Path Integral Probability Calculator
Path Integral Probability Calculator computes free particle probability amplitude from initial to final position over time, visualizing density for quantum studies.
Formulas Used in Path Integral Probability Calculator
The calculator uses the following formulas for a free particle in one dimension:
Free Particle Propagator:
\\[ K(x_f, t; x_i, 0) = \sqrt{\frac{m}{2\pi i \hbar t}} \exp\left( i \frac{m (x_f – x_i)^2}{2 \hbar t} \right) \\]Probability Density:
\\[ P = |K(x_f, t; x_i, 0)|^2 = \frac{m}{2\pi \hbar t} \\]Where:
- \\( m \\): Particle mass (kg)
- \\( \hbar \\): Reduced Planck constant (\\( 1.0545718 \times 10^{-34} \, \text{J·s} \\))
- \\( x_i \\): Initial position (m)
- \\( x_f \\): Final position (m)
- \\( t \\): Time (s)
- \\( P \\): Probability density (1/m)
Example Calculations
Example 1: Electron over Femtosecond
Input: Mass = 9.11e-31 kg, Initial Position = 0 m, Time = 1e-15 s
\\[
P = \frac{9.11 \times 10^{-31}}{2 \pi \cdot 1.0545718 \times 10^{-34} \cdot 10^{-15}} \approx 1.372 \times 10^{18} \, \text{m}^{-1}
\\]
Result: Probability Density at x_f = x_i: 1.372e18 m⁻¹
Example 2: Heavy Particle
Input: Mass = 1e-27 kg, Initial Position = 0 m, Time = 1e-15 s
\\[
P = \frac{10^{-27}}{2 \pi \cdot 1.0545718 \times 10^{-34} \cdot 10^{-15}} \approx 1.506 \times 10^{21} \, \text{m}^{-1}
\\]
Result: Probability Density at x_f = x_i: 1.506e21 m⁻¹
Example 3: Longer Time
Input: Mass = 9.11e-31 kg, Initial Position = 0 m, Time = 1e-12 s
\\[
P = \frac{9.11 \times 10^{-31}}{2 \pi \cdot 1.0545718 \times 10^{-34} \cdot 10^{-12}} \approx 1.372 \times 10^{15} \, \text{m}^{-1}
\\]
Result: Probability Density at x_f = x_i: 1.372e15 m⁻¹