state of hydrogen

[logistic_guy]

For $\displaystyle n = 2, l = 0$ state of hydrogen, what is the probability of finding the electron within a spherical shell of inner radius $\displaystyle 4.00r_0$ and outer radius $\displaystyle 5.00r_0$?

 

[khansaheb]

For $\displaystyle n = 2, l = 0$ state of hydrogen, what is the probability of finding the electron within a spherical shell of inner radius $\displaystyle 4.00r_0$ and outer radius $\displaystyle 5.00r_0$?

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[logistic_guy]

For $\displaystyle n = 2, l = 0$ state of hydrogen,

When $\displaystyle l = 0$, it means $\displaystyle m = 0$, so the wave function becomes:

$\displaystyle \psi_{nlm} = \psi_{200} = \frac{1}{\sqrt{32\pi r^3_0}}\left(2 - \frac{r}{r_0}\right)e^{-r/2r_0}$

Then, the probability is:

$\displaystyle P = \int P_r(r) \ dr = \int 4\pi r^2|\psi_{200}|^2 \ dr = \int_{4r_0}^{5r_0} r^2\frac{1}{8r^3_0}\left(2 - \frac{r}{r_0}\right)^2e^{-r/r_0} \ dr$


$\displaystyle u = \frac{r}{r_0}$

$\displaystyle du = \frac{1}{r_0} \ dr$


$\displaystyle P = \frac{1}{8}\int_{4}^{5} u^2\left(2 - u\right)^2e^{-u} \ du$

I am lazy to solve this integral so let us see what Lady Alpha says!

$\displaystyle P = 0.173$

This means that the probability is $\displaystyle 17.3\%$ to find the electron.