I am working through Griffiths’ *Introduction to Quantum Mechanics*. In chapter 1, he attempts to impose a condition such that
$$\frac{d}{dt}\int_{-\infty}^\infty\left|\psi(x,t)\right|^2dx=0$$
so that the normalization of a solution to the Schrödinger equation is independent of time. He derives that
$$\frac{d}{dt}\int_{-\infty}^\infty\left|\psi(x,t)\right|^2dx=\frac{i\hbar}{2m}\left.\left(\psi^\star\frac{\partial\psi}{\partial x}-\frac{\partial\psi^\star}{\partial x}\psi\right)\right|_{-\infty}^\infty.$$
Griffiths then concludes that a wave function must satisfy
$$\psi\rightarrow 0\qquad\textrm{as}\qquad x\rightarrow\pm\infty.$$

- Is this condition really enough?
- For example, are there square-integrable functions $\psi$ such that $$\psi\rightarrow 0,\frac{\partial\psi^\star}{\partial x}\rightarrow\infty\qquad\textrm{as}\qquad x\rightarrow\pm\infty?$$
- What would be a necessary condition to impose on $\psi$ to ensure that the above quantity is zero?
- The question Must the derivative of the wave function at infinity be zero? suggests that having a function with compact support is sufficient. Is this necessary?