# In how many possible ways can a photon be emitted?

I am currently studying atomic physics, and I encountered the question above.

I am posting this question because I can't afford to move on with even the tiniest bit of uncertainty in my understanding of the subject.

The question in question assumes that there is an atom with four (4) energy levels: E1, E2, E3, and E4, where E1 is the ground state level and E4 is the highest energy level of this particular atom.

Then, there is one electron at E4, and it asks in how many different ways a photon can be emitted as the electron transitions from the fourth energy level to a lower energy level.

Now, the textbook says that there are six (6) different ways a photon can be emitted from this atom. However, I always end up with seven (7) different ways a photon can be emitted, namely:

E4 to E3 (one occurrence of photon emission)

E4 to E2 (one occurrence of photon emission)

E4 to E1 (one occurrence of photon emission)

E4 to E3 to E2 (two occurrences of photon emission)

E4 to E3 to E1 (two occurrences of photon emission)

E4 to E2 to E1 (two occurrences of photon emission)

E4 to E3 to E2 to E1 (three occurrences of photon emission)

Is the textbook incorrect, or is it just my naïve understanding getting in the way?

P.S.: The model used here is the Bohr's model of the atom, which is an inaccurate description of the atom.

• Are you sure it isn't asking for the possible single photon processes with the electron starting in any level i.e. $E_4 \rightarrow E_3/E_2/E_1$, $E_3 \rightarrow E_2/E_1$ and $E_2 \rightarrow E_1$? I make that six. – John Rennie Jan 11 '15 at 10:33
• Hello, John. Thank you for the answer. If the question were referring to possible single photon processes in any level, yes, there would be six possibilities as you have pointed out. However, I just checked the textbook again, and here is the question in verbatim: How many possible ways can this atom emit a photon if the electron starts in the fourth energy level (E4)? – Sam Choi Jan 11 '15 at 11:24
• Not all transitions are "allowed," e.g. books.google.com/… – Carl Witthoft Jan 11 '15 at 13:06
• @Carl This is a useful link from a rather unexpected source! I've read the link and understand now that certain variables and rules need to be taken into consideration to determine whether transitions of a photon are allowed or not. Now, the explanation provided in my textbook did not include the transition from E4 to E3 to E2 to E1 as part of the solution (and it certainly did not elaborate on the "transitionability" of photons). This may be an unfair question to throw at you, but can we then conclude that this transition (E4 to E3 to E2 to E1) is disallowed for the atom in question? – Sam Choi Jan 11 '15 at 13:37

The textbook is correct.

The number of emissions = total number of possible combinations of any two energy levels i.e. (nCr) n choose 2.

4 choose 2 in this case = 6.

1) E4->E1

2) E4->E2

3) E4->E3

4) E3->E1

5) E3->E2

6) E2->E1

Hope this helps.

• @Muhammah Ali Thank you for the input, but the question assumes that the transition of an electron starts from the fourth energy level (E4). So, I think E3->E1, E3->E2, and E2->E1 are not valid answers, though I appreciate your calculation for possible combinations. – Sam Choi Jan 11 '15 at 13:44
• @SamChoi Muhammad Ali is right. The question asks for the possible (i.e. different) ways an atom can emit a (i.e. one) photon when the electron starts from $E_4$. The point is that your seventh possibility lists three possibilities which you have already counted: $E_4\to E_3$ is your first , $E_3\to E_2$ is contained in your fourth and $E_2\to E_1$ in your sixth possibility. So your seventh possibility doesn't count as a new way that this atom can emit a photon. – user42076 Jan 11 '15 at 15:52
• @SamChoi So the emphasis in the question is "How many possible ways can this atom emit a photon if the electron starts in the fourth energy level (E4)?" In this case there are really just six. Would the question have been "How many possible ways can this atom emit photons if the electron starts in the fourth energy level (E4)?", then you would be right. – user42076 Jan 11 '15 at 16:01
• @SamChoi Please post the wording of the question as it says in the book so that we can see what the question says and deal with your confusion. Or regard its answer as wrong in other case. – Muhammad Ali Jan 12 '15 at 21:48