As you increase the energy level of a hydrogen atom $n \rightarrow \infty$ I have learned that the energy of that energy level changes according to
$$ E_n =- \frac{13.6 {\rm eV}}{n^2},$$
and that there are technically an infinite set of these energy levels. As you get closer to $n=\infty$ the energy appears to decrease more and more rapidly, as it gets closer and closer to 0, at which point the the electron would no longer bound in quantized orbits of angular momentum. The angular momentum is given in the Bohr model as
$$ L = n\hbar,$$
and if you set this equal to an other quantity like $mrv$ or $r\times p$ you could get a radius for the orbit $r$. It is taught that this radius grows and technically also goes to infinity, is this always true? Is there any modifications that make the orbits converge at a certain distance as in something like a Poincare disk (see image)?
Best Answer
The Bohr model is far from exact, and the electrons don't strictly have 'radius of orbit' as they have a distribution function across space, but basically your intuition is not worng: as $n\to \infty$, the available angular momentum levels also grow, since $L<n$ (so for $n=15$, for example, the quantum angular momentum number can grow up to $14$). As $L$ grows, the probability to find the electron far from the center of the atom also grows, but quite slowly.
So yes, for an 'isolated' hydrogen atom, an electron at energy very close to zero from below will be spread throughout large volumes of the space surrounding the atom. This is not so surprising, maybe. If we employ again classical intuition, and think of the electron as an object that orbits the nucleus in a similar manner that planets orbit the sun, then even far away from the center the planets are bound by the potential.
In reality, though, this doesn't really happen. Electrons tend to try to stick to the lower levels, and will emit photons in order to decay and sit at the safety of the lowest levels. If we will try to excite the electron, it will more likely become completely unbound escape the atom altogether, ionizing it and leaving us with a free electron and a proton. The state at which the electron is at an eigenstate with very large $n$ is not realistic.