**2. Heisenberg uncertainty principle in R, M, and C**

The uncertainties in the probability set and universe **R** in position and momentum (Δ*xR* and Δ*pR*) are defined as being equal to the square root of their respective variances in **R**, so that:


#### **Table 4.**

*The momentum distribution characteristics for L* ¼ 200*, h* ¼ 1*, and n* ¼ 10,000*.*

$$
\Delta \mathbf{x}\_{\mathbb{R}} \times \Delta p\_{\mathbb{R}} = \sqrt{\mathbf{Var}\_{\mathbf{x}, \mathbb{R}}} \times \sqrt{\mathbf{Var}\_{p, \mathbb{R}}} = \sqrt{\frac{\mathbf{L}^2}{\mathbf{1}\mathbf{2}} \left(\mathbf{1} - \frac{\mathbf{6}}{n^2 \pi^2}\right)} \times \sqrt{\frac{\hbar^2 n^2 \pi^2}{\mathbf{L}^2}} = \frac{\hbar}{2} \sqrt{\frac{n^2 \pi^2}{3} - 2\pi}
$$

This product increases with increasing *n*, having a minimum value for *n* ¼ 1. The value of this product for *n* ¼ 1 is about equal to 0.568 ℏ which obeys the Heisenberg uncertainty principle, which states that:

$$
\Delta \mathbf{x} \times \Delta p \ge \frac{\hbar}{2} \\
\Leftrightarrow \forall n \ge 1 : \Delta \mathbf{x}\_R \times \Delta p\_R \ge \frac{\hbar}{2},
$$

The uncertainties in the probability set and universe **M** in position and momentum (Δ*xM* and Δ*pM*) are defined as being equal to the square root of their respective variances in **M**, so that:

$$
\Delta \mathbf{x}\_{\mathcal{M}} \times \Delta p\_{\mathcal{M}} = \sqrt{\mathbf{Var}\_{\mathbf{x}, \mathcal{M}}} \times \sqrt{\mathbf{Var}\_{p, \mathcal{M}}} \rightarrow \sqrt{i \left\{ \frac{L^2}{12} \left[ L - \left( 1 - \frac{6}{n^2 \pi^2} \right) \right] \right\}} \times \sqrt{+\infty} \rightarrow +\infty
$$

<sup>⇔</sup>∀*<sup>n</sup>* <sup>≥</sup><sup>1</sup> : <sup>Δ</sup>*xM* � <sup>Δ</sup>*pM* <sup>≥</sup> <sup>ℏ</sup> <sup>2</sup>, in accordance with the Heisenberg uncertainty principle.

The uncertainties in the probability set and universe **C** = **R** þ**M** in position and momentum (Δ*xC* and Δ*pC*) are defined as being equal to the square root of their respective variances in **C**, so that:

$$\begin{split} \Delta \mathbf{x}\_{\text{C}} \times \Delta p\_{\text{C}} &= \sqrt{\mathbf{Var}\_{\mathbf{x},\text{C}}} \times \sqrt{\mathbf{Var}\_{p,\text{C}}} \\ &\rightarrow \sqrt{\frac{L^{2}}{12} \left( 1 - \frac{6}{n^{2}\pi^{2}} \right) + i \left\{ \frac{L^{2}}{12} \left[ L - \left( 1 - \frac{6}{n^{2}\pi^{2}} \right) \right] \right\}} \times \sqrt{+\infty} \rightarrow +\infty \end{split}$$

<sup>⇔</sup>∀*<sup>n</sup>* <sup>≥</sup><sup>1</sup> : <sup>Δ</sup>*xC* � <sup>Δ</sup>*pC* <sup>≥</sup> <sup>ℏ</sup> <sup>2</sup>, in accordance with the Heisenberg uncertainty principle.

Consequently, the Heisenberg uncertainty principle is verified in the universe **R**, in the universe **M**, and the complex universe **C**.
