## Manipulating terms

The integral involving the kinetic energy can be written as, with the function $$\psi$$ vanishing strongly for large values of $$x,y,z$$ (given here by the limits $$a$$ and $$b$$), $$\int_a^b dxdydz \psi^* \left(-\frac{1}{2}\nabla^2\right) \psi dxdydz = \psi^*\nabla\psi|_a^b+\int_a^b dxdydz\frac{1}{2}\nabla\psi^*\nabla\psi.$$ We will drop the limits $$a$$ and $$b$$ in the remaining discussion. Inserting this expression into the expectation value for the energy and taking the variational minimum we obtain $$\delta E = \delta \left\{\int dxdydz\left( \frac{1}{2}\nabla\psi^*\nabla\psi+V\psi^*\psi\right)\right\} = 0.$$