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Gianfranco (talk | contribs) (Created page with "The expression for can be readily applied to the probabilities and positions as defined above, resulting in the first term given by <center> {| width="80%" | |- | width="33%" | | width="33%" |{{CD1}}<math>\langle x^2(t)\rangle=\sum_{j=1}^{92}P_j(t) x_j^2</math>{{CD2}} | width="33%" align="right" |<math>(8)</math> |} </center> And the second term given by the square of Eq. (4). The second term of <math>\Delta P_x</math> is given by the square of Eq. (5), but the f...") |
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The FINUFFT python wrapper was used to take the Fourier transform using a type 3, 2d non-uniform FFT<ref>Barnett AH, Magland J, Klinteberg LAF. A parallel nonuniform fast Fourier transform library based on an “Exponential of semicircle” kernel. SIAM J. Sci. Comput. 2019;41:C479–C504. doi: 10.1137/18M120885X. [CrossRef] [Google Scholar]</ref><ref>Barnett, A. H. Aliasing error of the kernel in the nonuniform fast Fourier transform. arXiv:2001.09405 [math.NA] (2020).</ref>, and the minimum value in time of the uncertainty relation was found. Points in momentum space were sampled on <math>p_x\in[-4,4]</math> and <math>p_y\in[-4,5]</math> along with the two additional points (<math>[-5,-4]</math>) and (<math>[-4,-5]</math>). | |||
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. | |||
[[File:Figure 4.jpeg|center|thumb|500x500px|<small>'''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.</small>]] | |||
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. | |||