In this Letter, we show how the relationships and can be found easily in the context of the decay of unstable particles into two photons.* The derivation uses the Doppler shift of light,1 the energy–frequency relationship for photons † and the relationship between energy and momentum for light ( ‡ with independent of reference frame). Here, we use to represent the (invariant) mass of the particle (sometimes denoted by in older works), for the speed of light, u for the speed of an object relative to an observer, and
Figure 1 shows a particle at rest in reference frame decaying into two photons that travel in opposite directions. We consider the special case that the problem is one-dimensional; that is, the momenta of these photons are along the
Reference frame (train car) travels with speed u with respect to reference frame S (train platform). The particle decay and the emitted photons are observed from both reference frames. The particle is stationary with respect to reference frame , and it decays into two photons, each with frequency ν0, traveling along the ±x′ directions. The particle is moving at speed u with respect to reference frame S, and it decays into two photons with frequencies ν1 and ν2 that travel along the ±x directions.
Reference frame (train car) travels with speed u with respect to reference frame S (train platform). The particle decay and the emitted photons are observed from both reference frames. The particle is stationary with respect to reference frame , and it decays into two photons, each with frequency ν0, traveling along the ±x′ directions. The particle is moving at speed u with respect to reference frame S, and it decays into two photons with frequencies ν1 and ν2 that travel along the ±x directions.
Since the particle was initially at rest in , then in that reference frame the two photons must have equal and opposite momenta and therefore equal frequencies ν0. Assuming that energy is conserved, the particle's energy before the decay was
This derivation is not completely novel; it resembles other derivations found in the literature. For example, Einstein used plane waves emitted in opposite directions to show that , but the use of plane waves rather than photons makes the calculation more difficult.2 Leo Sartori presented the momentum derivation, but without assuming that the mass energy was completely converted to photons, also making the derivation somewhat more complicated.3 The derivations by Lawrence Ruby and Fritz Rohrlich are very similar to this one, but the similarity is hard to recognize because those derivations do not use the relativistically invariant mass.4,5
Thus, we do not claim the originality of the concept;** however, we have tried to present it simply, so that it can easily be used in classrooms.
The neutral pion and the positronium atom are examples of such particles.
We find it useful to teach the photoelectric effect prior to teaching this lesson in order to introduce students to the concept of photons.
Most of our students have not yet studied electrodynamics, so we make this claim either with an appeal to their future studies of classical electrodynamic theory or by claiming it as an experimental fact.
Of course, it is challenging to measure the frequencies of the same pair of photons in different frames; one could instead imagine that the decay of many particles is observed.
We will note one original idea that was motivated by a comment from a reviewer: if the photons are emitted perpendicular to the relative reference frame motion, as observed in reference frame , then their frequencies in S are simply γν0. Thus, the energy analysis is trivial, but the momentum analysis is more challenging because their direction of propagation in the S frame must be calculated.
