Everybody knows that the wave function in momentum space is the Fourier transform of the wave function in the coordinate space and vice versa. So if we have the wave function in one of these spaces, we can derive the wave function in the other space. Now suppose instead of the wave function we have the absolute square of the wave function for example in momentum space, how can we obtain the wave function in coordinate space?.
The answer to the question in the heading is simple: the Fourier transform of the absolute square of the wavefunction, which is the probability density in one space, is the probability density in the dual space, since the Fourier transform is a unitary transformation.
However the probability density is the absolute square of the wavfunction, which expresses the fact that the phase of the wavefunction can't be recovered from knowledge only of its modulus. So one can compute all equal time moments, only, from it.
In general a probability distribution specifies a random variable. There is an alternate way of defining a random variable; which is via specifying it's characteristic function. The characteristic function is the Fourier transform of probability distribution function. If F(x) is the probability distribution function of a random variable x, then φ(t)=∫eitx F(x) dx, is the characteristic function. Integration limits are minus infinity to plus infinity. It's properties are well studied. For example different characteristic functions correspond to different probability density functions.
to reconsider is that measuring/observation influences the particles as well as estimating probability: by squaring the wave amplitude; for we could consider the inverse, i.e. the square root to realize that a particle cannot be located in two different places simultaneously (square root assigns to one value: two values: +/-).
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