, e. g., that for any functions of a given parity, the unique phase retrieval is impossible, as we cannot distinguish between the given function and the function obtained by complex conjugation.…”