I know that the chemical potential of a BEC can be calculated with $\mu=\frac{\partial E}{\partial N}$, where $E$ is the energy and $N$ is the number of particles. For the Gross-Pitaevskii equation with a time-dependent potential $V(x,t)$,

\begin{align} i\hbar\dot\psi=-\frac{\hbar^2}{2m}\nabla^2\psi+V(x,t)\psi^2+g\psi^4\ , \end{align}

we have the energy

\begin{align} E=\int d^3x\left[-\frac{\hbar^2}{2m}\nabla^2\psi+V(x,t)\psi^2+g\psi^4\right] \end{align}

and $N=\psi^2$. Therefore, $\mu=g|\phi|^2+V(x,t)$. If $V(x,t)=V(x)$, the chemical potential is apparently a constant (according to (1.39) in this thesis), but I am unsure how to see that. Is $\mu$ still constant even with time-dependence incorporated into the potential, $V$? I know that I'm missing something, but I'm not sure what I'm missing.

  • $\begingroup$ I'm not quite sure I'm able to follow your train of thought. If $\mu=g|\phi|^2+V(x,t)$ (but I doubt there should be an $x$ in there), then $\mu$ is clearly time dependent because of the potential. $\endgroup$ – noah May 10 at 22:54
  • $\begingroup$ What does 'BEC' stand for? $\endgroup$ – Gert May 10 at 23:13

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