Phy5646/Problem on Variational Method

Consider the one-dimensional potential

\psi (x)=\lambda \frac{x^{4}}{4}+\lambda a\frac{x^{3}}{4}-\lambda \frac{a^{2}x^{2}}{8}

$\psi (x)=\left({\frac {\beta }{\pi }}\right)^{\frac {1}{4}}e^{\frac {-\beta (x-x_{0})^{2}}{2}}$