Exploring Quantum Physics - Week 5 Question 2

Question:

What is the maximum value of the wavelength of light that will ionize the hydrogen atom?
Take the energy of the ground state of the hydrogen atom to be $-0.99946650834 R_\infty h c \,$, where $R_\infty$ is the Rydberg constant and $c$ is the speed of light.

Solution:

Recall the potential energy of an item in a gravitational field is define to have 0 potential energy if and only if it is at infinitely far. I guess the same idea apply here when we say the energy of the ground state of the hydrogen atom is a negative number.

So we need a photon with energy just enough of make the overall energy 0.

A photon has energy $E = hf = \frac{hc}{\lambda}$. Therefore we have $\lambda = \frac{hc}{0.99R_\infty hc} = 9.11 \times 10^-{8}$, or 91 nanometers.