# Probability current density

## Homework Statement

This isnt quite a question per se. But in my review im supposed to know how to derive a condition
$$\psi (\infty,t) = \psi^*(-\infty,t) = 0$$

In addition i'm supposed to show that

$$-\frac{d}{dt} \int_{V\infty} \rho(r,t) dV = \oint_{S\infty} J(r,t)\cdot dS' =0$$

where $$\rho(r,t) = |\Psi|^2$$
and J is the probability current.

2. The attempt at a solution
Im not sure how that condition can be used.
Can it used to how that the probability goes to zero as we approach infinity?

Integration of the probability current over an infinite space gives zero? Do i simply integrate the probability current by parts and use the condition given??

Also one question... is it possible to derive the momentum operator?? Can i do the same for the position operator?

Could i use $$<p> = m \frac{d}{dt} <r>$$
where $$<r> = \int r |\Psi|^2 d^3 r$$

When doing the time derivative of the integrand, am i allowed to assume that dr/dt = 0?? After all the operators are not supposed to vary with time... right?
Thanks for the help!

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