If an atom has 'infinite energy levels', how does the photoelectric effect exist?

Question

I don't quite get claims that an atom has 'infinite energy levels'. The photo-electric effect tells us that if photons of a certain threshold of energy are absorbed, the electrons will be freed from the pull of the nucleus, but if there are infinite energy levels, how does this work?

Is it something to do with the fact that energy levels become exponentially 'closer' together as n increases, meaning if enough energy is given, the electron simply jumps beyond these infinite layers? Perhaps like how there is infinite space between say 0 and 1, yet you can add 2 to 0 and completely skip this.

Answer 1

I think the photoelectric effect is a distraction here. You are basically asking: if an atom has infinitely many bound states, how can you ionize an atom?

The point is that there is a maximum energy you can give the atom and have it remain in a bound state. With a hydrogen atom, for example, all of the bound states are less than 13.6 eV above the ground state. If you add 13.6 eV or more of energy to a hydrogen atom, you can excite it to a state in which the electron is no longer bound to the nucleus.

As you say, it is similar to how there are infinitely many numbers between 0 and 1, but you can still add a finite number to zero and get a number greater than or equal to 1.

Answer 2

Without anything else in the universe, a lone hydrogen atom has energy levels which scale as $n^{-2}$ and hence approaches a free state in the limit of large $n$. In reality though there are other things in the universe and so the energy levels are typically truncated after some value of $n$ since the average radial position of the electron scales as $n ^2$.