5.2.3.1 So­lu­tion hmolc-a

Ques­tion:

Sup­pose, given the wave func­tion $\psi_{\rm {l}}({\skew0\vec r}_1)\psi_{\rm {r}}({\skew0\vec r}_2)$, that you found an elec­tron near the left pro­ton. What elec­tron would it prob­a­bly be? Sup­pose you found an elec­tron at the point halfway in be­tween the pro­tons. What elec­tron would that likely be?

An­swer:

The to­tal prob­a­bil­ity of find­ing elec­tron 1 at a po­si­tion ${\skew0\vec r}$ is

$$\int\vert\Psi({\skew0\vec r}, {\skew0\vec r}_2)\vert^2{ \rm... ...3{\skew0\vec r}_2 = \vert\psi_{\rm {l}}({\skew0\vec r})\vert^2$$

since $\psi_{\rm {r}}$ is nor­mal­ized. Sim­i­larly, the prob­a­bil­ity of find­ing elec­tron 2 at po­si­tion ${\skew0\vec r}$ is $\vert\psi_{\rm {r}}({\skew0\vec r})\vert^2$.

If ${\skew0\vec r}$ is close to the left pro­ton, $\vert\psi_{\rm {l}}({\skew0\vec r})\vert^2$ is sig­nif­i­cant, but $\vert\psi_{\rm {r}}({\skew0\vec r})\vert^2$ is small, so you are much more likely to find elec­tron 1 there than elec­tron 2.

But at the point halfway in be­tween the pro­tons, $\vert\psi_{\rm {l}}({\skew0\vec r})\vert^2$ and $\vert\psi_{\rm {r}}({\skew0\vec r})\vert^2$ are equal by sym­me­try, and you are just as likely to find elec­tron 1 there as elec­tron 2.