Math 2.0

Fluid mechanics Definition

Quantum vortex

Builds on SuperfluidityCirculationWinding numberGross–Pitaevskii equation

Definition

Let a superfluid of particles of mass mm be described by a smooth wave function ψ=n eiθ\psi = \sqrt{n} \, e^{i \theta} on a region Ω⊆ℝ3\Omega \subseteq \R^3. Where n>0n > 0, its superfluid velocity is 𝐯s=(ℏ/m)∇θ\mathbf{v}_s = (\hbar / m) \nabla \theta. A quantum vortex is a curve Γ⊂Ω\Gamma \subset \Omega such that:

  1. ψ=0\psi = 0 on Γ\Gamma, and ψ≠0\psi \ne 0 near Γ\Gamma off Γ\Gamma;
  2. along a small loop CC around Γ\Gamma, the phase θ\theta changes by 2πq2\pi q for a nonzero integer qq, the winding number of ψ\psi along CC.

The integer qq is the charge of the vortex. In two dimensions, a quantum vortex is an isolated zero of ψ\psi with nonzero winding number.

Remarks

The circulation around the vortex is quantised: ∮C𝐯s⋅d𝐥=ℏm∮C∇θ⋅d𝐥=q hm.\oint_C \mathbf{v}_s \cdot d\mathbf{l} = \frac{\hbar}{m} \oint_C \nabla \theta \cdot d\mathbf{l} = q \, \frac{h}{m}. Example: in the plane, the Gross–Pitaevskii equation has solutions ψ=f(r)eiqφ\psi = f(r) e^{i q \varphi} in polar coordinates (r,φ)(r, \varphi), with f(0)=0f(0) = 0 and f(r)→n∞f(r) \to \sqrt{n_\infty} as r→∞r \to \infty. There 𝐯s=qℏ/(mr) 𝐞φ\mathbf{v}_s = q \hbar / (m r) \, \mathbf{e}_\varphi, and the circulation around any circle centred at 00 equals 2πr⋅qℏ/(mr)=qh/m2\pi r \cdot q\hbar/(m r) = q h / m. Non-example: a classical Rankine vortex can have any circulation, and its core rotates as a solid body with nonzero vorticity. Away from its core, the flow of a quantum vortex is irrotational, like a point vortex in an ideal fluid.

Reading path

To understand this page from the ground up, read these 5 pages in order. The foundations come first.

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Circulation, concept, no evidence levelCirculationconceptGross–Pitaevskii equation, concept, no evidence levelGross–Pitaevskii equa…conceptSuperfluidity, concept, no evidence levelSuperfluidityconceptWinding number, concept, no evidence levelWinding numberconceptQuantum vortex, definition, no evidence levelQuantum vortexdefinition
Each page uses the pages above it. The colour of the bar on a box shows how strong the evidence is, and the box also gives it as text. The list has the same pages.
  • no evidence level
  1. Circulation concept
  2. Gross–Pitaevskii equation concept
  3. Superfluidity concept
  4. Winding number concept
  5. Quantum vortex definition

Further reading

  1. Web https://en.wikipedia.org/wiki/Quantum_vortex
About this page
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Kinddefinition
WritingW0 stub
MSC76Y05, 82D50, 76B47
DefinesQuantum vortex
UsesSuperfluidity, Circulation, Winding number, Gross–Pitaevskii equation
Changed3 November 2026

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