How to understand whether an electron in an atom is in superposition, ground or excited state?

Question

As I understand it, defining the orbital of an electron implies that it can be detected with a certain probability in a certain place. But if we take an atom and it is in a superposition state, then it turns out that all orbitals are multiplied by coefficients and added to form a linear combination of states? Where do we find the electron in this case? Or did I misunderstand something? Are there any non-superpositioned atoms at all? How to understand that an atom is in superpositioned state or not?

Answer 1

The probability density of the electron is always given by:

$$ P(\mathbf{r}, t) = \Psi(\mathbf{r}, t){}^* \Psi(\mathbf{r}, t) $$

This equation is the same whether the wavefunction of the electron $\Psi$ is an eigenstate like the $1\textrm{s}$, $2\textrm{s}$, etc orbitals or a superposition like $\Psi = a\psi_{1s} + b\psi_{2s} + ...$. The only difference is that for an eigenstate the probability density is time independent while for a superposition the probability density changes with time.

Answer 2

In classical physics the evolution of a system is described in terms of variables such as a position vector $\mathbf{r}(t)=(x(t),y(t),z(t))$. If you measure the $x$ position of the particle at time $t$ you get the value $x(t)$.

In quantum physics the evolution of a system is described in terms of mathematical operators called observables that can be represented by matrices. The possible results of measuring an observable are its eigenvalues. Quantum theory predicts the probability for each possible value of an observable. If the probability for one particular value of an observable $\hat{A}$ is 1 then that observable is said to be sharp. For some pairs of observables $\hat{A},\hat{B}$ if $\hat{A}$ is sharp then $\hat{B}$ isn't and vice versa. So in general whether a system is in a superposition depends on what observable you're talking about.

Orbitals are energy eigenstates: to a good approximation they have a sharp energy and their expectation values don't change over time. But in general those states have a significant probability for the electron to be found in a region of a size roughly of the order of $10^{-10}m$.

The state a system will be found in depends on how you interact with it to measure it. In everyday life systems interact with their environment on a timescale much shorter than the timescale over which they change significantly and this suppresses quantum effects in everyday life- this is called decoherence:

https://arxiv.org/abs/1911.06282

The evolution of the atom is dominated by the interaction between the proton and the electron because it interacts weakly with its environment. Such systems are usually found in energy eigenstates as a result of decoherence:

https://arxiv.org/abs/quant-ph/9811026

So electrons are usually found in orbitals.

Answer 3

As quantum mechanics teaches as: if we measure a quantity $A$, and have a system in a superposition of eigenstates of the corresponding operator $$ |\psi\rangle=\sum_ic_i[\phi_i\rangle,\\ A|\phi_i\rangle=a_i|\phi_i\rangle, $$ then our measurement will give value $a_i$ with a probability $|c_i|^2$.

What is implicit in the discussion of atoms is that in typical spectroscopic we measure the atoms energy (and possibly some other quantities, like angular momentum, and spin, which commute with the energy operator.) The rest is basic QM.

Answer 4

I'm a mathematician, not a physicist, so take this with a sizeable amount of salt, but the way I understand it, we can never measure a superposition. Interacting with a system in a superposition causes the superposition to decohere and we end up measuring only one of the states. We can put a particle or quantum system into a superposition, but this only means there's some non-zero chance we will measure one state and a non-zero chance will measure the/an other state. (It's only in the superposition when we're not measuring / interacting with it.) We convince ourselves that superpositions are a real thing (even though they exist only when we're not looking), by repeating the process of a putting a system into a superposition and then interacting with it and measuring one of its states. The statistical results we get are only sensible if we assume the system was in a superposition before the measurements.