Probability to fall back to ground state of hydrogen atom I would like to show mathematically that electrons always fall back to ground state when in the excited state. I think that the best way would be to find the probability. Once the excited state is mentioned we know the wave function of it from the Schrödinger equation solution. But I am not sure how to find it. Could someone guide me?
 A: The process that you want to describe is more generally called as spontaneous emission (if it occurs by emitting a photon, although one can imagine other relaxation processes, if the atom is not isolated). To describe it one has to consider the system atom+electromagnetic field, e.g., by writing the Schrödinger equation for the joint wave function of this system. Importantly, the field has to be described as a quantum field, that is this seemingly simply problem goes beyond typical quantum mechanics course (although some QM books do introduce field quantization in later chapters).
Assuming that you have the corresponding background, the joint wave function can be written as:
$$
|\psi(t)\rangle = c_e(t)|e\rangle\otimes |0\rangle + \sum_k c_k(t)|g\rangle\otimes |1_k\rangle,
$$
where $|g\rangle, |e\rangle$ are the ground and the excited states of the atom, $|0\rangle$ is the electromagnetic vacuum, $|1_k\rangle$ is the vacuum with a photon in mode $k$. (Note that I assume zero temperature here. Description for finite temperature is more involved and requires description in terms of the density matrix or more evolved formalism.) One can now write the Scrödinger equation, which has to include the appropriate electron-field interaction term, likely in the dipole approximation, solve it for $c_e(t)$ and take the thermodynamic limit, i.e. take the number of photon modes to infinity.
A: What you are describing is spontaneous emission, which can only be described in the context of quantum field theory, specifically quantum electrodynamics (QED), because the electromagnetic field has to be treated quantum mechanically, not classically. This is because both the electronic energy levels and the EM field both have to be quantized. Due to this fact, spontaneous emissions only occur in the presence of an external, quantized electromagnetic field. In the absence of an EM field, an excited atom will not decay to ground state. However, it is important to note that spontaneous transitions can also occur in free space far away from an EM field because the atom will still be subject to the vacuum fluctuations of the EM field. If you wanted to calculate the probability of whether or not spontaneous emission has occurred at a certain time $t$, you could do so through the utilization of something called Einstein Coefficients. Lastly, it is important to note that this is a condensed explanation of what you are talking about, and I would strongly urge you to consider doing more research and learning on your own before moving forward.
Here's the link on Einstein coefficients
