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?
Best Answer
No this is not enough. You were given counterexamples on the web site you mention in 4.
Yes, there are such functions, for example $(\sin x^3)/x$
It is hard to tell what is necessary, besides the trivial condition $\psi\psi'\to 0$.
Compact support is sufficient but not necessary.
You wrote the formula incorrectly: your RHS is $0$.
Physicists frequently do not state their conditions precisely, you should accept this when you read physics literature.