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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

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \langle x^2(t)\rangle=\sum_{j=1}^{92}P_j(t) x_j^2}

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle (8)}

And the second term given by the square of Eq. (4). The second term of Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \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 Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle P_x^2(t)} .

Denoting Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \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,

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \langle p_x(t)\rangle=\sum_{j=1}^{92}\tilde{P}_j(t)p_j}

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle (9)}

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

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \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}

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle (10)}