Discretize paths, sample trajectories, and evaluate quantum amplitudes. Switch potentials, tune slices, inspect action terms. Download clean CSV or PDF for reports and sharing.
| Case | Potential | m | ħ | β | x₀ → x_T | N | Samples | Expected behavior |
|---|---|---|---|---|---|---|---|---|
| A | Free | 1 | 1 | 1 | 0 → 1 | 64 | 5000 | Matches closed-form ρ_free. |
| B | Harmonic (ω=1) | 1 | 1 | 1 | 0 → 0 | 96 | 8000 | Should approach analytic harmonic kernel. |
| C | Quartic (λ=0.1) | 1 | 1 | 0.5 | -1 → 1 | 128 | 12000 | Needs more samples; weights vary widely. |
This tool evaluates a discretized Euclidean (imaginary-time) path integral for a one-dimensional particle between fixed endpoints x₀ and x_T over time β.
Using a time slicing with N steps (ε=β/N), and sampling free Brownian bridges, the interacting density matrix is estimated by reweighting:
The free normalization ρ_free is known exactly, so the estimator does not require an unknown overall constant. Statistical uncertainty is reported from the sample variance of the reweight factor.
This calculator estimates the Euclidean density matrix ρ(xT,β;x0,0) by sampling paths and averaging a reweighting factor. In many workflows, β plays the role of inverse temperature, and the same kernel is used to extract ground-state information when β is large.
The time interval is split into N slices, with step size ε = β/N. With the default example β=1 and N=64, the step is ε≈0.015625. Increasing N reduces Trotter discretization error, but increases compute cost roughly linearly in N.
Paths are generated as a Gaussian random walk and then corrected to satisfy the endpoint constraint x(0)=x0, x(β)=xT. The per-slice diffusion scale is σ = √(ħ ε / m), so doubling m reduces typical fluctuations by about √2 at fixed β and N.
The potential contribution is integrated using a midpoint rule, ∫V dτ ≈ ε Σ V((xj+xj+1)/2). This midpoint choice improves accuracy over left-point sampling for smooth potentials and makes the estimator less sensitive to oscillations between adjacent slices.
The interface supports free, harmonic, quartic, double-well, linear, square-well, and a polynomial up to x⁶. For the harmonic model, the analytic kernel is also computed, enabling immediate validation. For stiff models (large ω, a, or V0), the weight distribution can become extremely broad.
The estimator uses ρ = ρfree · ⟨A⟩, where A = exp(-∫V/ħ). The reported standard error scales approximately as 1/√S with S samples, so increasing samples from 5,000 to 20,000 typically reduces Monte Carlo noise by about a factor of 2.
Along with ρ, the calculator reports the mean and spread of V(mid) and the mean of ∫V dτ. Large standard deviation of V(mid) or warnings about weight spread indicate slow convergence and a need for higher N and S.
Start with moderate settings (for example N=64–128, S≥5000), then increase N until the result changes less than your target tolerance. Next, increase S until the reported error bars are acceptably small. For the harmonic case, aim for agreement with the analytic kernel within the Monte Carlo uncertainty before moving to more complex potentials.
No. It evaluates an imaginary-time (Euclidean) kernel. For real-time propagation, you typically need contour methods or analytic continuation, which is numerically delicate for noisy Monte Carlo estimates.
Free bridges are easy to sample and have an exact normalization. The potential is included by reweighting with exp(-∫V/ħ), which turns the path integral into a statistically estimable average.
Increase N until results stabilize. A common pattern is doubling N (64→128→256) and checking that changes are smaller than your acceptable error, while monitoring computation time.
Large positive potentials, large β, or small ħ can make exp(-∫V/ħ) tiny. Reduce potential strength, reduce β, or increase samples to improve effective statistics.
It is the sampled average of the potential integral along paths. Large values generally correspond to stronger suppression of amplitudes and can signal that the kernel will be very small and harder to estimate accurately.
Yes. Select the harmonic potential and compare ρ_est against the displayed analytic harmonic kernel. Adjust N and samples until the difference is within the reported Monte Carlo uncertainty.
This implementation is one-dimensional. Extending to multiple dimensions requires vector paths, a higher computational load, and careful tuning of sampling and variance reduction to keep weight fluctuations manageable.
Important Note: All the Calculators listed in this site are for educational purpose only and we do not guarentee the accuracy of results. Please do consult with other sources as well.