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The expression for can be readily applied to the probabilities and positions as defined above, resulting in the first term given by

$\langle x^{2}(t)\rangle =\sum _{j=1}^{92}P_{j}(t)x_{j}^{2}$

$(8)$

And the second term given by the square of Eq. (4). The second term of $\Delta P_{x}$ is given by the square of Eq. (5), but the first term is more nuanced. This is owing to the complex number returned when acting the derivative operator twice on the probability. To overcome this, Fourier transforms were used to change Eq. (5) into the momentum basis which then allowed for the efficient calculation of $P_{x}^{2}(t)$.

Denoting ${\tilde {P}}_{j}(t)$ as the momentum-space probability obtained through a 2-dimensional, non-uniform Fourier transform of the position space pseudo-wavefunction, Eq. (5) can be rewritten as,

$\langle p_{x}(t)\rangle =\sum _{j=1}^{92}{\tilde {P}}_{j}(t)p_{j}$

$(9)$

Leading to the first term in the expression to be written as,

$\langle p_{x}^{2}(t)\rangle =m^{2}\sum _{j=1}^{92}{\tfrac {x_{j}^{2}}{{\tilde {p}}_{j}(t)}}[{d \over dt}P_{j}(t)]^{2}$

$(10)$

The FINUFFT python wrapper was used to take the Fourier transform using a type 3, 2d non-uniform FFT, and the minimum value in time of the uncertainty relation was found. Points in momentum space were sampled on $p_{x}\in [-4,4]$ and $p_{y}\in [-4,5]$ along with the two additional points ($[-5,-4]$) and ($[-4,-5]$).

Figure 4 shows the position and momentum probabilities respectively in their own basis. An animation showing how these evolve in time for the different conditions is presented in Supplementary Material 2.

Figure 4: (A) Probability distribution for a single subject in the position basis. (B) Momentum basis probability distribution for a single subject. The momentum values used for the Fourier transform are indicated by the point locations. Points are colour-/size-coded to represent the probability value at that location.

To compute the values reported in Table 2, the corresponding value was found for each subject, and these were used to calculate the group average reported here.