1. Quantum probabilities and probability density
2. 3D nature of hydrogen atom wave functions
3. Orthonormality

Estimated Time: 45 minutes

Students work in groups to determine the probability that an electron in the $1s$ state of hydrogen would be found within one Bohr radius of the center.

• Eigenstates for the hydrogen atom
• Calculating probabilities in wavefunction notation

This works really well when prefaced with other activities dealing with probabilities in wave function notation (Energy and Angular Momentum for a Particle on a Ring) and Quantum Calculations on the Hydrogen Atom.

• Tabletop Whiteboard with markers
• A handout for each student

Ask students to work in groups to determine the probability that an electron in the $1s$ state of hydrogen, $$\psi_{100}(r,\theta,\phi)={1\over \sqrt{a_o^3\pi}}e^{-r/a_0}$$ would be found within one Bohr radius $\left(P_{r<a_0}\right)$ of the center.

1. Probability v. Probability Density:
   • Students struggle with the two different ways of finding probability: for discrete and continuous measurements.
   • Most recognize that they need to do an integral, but are not sure whether you square the norm before or after you do the integral.
   • Most groups will eventually figure this out if you let them flounder a little, but it is a good idea to reiterate in the wrap-up.
2. 1D v. 3D:
   • Since the $\vert 100\rangle$ state has no explicit dependence on $\theta$ or $\phi$, many students will simply use $dr$ instead of $dV$, which means that they are missing a factor of $r^2$ in the $dr$ integral.
   • Once they realize it is three dimensional, most know that $dV=r^2 \sin\theta d\theta d\phi$ and what the appropriate limits are for the angular integrals.
   • Many groups will do the $d\theta$ and $d\phi$ integrals explicitly and it is a good idea to let them do so, but it is nice in the wrap-up to address the fact that they do not have to do so because of the orthonormality of the $Y_l^m(\theta,\phi)$'s.
     • Also, for groups that finish early, you can ask them to show this explicitly.
3. Probability v. Expectation value
   • Several groups will try to start with $\langle\psi \vert r\vert \psi\rangle$ - one good way to help them to think about the difference between this and the probability is to ask them about the dimensions. Most students know that probability should be dimensionless and $\langle\psi \vert r\vert \psi\rangle$ will give them dimension of length.

1. It is a good idea to reiterate the two ways of finding a probability and how they are connected.
   • For discrete measurements:$$P_{a_n}=\vert\langle a_n \vert \psi\rangle \vert^2\doteq\left|\int_{-\infty}^{+\infty}\Phi^*(x)\psi(x)dx\right|$$
   • For continuous measurements:$$P_{a<x<b}=\sum_{x=a}^b\vert\langle x \vert \psi\rangle \vert^2 \doteq\int_a^b\vert\psi(x) \vert^2 dx$$
2. Most students, once they realize that they need to do 3D will do the $d\theta$ and $d\phi$ integrals explicitly. The wrap-up is a good time to reiterate the orthonormality of the $Y_l^m(\theta,\phi)$'s, which allow you to separate out the $d\theta$ and $d\phi$ integrals and only deal with the $dr$ integral.

$$\int \left|\psi_{nlm}(r,\theta,\phi)\right|^2 dV=\int \left|R_{nl}(r)\right|^2 r^2 dr \int \left|Y_l^m(\theta,\phi)\right|^2 d\Omega=\int \left|R_{nl}(r)\right|^2 r^2 dr $$