A equation about Lambert $W$ function

Question

$$(\Delta X)(\Delta P)=\frac{\hbar}{2}\exp\left[\frac{\alpha^{2n}l_{PI}^{2n}}{\hbar^{2n}}\left(\langle\hat{P}\rangle^2+(\Delta P)^2\right)^n\right]$$ I want to solve the equation to get $\Delta P$ represented by $\Delta X$ with the method of Lambert $W$ function when $\langle\hat{P}\rangle=0$. How can I do?

Answer 1

Tymas comment is right. $x p=h e^{a p^{2n}}$ has solution for $p$ $$ p=\frac {h}{x} \exp\left({-{\frac {1}{2\,n}{\rm W} \left(-2\,an{{\rm e}^{ -2\,n\,\ln \left( x/h \right)}}\right)}}\right) $$ according to Maple.

But $x p=h e^{a (1+p)^{2n}}$ does not have solution for $p$ using $W$ (as far as Maple found).