Superfluid Hydrodynamics

We may transform the GPE into a hydronamic equation for a superfluid,

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \frac{\partial |\psi|^2}{\partial t} + \nabla \frac{\hbar}{2mi} \left( { \psi^*\nabla \psi - \psi \nabla\psi^* } \right) \,, }

by introducing flow, from current $j$,

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle v = \frac{j}{n} = \frac{\psi^*\nabla \psi - \psi \nabla\psi^*}{2m i |\psi|^2} \,. }

This gives the continuity equation

${\frac {\partial n}{\partial t}}+\nabla (nv)=0\,.$

Writing $\psi =fe^{i\phi }$, and noting that the gradient of the phase gives us the velocity field, we get equations of motion for $f$ and $\phi$,

$-\hbar {\frac {\partial \phi }{\partial t}}=-frac{\hbar ^{2}}{2mf}\nabla ^{2}f+{\frac {1}{2}}mv^{2}+V(r)+U_{0}f^{2}\,.$

This reduces to

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \begin{array}{rcl} m \frac{\partial v}{\partial t} &=& - \nabla (\delta \mu + \frac{1}{2} mv^2 \\ \delta \mu &=& v + U_0 n \frac{\hbar^2}{2m\sqrt{n}} \nabla ^2 \sqrt{n}-\mu_0 \,. \end{array}}

The Thomas-Fermi approximation is now applied, neglecting Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \nabla f} , but keeping Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \nabla \phi} , giving

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle m \frac{\partial^2 \delta n}{\partial t^2} = U_0 \nabla (n_0 \nabla (\delta n)) \,, }

a wave equation for the density. For Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle n_0} constant, $U_{0}$ is the speed of sound squared, Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle c = \sqrt{U_0/m}} . The Thomas-Fermi solution for Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle n_0} gives collective modes of the condensate. A droplet of condensate can have shape resonances, waves, and many other physical behaviors, captured by these solutions.