Calculator Inputs
Example Data Table
Use these values to test the calculator and compare behavior.
| Mode | Potential | m | ħ | xa | xb | T | N | ω | Samples |
|---|---|---|---|---|---|---|---|---|---|
| Euclidean | Free | 1 | 1 | 0 | 1 | 1 | 20 | – | 2000 |
| Euclidean | Harmonic | 1 | 1 | 0 | 0 | 1 | 30 | 1 | 4000 |
| Real-time | Free | 1 | 1 | -1 | 1 | 0.8 | 25 | – | 5000 |
Formula Used
The calculator uses a time-sliced path integral in one spatial dimension. For imaginary time (Euclidean) with total time T and N slices, the step is ε = T/N.
The Euclidean action for a discretized path x_0 = x_a, x_N = x_b is:
S_E \approx \sum_{j=0}^{N-1}\left[\frac{m}{2}\frac{(x_{j+1}-x_j)^2}{\varepsilon}+V(x_j)\,\varepsilon\right]
The Euclidean propagator estimate is computed as:
K_E \approx \left(\frac{m}{2\pi\hbar\varepsilon}\right)^{N/2}\,\langle \exp(-S_E/\hbar) \rangle
Potential options: Free uses V(x)=0. Harmonic uses V(x)=\tfrac12 m\omega^2 x^2.
How to Use This Calculator
- Choose Euclidean for stable numerical sampling.
- Select Free or Harmonic potential for V(x).
- Enter m, ħ, xa, xb, and T.
- Set N slices; larger N improves time resolution.
- Increase samples to reduce Monte Carlo noise.
- Click Calculate to show results above the form.
- Use Download CSV or Download PDF to export.
Tip: If values overflow or collapse to zero, reduce N or increase ħ. For harmonic cases, adjust ω to explore stronger confinement.
Article: Understanding the Numerical Path Integral
1) Why path integrals are used
In the path integral picture, propagation is a weighted sum over histories connecting xa to xb in time T. This calculator estimates the propagator by averaging an exponential of the discretized action over many sampled paths, which is useful when analytic solutions are inconvenient.
2) Time slicing and discretization data
The continuous path is replaced by N slices with step ε = T/N. Larger N reduces discretization error but raises cost because each sample evaluates N increments. For exploratory studies, N = 20–60 often provides stable trends with manageable runtime.
3) Brownian bridge sampling strategy
In Euclidean sampling, the kinetic term behaves like a Gaussian random walk. The calculator draws a path and applies a bridge correction so the endpoint matches xb. The step scale is σ = √(ħ ε / m); reporting σ helps you gauge typical per-slice fluctuations.
4) Free particle as a reference case
With V(x)=0, the free particle is a baseline check. At fixed m, ħ, T, and endpoints, increasing samples should tighten the estimate, while increasing N should shift results smoothly rather than abruptly. Use this case to validate settings before adding confinement.
5) Harmonic oscillator data and confinement
For V(x)=½ m ω² x², larger ω penalizes excursions and usually lowers path spread in Euclidean mode. The potential term adds ε-weighted contributions along the path, so action statistics rise as confinement strengthens. Comparing ω = 0.5, 1, 2 highlights the transition.
6) Monte Carlo variance and sample sizing
The estimator quality depends strongly on samples. In well-behaved Euclidean runs, error shrinks roughly like 1/√samples. If |K| jumps between runs, increase samples first, then refine N. For quick checks, 2000–5000 samples are a practical starting range.
7) Reading the output metrics
The calculator reports the complex estimate K (real and imaginary parts), its magnitude |K|, and phase in radians. In Euclidean mode the imaginary part should be near zero because the weight is real. Mean action and action standard deviation summarize typical path costs and ensemble spread.
8) Practical tuning recommendations
If exp(-S/ħ) underflows to zero, reduce N, increase ħ, or shorten T. For noisy results, raise samples before changing physics parameters. To probe endpoint sensitivity, vary xa and xb at fixed T, then compare trends in |K| and action statistics.
FAQs
1) What does this calculator actually estimate?
It estimates a 1D time-sliced propagator by averaging the exponential weight of the discretized action over many sampled paths, then multiplying by the standard prefactor.
2) Why is Euclidean mode recommended?
Euclidean weights are positive and exponentially damped, which makes Monte Carlo sampling stable. Real-time weights oscillate and can cancel strongly, causing large variance.
3) How should I choose the number of slices N?
Start with 20–40 slices. If results shift noticeably when you increase N, raise it gradually. If the estimate becomes unstable, reduce N and increase samples.
4) What is the role of the random seed?
The seed controls the pseudo-random sequence used to build paths. Keeping it fixed makes runs reproducible; changing it helps verify that results are not tied to one random draw.
5) Why do I see a complex value for K?
Real-time propagation is naturally complex. In Euclidean mode the weight is real, so the imaginary part should be very small; any residual value is numerical noise.
6) How do ω and m change the harmonic case?
Larger ω strengthens confinement and increases the potential contribution to the action. Larger m reduces step fluctuations through σ = √(ħ ε / m), typically tightening sampled paths.
7) How can I reduce noisy or inconsistent outputs?
Increase samples first, then test sensitivity to N. Avoid extreme parameters that cause underflow. Compare multiple seeds and use Euclidean mode when you need dependable numerical behavior.