Editing Quantum turbulence (section)

=== Quantized circulation ===
The property of quantized circulation arises as a consequence of the existence and uniqueness of a complex macroscopic [[Wave function|wavefunction]] <math>\Psi</math>, which affects the [[vorticity]] (local rotation) in a very profound way, making it crucial for quantum turbulence.

The velocity and density of the fluid can be recovered from the wavefunction <math>\Psi(\mathbf{x},t)</math> by writing it in [[Complex number|polar form]] <math>\Psi(\mathbf{x},t) = |\Psi(\mathbf{x},t)|e^{i\phi(\mathbf{x},t)}</math>, where <math>|\Psi|</math> is the magnitude of <math>\Psi</math> and <math>\phi</math> is the phase. The velocity of the fluid is then <math>\mathbf{v}(\mathbf{x},t) = (\hbar/m)\nabla \phi</math>, and the number density is <math>n(\mathbf{x},t) = |\Psi|^2</math>. The mass density is related to the number density by <math>\rho(\mathbf{x},t) = mn</math>, where <math>m</math> is the mass of one [[boson]].

The [[Circulation (physics)|circulation]] <math>\Gamma</math> is defined to be the line integral along a simple closed path <math>C</math> within the fluid

<math>\Gamma = \oint_C \mathbf{v} \cdot \mathbf{dr}</math>

For a [[Simply connected space|simply-connected]] surface <math>S</math>, [[Generalized Stokes theorem|Stokes theorem]] holds, and the circulation vanishes, as the velocity can be expressed as the gradient of the phase. For a multiply-connected surface, the phase difference between an arbitrary initial point on the curve <math>C</math> and the final point (same as initial point as <math>C</math> is closed) must be <math>2\pi q</math>, where <math>q=0,1,2,\cdots</math> in order for the wavefunction to be single-valued. This leads to a quantized value for the circulation

<math>\Gamma = \frac{\hbar}{m} \oint_C \nabla S \cdot \mathbf{dr} = q\kappa</math>

where <math>\kappa = h/m </math> is the ''quantum of circulation'', and the integer <math>q</math> is the charge (or winding number) of the vortex. Multiply charged vortices (<math>q>1</math>) in helium II are unstable and for this reason in most practical applications <math>q=1</math>. It is energetically favourable for the fluid to form <math>q</math> singly-charged vortices rather than a single vortex of charge <math>q</math>, and so a multiply-charged vortex would split into singly-charged vortices. Under certain conditions, it is possible to generate certain vortices with a charge higher than 1.