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[tex] \lim_{x \to +\infty} \left(x \Psi^* \frac{\partial \Psi}{\partial x} \right) = 0 [/tex]

This is what i've done so far:

Since [itex]\Psi[/itex] must go to zero faster than [itex]x^{-1/2}[/itex] as [itex]x \to +\infty[/itex] we have

[tex]

\begin{align*}

\frac{\partial \Psi}{\partial x} &< \frac{d}{dx} (x^{-1/2}) \\

&= -\frac{1}{2} x^{-3/2}

\end{align*}

[/tex]

So

[tex]

\lim_{x \to +\infty} \left(x \Psi^* \frac{\partial \Psi}{\partial x} \right) < \lim_{x \to +\infty} \left(-\frac{1}{2} \Psi^* x^{-1/2} \right) = 0

[/tex]

Since [itex]\Psi^* \to 0[/itex] as [itex]x \to \infty[/itex].

I think this proves that the limit must be smaller than zero, but here I'm stuck.

Help is much appreciated,

A_B