Store:Exel14

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,

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

$(10)$