No need of Fourier

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In Quantum Mechanics, position and momentum are conjugate variables. The probability density function in momentum space, φ(p;t), is the Fourier transform of the wave function in position space, ψ(x;t). A consequence of that is the Kennard inequality that describes the Uncertainty principle. A related consequence is that position and momentum operators are noncommuting, thus these observables are said to be incompatible and cannot in general be measured simultaneously.

In our local-realistic model, there is no need of the abstract mathematical formulation of standard quantum mechanics. In the latest post we have seen that the Schrödinger equation and the Born rule can be retrieved without appealing to complex wave functions. Indeed, we have already evaluated the position probability density function (in the case of a free particle with any type of source preparation) from the model's elementary rules of motion. The key role in this evaluation is played by expressing the position x as a stochastic function of the source momentum v0, itself a stochastic process with known pdf (uniform), then using the probability chain rule.

We can also evaluate the momentum pdf from the relation between the source momentum and the momentum propensity, after replacing V for x. By further neglecting the terms containing the source positions (after a large time, source positions become undistinguishable w.r.t. current position), we obtain


Clearly, this pdf is time-independent, as a consequence of the fact that we have taken the steady-state values of the momentum propensity in the derivations above. The standard QM evaluation of this density as |φ(p;t)|2 is also time-independent because time only enters φ(p;t) as a phase factor.

As an example, the figures below show the distribution of momentum propensity as a function of source momentum (top) and the momentum propensity pdf (bottom) for the two-slit scenario. Theoretical values are shown in red. The agreement tends to be perfect.



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