Question

The hydrogen atom$1_s$wave function is a maximum at$r=0$. But the$1_s$ radial probability density, shown peaks at $r=a_B$ and is zero at$r=0$. Explain this paradox.

Short answer

The given statement is proved.

Step-by-step solution

Given Information

We need to prove peaks at $r=a_B$and is zero at$r=0$.

Explanation

The radial probability density $P_r\left(r\right)=4\pi r^2\left|R_{nl}\left(r\right)\right|$is the probability density for finding the electron at a distance from the nucleus. The radial wave function $R_{ls}\left(r\right)$is a maximum at the origin. the origin is the point for finding the electron. The probability density $\text{P}\left(\text{r}\right)$is smaller at any one point $r=a_B$than the origin. Since there are many points having $r=a_B$only a single point $r=0\text{m}$The increased number of points more than compensates for the decreased probability per point. As a result, finding the electron at a distance $r=a_B$So, it is at a distance $r=0\text{m}$.