Let’s say that we have a fictitious atom that has an electron with three orbitals. Lets say that the potential between the ground state and orbital 1 is some amount of energy e1 and similarly the potential between orbital 1 and 2 is e2 and between 2 and 3 is e3. Is it true that this atom can only absorb photons whose energy given by E = h-bar * lambda is equal to e1, e1 + e2 and e1 + e2 + e3? If an electron has a potential of e1, can it absorb a photon of energy e2 or e2 + e3? Is the time of electron’s return to the ground state random? Does any photon with energy greater than e1 + e2 + e3 ionize the atom? When the electron returns to the ground state, can it give off a photon of energy e3, a photon of energy e2 and a photon of energy e1?
Sweet merciful McGillicutty, that’s a lot of questions. Let’s see what we can do.
The answer to all of these questions is “yes”, with one caveat: any one of these transitions might be heavily disfavoured by whatever selection rules are in play in your fictitious atom. For example, it might be much more likely for an atom to go directly from excited state #3 to state #1 (emitting a photon w/ energy e2 + e3) than to emit two photons, one of energy e2 and one of energy e3. The selection rules for your system will generally have to do with the detailed properties of the wavefunctions of the ground state and the excited states; and they generally won’t say that a given transition is impossible, merely that it’s very unlikely compared to other ways for the atom to decay.
Yes and no. It’s predictable and unpredictable in the same way a nuclear decay is both predictable and unpredictable: we don’t know exactly when an individual excited atom will decay, but each excited state has a well-defined half-life, and so we can predict fairly accurately how long it’ll take half of a sample of excited atoms to decay to the ground state. The lifetime of the first excited state of hydrogen, for example, is about a nanosecond.
Only if the “potential well” for the fictitious atom is bounded from above. If not, then no amount of energy will allow the electron to escape. Real-world situations are not generally like this, of course, but it can be a useful approximation to make sometimes.
Theoretically, yes. However, the probability of it happening is ridiculously, utterly tiny. If you think of a state with a lifetime of one nanosecond, the chance is about one in 10[sup]10[sup]22[/sup][/sup] that it will remain in that state for one million years. Hell, if you converted every proton in the Universe into a hydrogen atom, and somehow excited them all at once, you’d be lucky if you found one still excited a microsecond later.
Yeah, what I said there was no miracle of clarity. Let’s try again.
When we find the energy levels of a quantum particle (such as an electron) moving in space, we need to know the potential energy of that particle as a function of position. For the hydrogen atom, we have a potential energy that’s bounded above: no matter how far you get from the proton, the electrostatic potential energy won’t go above a certain amount. In such cases, the quantum particle will become free if you give it enough energy. Note, however, that this amount of energy is not necessarily “any amount greater than the energy of the highest state”; all we know is that it’s finite, and must at least equal the energy of the highest bound state. How much extra energy is required depends on the fine details of the potential energy function.
However, there are some situations in which we might think of a potential as being unbounded. For example, consider a particle attached to a spring. The harder you pull on it, the more force the spring is going to pull back with. You can write down a potential energy for the spring, but it turns out that this potential energy gets larger and larger the farther you get from the centre. This means that no amount of energy you give it will allow you to liberate the particle from this spring, and (if the particle was also charged) no photon, no matter how large its energy would allow you to “ionize the atom”. (However, I believe that if this were the case, then you’d have an infinite number of bound states anyhow; so this really wouldn’t be the type of situation you’re implicitly thinking about.)