Transitions between states that are in a superposition of orbitals

Question

In a, say, Hydrogen atom, why don’t we ever consider wavefunctions that are linear combinations of atomic orbitals?I do not immediately see why an electron cannot be in such a state. More interestingly, why don’t we consider transitions between these states and what would be the selection rules for such transitions? Are such transitions observable experimentally?

Answer 1

These states contain an oscillating electron density because unlike the eigenstates they are time dependent. For a nice description of this see Emilio Pisanty's answer to Is there oscillating charge in a hydrogen atom?

Since these superpositions contain an oscillating electron density they will emit EM waves i.e. they will emit a photon and decay to one of the lower energy eigenstates. So while the electron can be in a superposition of eigenstates it cannot stay there for very long.

Answer 2

The word "transition" implicitly contains a reference to observations, but this reference is not explicit. When you observe a hydrogen atom, what you normally get to observe is the photons it emits. Typically you observe these with a detector that carries out an energy measurement (or a wavelength measurement, which amounts to the same thing). By measuring the energy of the photon, you force its wavefunction to collapse to a state of definite energy. But if the photon is in a state of definite energy, the difference between the initial and final states of the hydrogen atom must have that value, and typically that means that the hydrogen atom was in an energy eigenstate to start with and ended in another energy eigenstate.

If you observe such a photon and measure its energy, it is still absolutely possible that the initial state was in a superposition of states all having energy $E_1$ and the final state was some superposition of states all having energy $E_2$.

If you observe a photon but measure some other observable rather than energy, e.g., the photon's polarization, then there is no particular reason to think that the atom went from an energy eigenstate to another energy eigenstate.

Answer 3

In a, say, Hydrogen atom, why don’t we ever consider wavefunctions that are linear combinations of atomic orbitals?

We do. More specifically, I do. And since I am part of "we," we do.

For example, the real $p$ orbitals, $p_x$, $p_y$, and $p_z$ are linear combinations of the $|n=2,\ell=1, m\rangle$ atomic orbitals. (Whether the former or the latter are consider the atomic orbitals versus the linear combinations is somewhat a matter of taste.)

I do not immediately see why an electron cannot be in such a state.

The state of an electron can be described by whatever state we choose. We are free in theory to choose the state to be any normalizable wavefunction whatsoever.

More interestingly, why don’t we consider transitions between these states and what would be the selection rules for such transitions?

We can and do.

Are such transitions observable experimentally?

If you can manage to set up the physical system in one of your proposed "linear combinations of atomic orbitals," you can presumably also manage to observe the transitions out of that state.

However, as a warning, you should remember that atomic orbitals are a convenient fiction, since real electrons interact with each other.

Further, it may be quite difficult to set up the initial state to be anything other than the ground state.

Further, as for the final states, we are potentially not able to control those states and may have to consider all of them, in which can whatever complete basis we use will give the same answer.

Answer 4

I think you could benefit from considering how a few actual experiments work and how they are modelled. You'll find that people actually working with energy levels in atoms DO frequently consider superpositions. You may also be forgetting about the phenomonology of decays to lower states. This effect isn't captured when you solve the Scrhodinger equation for the hydrogen atom in intro quantum classes (you ignore the interaction with the quantized electromagnetic field), but it strongly influences how we discuss energy levels.

The simplest experiment that exhibits quantum energy levels is a spectroscopy experiment. You shine a laser with a variable frequency into a gas cell, and you see when the atoms start scattering photons from the laser. The atoms are nominally in a pure groundstate, and when they are interacting with a resonant laser field, the atoms are modelled as oscillating between the groundstate and the excited state, and when they are in the excited state they can also "decay", which sends a photon in a random direction, and this is what we call the "scattered light." We call this "cycling" between two states. So even in this situation the atoms are being modelled as being in a perpetually changing superposition.

Typically, electronically excited states last a very short time (unless they are "quasistable states" like the $2$S state of hydrogen), but states separated by hyperfine structure last essentially forever from an experimental perspective. So you will definitely hear an experimentalist say "we put our hydrogen atoms in the $F=1$, $m_F=0$ hyperfine state," but you won't really hear an experimentalist say "we put our hydrogen atoms in the 2P state," because that only lasts a nanosecond. Rather, the electronically excited states are only really discussed as explanations for why certain frequencies of light interact with the atoms.

Now let me explain Ramsey spectroscopy where the atoms are definitely in superpositions. For example, in the cesium microwave clock (which defines the second), atoms are put in a superposition of two hyperfine states with a precisely timed microwave pulse (which never decays), and then at some later time, they are hit with another pulse. Then the probability that they are now in the upper versus lower hyperfine state now depends on the phase that the superposition gained relative to the phase of the microwaves. So you see that Ramsey spectroscopy, a key tool in precision measurements, intimitely depends on having states in superpositions, but it only really makes sense to discuss superpositions between long-lived states. In the same way, optical atomic clocks make superpositions between electronically excited states, but only if the excited state is very long lived.