Gross–Pitaevskii equation

Scalar contact form

For a particle-number-normalized field, the baseline form is iℏ∂tψ = [−ℏ²∇²/(2m) + V + g|ψ|²]ψ. Record departures term by term. See [Pitaevskii and Stringari, Nonuniform Bose Gases at Zero Temperature](https://academic.oup.com/book/26678/chapter-abstract/195458863).

1. What is the exact operator equation, including every sign, coefficient, and additional term? definition
2. Which reference supports this formulation and its interpretation of the field? provenance

Field and coefficient conventions

Require an explicit normalization integral and length, time, energy, and field scales before comparing coefficients or observables.

1. Does the field integrate to unity, condensate number, or another declared quantity, and where is the particle-number factor placed? definition
2. What transformation recovers physical density, time, and energy from the dimensionless variables? measurement

Regime evidence

The standard treatment concerns weakly interacting, dilute condensates near the zero-temperature limit; applicability needs evidence for the actual system. See [Pethick and Smith, Theory of the condensed state](https://www.cambridge.org/core/books/abs/boseeinstein-condensation-in-dilute-gases/theory-of-the-condensed-state/19D43707671194C5704F4B1FE10055DA).

1. What density and scattering-length estimates determine the relevant diluteness parameter across the cloud? measurement
2. What evidence bounds thermal population, quantum depletion, and correlations sufficiently for the intended observable? boundary

Reduction and extension contract

Require a derivation or cited justification for effective dimensionality and identify nonlocal interactions, multiple components, dissipation, noise, or fluctuation corrections explicitly.

1. Which confinement and energy-scale conditions justify eliminating spatial directions, and how was the effective coupling obtained? boundary
2. Which omitted physical effect would invalidate the baseline equation for this task, and which linked formulation addresses it? action

Parameter and potential definition

Specify mass, interaction coefficient, particle number, potential, and reference frame with their provenance and uncertainty.

1. What values and uncertainties define mass, scattering length, condensate number, and any effective coupling? measurement
2. What function defines confinement or driving, and is any rotational term expressed in a declared rotating frame? definition

Problem closure

Treat phase information, topology, normalization, and boundary conditions as substantive inputs rather than solver defaults.

1. What initial complex field or stationary-state constraints select the intended physical state? definition
2. Which boundary conditions represent the physical system, and which boundaries merely truncate the computational domain? boundary

Chemical potential and energy

Record the stationary ansatz ψ(r,t) = φ(r)exp(−iμt/ℏ), the associated energy functional, and the constraints used to select a solution.

1. Is the target a ground state, a constrained minimum, or an excited stationary branch? definition
2. How are chemical potential and total energy computed separately under the chosen normalization? measurement

Condensate observables

Attach explicit extraction rules to density, phase, current, collective motion, and vortices. These are established mean-field applications discussed in [Dalfovo et al., Theory of Bose-Einstein condensation in trapped gases](https://arxiv.org/abs/cond-mat/9806038).

1. How are density, current, and phase-derived quantities calculated, including behavior near field zeros? measurement
2. What phase-winding and density evidence distinguishes a vortex from a numerical artifact? measurement

Stability evidence

Keep energetic stability, linear dynamical stability, and finite-time persistence as separate assessments.

1. What constrained energy comparison or linearized perturbation analysis supports the stated stability classification? measurement
2. Which perturbations and observation times have been tested before using this solution to predict persistent behavior? action

Method and resolution evidence

Require convergence evidence for relevant spatial and temporal scales; imaginary-time relaxation must be identified as a state-finding method.

1. Does the selected method find the requested branch or evolution, and what initialization or constraints could bias its result? action
2. How do the target observables change under mesh, time-step, solver-tolerance, and domain-size refinement? measurement

Acceptance and escalation

Define task-specific acceptance criteria using equation residuals, applicable invariants, benchmark limits, and independent physical evidence.

1. Which norm and energy conservation or balance laws apply to this potential, boundary treatment, and equation variant? boundary
2. What residual, benchmark agreement, and observable uncertainty must be achieved before accepting the result or escalating to another theory? action

Check these first

Recalled without web access and unsourced; every item is a lead to verify.

1. Which of these check these first hold for the sense of Gross-Pitaevskii equation this model covers, and on what evidence? provenance

Kinds and varieties

Recalled without web access and unsourced; every item is a lead to verify.

1. Which of these kinds and varieties hold for the sense of Gross-Pitaevskii equation this model covers, and on what evidence? provenance

Real-world use

Recalled without web access and unsourced; every item is a lead to verify.

1. Which of these real-world use hold for the sense of Gross-Pitaevskii equation this model covers, and on what evidence? provenance

Failure modes and hazards

Recalled without web access and unsourced; every item is a lead to verify.

1. Which of these failure modes and hazards hold for the sense of Gross-Pitaevskii equation this model covers, and on what evidence? provenance

Neighbouring kinds and how to tell them apart

Recalled without web access and unsourced; every item is a lead to verify.

1. Which of these neighbouring kinds and how to tell them apart hold for the sense of Gross-Pitaevskii equation this model covers, and on what evidence? provenance