= Solution
Direct substitution gives $g^sg^r=g^{s+r}$. Differentiating at $s=0$ gives $t\partial_x+ix(ψ\partial_ψ-ψ^*\partial_{ψ^*})$.
Writing the transformed field in transformed coordinates gives
$$
\widetilde ψ(t,\widetilde x)=e^{i(s\widetilde x-s^2t/2)}ψ(t,\widetilde x-st).
$$
Consequently
$$
\widetilde ψ_t=e^{i\Phi}(ψ_t-sψ_x-is^2ψ/2),
$$
$$
\widetilde ψ_x=e^{i\Phi}(ψ_x+isψ),\qquad
\widetilde ψ_{xx}=e^{i\Phi}(ψ_{xx}+2isψ_x-s^2ψ),
$$
where $\Phi=s\widetilde x-s^2t/2$. These formulas, together with their complex conjugates and the analogous mixed derivatives, give the second prolongation. Differentiating at zero yields
$$
\boxed{η^0=ixψ_t-ψ_x,\qquad η^{11}=ixψ_{xx}+2iψ_x.}
$$
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