Heisenberg Uncertainty Priniciple Calculator

Heisenberg Uncertainty Priniciple Calculator computes position and momentum uncertainties, verifying the principle for quantum systems.

x1 (m):

p1 (kg·m/s):

x2 (m):

p2 (kg·m/s):

x3 (m):

p3 (kg·m/s):

x4 (m):

p4 (kg·m/s):

x5 (m):

p5 (kg·m/s):

Formulas Used in Heisenberg Uncertainty Calculator

The calculator verifies the Heisenberg Uncertainty Principle using the following formulas:

Heisenberg Uncertainty Principle:

\\[ \Delta x \cdot \Delta p \geq \frac{\hbar}{2} \\]

Uncertainty Product:

\\[ U = \Delta x \cdot \Delta p \\]

Relative Uncertainty Ratio:

\\[ R = \frac{\Delta x \cdot \Delta p}{\hbar / 2} \cdot 100 \\]

Standard Deviations:

\\[ \Delta x = \sqrt{\langle x^2 \rangle – \langle x \rangle^2}, \quad \Delta p = \sqrt{\langle p^2 \rangle – \langle p \rangle^2} \\]

Where:

\\( \hbar \\): Reduced Planck constant (\\( 1.0545718 \times 10^{-34} \, \text{J·s} \\))
\\( \Delta x \\): Uncertainty in position (m)
\\( \Delta p \\): Uncertainty in momentum (kg·m/s)
\\( U \\): Uncertainty product (J·s)
\\( R \\): Relative uncertainty ratio (%)
\\( \langle x \rangle, \langle p \rangle \\): Mean position and momentum
\\( \langle x^2 \rangle, \langle p^2 \rangle \\): Mean squared position and momentum

Example Calculations

Example 1: Electron with Minimal Uncertainty

Input: Points: (0, 0), (0.1, 1e-34), (-0.1, -1e-34)

\\[ \langle x \rangle = \frac{0 + 0.1 + (-0.1)}{3} = 0, \quad \langle x^2 \rangle = \frac{0^2 + 0.1^2 + (-0.1)^2}{3} = 0.006667 \\] \\[ \Delta x = \sqrt{0.006667 – 0^2} \approx 0.08165 \, \text{m} \\] \\[ \langle p \rangle = \frac{0 + 1 \times 10^{-34} + (-1 \times 10^{-34})}{3} = 0, \quad \langle p^2 \rangle = \frac{0^2 + (1 \times 10^{-34})^2 + (-1 \times 10^{-34})^2}{3} \approx 6.667 \times 10^{-69} \\] \\[ \Delta p = \sqrt{6.667 \times 10^{-69} – 0^2} \approx 8.165 \times 10^{-35} \, \text{kg·m/s} \\] \\[ U = \Delta x \cdot \Delta p \approx 6.667 \times 10^{-36} \, \text{J·s}, \quad R = \frac{6.667 \times 10^{-36}}{5.272859 \times 10^{-35} / 2} \cdot 100 \approx 25.30 \, \% \\]

Result: \\( \Delta x \approx 0.08165 \, \text{m}, \Delta p \approx 8.165 \times 10^{-35} \, \text{kg·m/s}, U \approx 6.667 \times 10^{-36} \, \text{J·s}, R \approx 25.30 \, \% \\)

Example 2: Electron with Moderate Uncertainty

Input: Points: (0, 0), (0.5, 5e-34), (-0.5, -5e-34)

\\[ \Delta x \approx 0.40825 \, \text{m}, \quad \Delta p \approx 4.082 \times 10^{-34} \, \text{kg·m/s}, \quad U \approx 1.667 \times 10^{-34} \, \text{J·s}, \quad R \approx 632.46 \, \% \\]

Result: \\( \Delta x \approx 0.40825 \, \text{m}, \Delta p \approx 4.082 \times 10^{-34} \, \text{kg·m/s}, U \approx 1.667 \times 10^{-34} \, \text{J·s}, R \approx 632.46 \, \% \\)

Example 3: Macroscopic Particle with Large Uncertainty

Input: Points: (0, 0), (1, 1e-33), (-1, -1e-33)

\\[ \Delta x \approx 0.8165 \, \text{m}, \quad \Delta p \approx 8.165 \times 10^{-34} \, \text{kg·m/s}, \quad U \approx 6.667 \times 10^{-34} \, \text{J·s}, \quad R \approx 2529.82 \, \% \\]

Result: \\( \Delta x \approx 0.8165 \, \text{m}, \Delta p \approx 8.165 \times 10^{-34} \, \text{kg·m/s}, U \approx 6.667 \times 10^{-34} \, \text{J·s}, R \approx 2529.82 \, \% \\)