Question

Let $P_{a b}(t)$ be the probability of finding the particle in the range $(a<x<b)$, at time $t$. (a) Show that $$\frac{d P_{a b}}{d t}=J(a, t)-J(b, t)$$ where $$J(x, t) \equiv \frac{i \hbar}{2 m}\left(\Psi \frac{\partial \Psi^{}}{\partial x}-\Psi^{} \frac{\partial \Psi}{\partial x}\right)$$ What are the units of $J(x, t) ?[J$ is called the probability current, because it tells you the rate at which probability is "flowing" past the point $x$. If $P_{a b}(t)$ is increasing, then more probability is flowing into the region at one end than flows out at the other.] (b) Find the probability current for the wave function in the previous problem. (This is not a very pithy example, I'm afraid; we'll encounter some more substantial ones in due course.)

          Let $P_{a b}(t)$ be the probability of finding the particle in the range $(a<x<b)$, at time $t$.
(a) Show that
$$\frac{d P_{a b}}{d t}=J(a, t)-J(b, t)$$
where
$$J(x, t) \equiv \frac{i \hbar}{2 m}\left(\Psi \frac{\partial \Psi^{*}}{\partial x}-\Psi^{*} \frac{\partial \Psi}{\partial x}\right)$$
What are the units of $J(x, t) ?[J$ is called the probability current, because it tells you the rate at which probability is "flowing" past the point $x$. If $P_{a b}(t)$ is increasing, then more probability is flowing into the region at one end than flows out at the other.] 
(b) Find the probability current for the wave function in the previous problem. (This is not a very pithy example, I'm afraid; we'll encounter some more substantial ones in due course.)

Added by Gloria H.

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