# 10.5: Problems

Problem $$\PageIndex{1}$$

The wave function describing the state of an electron in the 1s orbital of the hydrogen atom is:

$\psi_{1s}=Ae^{-r/a_0}, \nonumber$

where $$a_0$$ is Bohr’s radius (units of distance), and $$A$$ is a normalization constant.

1. Calculate $$A$$
2. calculate $$\left \langle r\right \rangle$$, the average value of the distance of the electron from the nucleus.
3. The radius of the hydrogen atom is taken as the most probable value of $$r$$ for the 1s orbital. Calculate the radius of the hydrogen atom.
4. What is the probability that the electron is found at a distance from the nucleus equal to $$a_0/2$$?
5. What is the probability that the electron is found at a distance from the nucleus less than $$a_0/2$$?
6. We know that the probability that the electron is found at a distance from the nucleus $$0 < r < \infty$$ is 1. Using this fact and the result of the previous question, calculate the probability that the electron is found at a distance from the nucleus greater than $$a_0/2$$.

Hint:$$\int x^2 e^{ax}dx=e^{ax}\frac{\left ( 2-2ax+a^2x^2 \right )}{a^3}$$

Note: Be sure you show all the steps!

Problem $$\PageIndex{2}$$

The wave function describing the state of an electron in the 2s orbital of the hydrogen atom is:

$\psi_{2s}=Ae^{-r/2a_0}\left(2-\frac{r}{a_0}\right) \nonumber$

where $$a_0$$ is Bohr’s radius (units of distance), and A is a normalization constant.

• Calculate $$A$$
• Calculate $$\left \langle r\right \rangle$$, the average value of the distance of the electron from the nucleus.

Problem $$\PageIndex{3}$$

Calculate the normalization constant of each of the following orbitals:

$\psi_{2p+1}=A_1 r e^{-r/2a_0}\sin \theta e^{i\phi} \nonumber$

$\psi_{2p-1}=A_2 r e^{-r/2a_0}\sin \theta e^{-i\phi} \nonumber$

1The integral in $$r$$ was solved using the formula sheet

2If you find this strange think about a situation where 20 18-year olds gather in a room with 4 60-year olds. The average age in the room is 25, but the most probable age is 18