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An example showing the non-uniqueness of a wavefunction given its position and momentum distributions

The author presents an elementary example which shows that position and momentum distributions do not uniquely determine the wavefunction. In particular, the example shows that a wavefunction consisting of two gaussian wavepackets with momenta k1 and k2 has the same position and momentum distributions as the wavefunction obtained by interchanging k1 and k2. This interpretation relies on viewing the wavefunction as that of a free particle but, of course, the abstract result is simply that for f in L2(R) knowledge of mod f(x) mod and mod f(k) mod is not always sufficient to determine f(x) (or f(k); here f denotes the Fourier transform of f). In addition, the bearing of the example presented here upon certain interesting questions raised by Vogt (1978) is discussed.