The Extra Energy of a Doppler-Shifted Photon: Where Does it Go?

[Zak]

I understand mathematically etc how light can doppler shift when looked at as a wave but can it be understood how light doppler shifts as a photon i.e. could just one photon be doppler shifted?

For example;
Say there is an atom with energy levels 1 and 2. One observer, stationary relative to the atom, observes a photon of energy level 1 excite the atom (to level 1 of course).
Another observer is moving relative to the atom such that they observe the photon to have enough energy, due to the doppler shift, to excite the atom to level 2. Surely, then, they would observe the atom excite to level 2? However, this does not agree with the first observer.

I feel that there must be something fundamentally wrong with my understanding of this process and that before I seek the answer to this particular question there is more to be understood about the process of Doppler shifting etc.

 

[craigi]

For what it's worth, I'm going to make a complete guess that the energy levels are shifted by exactly the same amount as the doppler shift.

 

[The_Duck]

Say there is an atom with energy levels 1 and 2. One observer, stationary relative to the atom, observes a photon of energy level 1 excite the atom (to level 1 of course).
Another observer is moving relative to the atom such that they observe the photon to have enough energy, due to the doppler shift, to excite the atom to level 2. Surely, then, they would observe the atom excite to level 2? However, this does not agree with the first observer.

The observers will never disagree about which energy level the atom is excited to. So where does the extra energy go, according to the second observer? It goes into the recoil of the atom. When the atom absorbs the photon it also absorbs the photon's momentum and so recoils slightly: the atom's kinetic energy increases. The second observer sees a larger increase in kinetic energy, which exactly accounts for the excess energy of the photon in their frame.

Why does the second observer see a larger increase in kinetic energy? Say the atom is initially stationary in the first observer's frame, and after absorbing the photon it recoils at a speed of 1 m/s. Meanwhile in the second observers frame it starts out with a speed of 1000 m/s (say) and after absorbing the photon has a speed of 1001 m/s. Plugging these numbers into the formula $K.E. = mv^2/2$, you will see that the increase in kinetic energy is much larger in the second observer's frame.