A particle with momentum $\hat{p}$ moves in a one-dimensional real potential with Hamiltonian given by

$$\hat{H}=\frac{1}{2 m}(\hat{p}+i s A)(\hat{p}-i s A), \quad-\infty<x<\infty$$

where $A$ is a real function and $s \in \mathbb{R}^{+}$. Obtain the potential energy of the system. Find $\chi(x)$ such that $(\hat{p}-i s A) \chi(x)=0$. Now, putting $A=x^{n}$, for $n \in \mathbb{Z}^{+}$, show that $\chi(x)$ can be normalised only if $n$ is odd. Letting $n=1$, use the inequality

$$\int_{-\infty}^{\infty} \psi^{*}(x) \hat{H} \psi(x) d x \geqslant 0$$

to show that

$$\Delta x \Delta p \geqslant \frac{\hbar}{2}$$

assuming that both $\langle\hat{p}\rangle$ and $\langle\hat{x}\rangle$ vanish.