# What are appropriate boundary conditions for the quantum velocity field in a time-dependent infinite square well

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Consider the 1D  particle-in-a-box problem with time-dependent box length $L(t)>0$ (with $\hbar = m =1$)

\begin{align} i\psi _t &= \textstyle -\frac{1}{2} \psi _{xx}  \\ \psi (t,0) &=\psi (t,L(t)) = 0 , \end{align} The Madelung transformation

$\psi (t,x) = e^{iS(t,x)} \sqrt{\rho (t,x)}$

formally gives the fluid-dynamical formulation: \begin{eqnarray*} v_t+v\cdot (\nabla v) &=&  \textstyle \frac{1}{2} \nabla \frac{\Delta\sqrt{\rho } }{\sqrt{\rho }}   \\ \rho _t +\nabla \cdot (\rho v)&=& 0, \end{eqnarray*}
where $v:=\nabla S$ (and $\nabla$ and $\Delta$ may be read as $\partial /\partial _x$ and $\partial ^2/\partial _x ^2$).

Clearly, we have $\rho (t,0)= \rho (t,L(t)) =0$. But what are the appropriate boundary conditions
for $v$ (if there are any)? Is the (naive?) guess
$v(t,0) = 0, \qquad v(t,L(t)) \sim \frac{d}{dt} L(t)$ reasonable?

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