Past exam of the mathematics course of the University of Cambridge / 2007 / iii / Paper 84 / 1

Treat $\psi$ and its complex conjugate as independent variables in the functional derivative. Integrating the gradient variation by parts, with vanishing boundary terms, givesThe factors follow from ${\partial }_{{\psi }^{\ast }}|\psi {|}^4=2|\psi {|}^2\psi$ and ${\partial }_{{\psi }^{\ast }}|\psi {|}^6=3|\psi {|}^4\psi$. Thus the cubic-quintic Gross–Pitaevskii equation is

At fixed particle number $N_p=\int |\psi {|}^{2}d\mathbf{x}$, an equilibrium wavefunction is a constrained critical point of the energy functional. Introduce the chemical potential $\mu$ as the Lagrange multiplier for $N_p$, and impose $\delta (E-\mu N_p)=0$. The resulting stationary cubic-quintic Gross–Pitaevskii equation isThe time-dependent equilibrium has $\psi (\mathbf{x},t)=e^{-i\mu t/\hslash }{\psi }_0(\mathbf{x})$, so its number density is stationary even though its quantum phase changes uniformly. The specified particle number fixes $\mu$ together with the boundary conditions. In an infinite uniform bulk, particle number is understood through a finite volume or a prescribed bulk density.

For a nonzero uniform wavefunction, the Laplacian vanishes. Dividing the stationary cubic-quintic Gross–Pitaevskii equation by ${\psi }_0$ yieldsEquivalently, the interaction energy density is $e(n)=V_0{n}^2/2+W_0{n}^3/3$ and its derivative $e^{\prime }(n_0)$ is the chemical potential. Positive local compressibility requires ${\mu }^{\prime }(n_0)=V_0+2W_0{n}_0>0$. This stability condition is distinct from the sign of $\mu$ itself.

For the displayed scaling, assume $\mu >0$ and $n_0>0$. SetSubstituting in the stationary cubic-quintic Gross–Pitaevskii equation and dividing by $\mu \sqrt{n_0}$ givesThe uniform-state relation proves $\alpha +\beta =1$, so the real bulk amplitude is one; its constant quantum phase can be chosen to make $\tilde{\psi }\to 1$ in the zero-winding sector. The unit of distance $\ell$ is the kinetic-to-chemical-potential scale, reducing to the stated healing length convention for purely repulsive two-body coupling.

The positive-$\mu$ assumption is necessary for the printed sign convention. If $\mu <0$, the real length is $\hslash /\sqrt{2m|\mu |}$, and the dimensionless linear term is $-1$, with interaction coefficients divided by $|\mu |$; the displayed equation with linear term $+1$ does not follow. At $\mu =0$ this scaling is undefined. These are limitations of this particular nondimensionalization, not a claim that all cubic-quintic bulk states have positive chemical potential.

A straight quantum vortex with integer winding number $\mathcal{N}$ has $\tilde{\psi }=R(r)e^{i\mathcal{N}\theta }$, independent of $z$. Here $r$ is measured in the unit of distance from the preceding part. The polar Laplacian supplies the centrifugal term, givingFor $\mathcal{N}\ne 0$, regularity requires $R\sim c{r}^{|\mathcal{N}|}$ near the axis; the bulk condition is $R\to 1$. The full wavefunction tends to $e^{i\mathcal{N}\theta }$ rather than one. This is the constant far-field phase excludes net vortex winding distinction: the equation remains applicable to a vortex, but the nonwinding boundary condition must be interpreted as an amplitude condition.

Let $F(R)=R-\alpha R^3-\beta R^5$. Since $\alpha +\beta =1$, one has $F(1)=0$ and $F^{\prime }(1)=1-3\alpha -5\beta =-2(1+\beta )$. Insert $R=1-p/r^2+\cdots$. Its derivative terms start at order $r^{-4}$, whileCancellation of the order-$r^{-2}$ terms proves the cubic-quintic quantum vortex tailFor a stable nondegenerate bulk state in this scaling, $1+\beta =n_0(V_0+2W_0{n}_0)/\mu >0$. At $\beta =-1$ the amplitude restoring derivative vanishes, and an inverse-square perturbation cannot cancel the centrifugal term, so the stated expansion fails. For zero winding the uniform solution has $p=0$.