[Physics] Path taken by electron returning to ground state

electronsenergyquantum mechanics

Sometimes electrons are excited and return to their ground state in one step, while sometimes they take multiple jumps. What determines the path they take?

Best Answer

'Path' is perhaps a misleading word to use here. Path typically means a physical path through space - a trajectory. You can use path in the context of a path through a sequence of energy states, talking about which energy states an electron is in in which order, but you have to establish the context to use the word like that before hand.

Back to the actual question: what determines how many photons are emitted? It is random.

Consider a simple system where an electron can have three energy states, the lowest being $E_{0}$, then $E_{1}$ and $E_{2}$. An electron in state $E_{2}$ can either transition directly to $E_{0}$ emitting a photon where $h\nu=E_{2}-E_{0}$, or it can transition to $E_{1}$ and then $E_{0}$, emitting two photons (of energy $E_{2}-E_{1}$ and $E_{1}-E_{0}$).

It is possible to calculate transition probabilities per unit time for each possible transition. So in any given time period there will be a certain probabilty $p_{20}$ that the electron drops to state $E_{0}$ and a probability $p_{21}$ that the electron drops to state $E_{1}$. And of course, a probability that it remains in its current state.

Once those probabilities have been calculated, you can figure out how often an electron will return to the ground state via emitting a single photon, vs two photons (in our simple system). But there is nothing deterministic to say which path will be taken (see, now we can use path without anyone arguing about what it means... :) ). It is entirely random. All we can do is figure out the probabilties involved.

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