Yesterday I read that we can affect the path and the 'form' (particle or wave) of a photon after the fact (Wheeler's delayed choice experiment). Part of what is puzzling me is the beam-splitter. Are the individual photons actually being split into two new photons of lesser energy?
This question implies that you cannot split a photon but it seems that beam splitters do exactly that.
The crucial word is "beam", in "beam splitter". Beam means an ensemble, in contrast to "photon" which is an individual particle.
A light beam is an ensemble of photons and if it is of a single frequency $\nu$, all photons have energy $E= h*\nu$. A light beam can be split in a beam spliter, i.e. the ensemble of photons can be split into two streams of photons: the intensity of the beam goes down, but the individual photons still have frequency $h*\nu$.
Now one can think of impinging photons one by one on a beam splitter. A photon is described by a wavefunction which when squared will give the probability of finding the photon in a particular (x,y,z). It will go either where one stream went or the other according to the probabilities, but it will be seen as a whole photon of energy $E=h*\nu$.
A single photon is a quantised packet of Electromagnetic energy, the smallest indivisible unit imposed by boundary conditions according to quantum mechanics. In this regime I find it easier to think of the photon as a particle with a 'polarization' degree of freedom which can be horizontal $\left|H\right>$,vertical $\left|V\right>$ or in any linear superposition of the two.
When the photon meets a beam splitter it acts like a quantum particle and takes both paths with some probabilities. Much like the electron taking both paths in Young's famous double slit experiment.
It is not the actual photon being split into two new ones, only the 'particle-like' position wavefunction holds the information about where the photon is. Quantum mechanically, we can treat photons exactly like any other bosonic particle.
In short, no, the individual photons are not actually being split into two new photons of lesser energy. It is true that in the classical interpretation of light, an ensemble of many distinct waves of photons, will create destructive and constructive interference patterns ( e.g. Young's Double Slit Experiment ). It is even more intriguing that these constructive and destructive interference effects are present even in the case of a single photon.
As an aside, when we mention photons, we mean the abstract idea that includes the wave/particle features inherent to an unobserved/detected electromagnetic wave. This is to say that until we make a measurement ( think delayed-choice ), these photons can and will behave as both.
To elaborate a bit on your question, we could ask, "Well, what about this individual photon is being split?"
Here is what I hope is an answer that intrigues you to keep asking more questions about this subject: In our attempt to control, predict, and describe our universe, we run into a problem when attempting to explain really small and/or fast things. This was first theorized with deBroglie's wavelength and Heisenberg's Uncertainty Principle accurately describes why we can not simultaneously know position and momentum with infinite precision.
So, what does this mean? It means that when physicists use quantum mechanics to predict things, they inevitably and unavoidably must use some notion of probabilities. Which by now, I'm sure you've started to guess at what will become split after a photon meets a beam-splitter. If not, that's ok, it's quantum mechanics...
In summary, a single photon incident ( fancy word for arriving at ) at a 50/50 beam-splitter ( common in quantum mechanics, and meaning that there is a 50% chance that the photon will transmit directly through, and a 50% chance that it was be reflected, and for the guys checking me: the reflected wave's phase will be shifted by 180 degree ) will emerge from the beam-splitter with a 1/2 probability of being reflected and 1/2 probability of being transmitted. So, it is the probability of the photon being in some output port that is split.
Very short and "axiomatic" answer: You indeed can "split" one particle. In QM particles are treated as a "wave functions", maybe it will be more easy for you to imagine a splitting wave. However, only at the point when the photon is detected the particle is measured in one point of space. This is the very foundation of QM and I agree that it's hard to grasp the concept.
Photon is quantum. Experiences with photons are quantum experiences. In quantum experiences, you are calculating transitions probabilities .
To calculate transitions probabilities $p$, you will have to use quantum complex amplitudes probabilities $A$, with $p$ = $|A|$
To calculate transition amplitudes $A$, you will have to consider all the paths that are available to the photon field, and sum other them :
$A$ = $\sum_{paths} A_{path} = \sum_{paths} e^{- i S_{path}} $,
where $S_{path}$ is the action of the photon on the considered path.
So, you will have to get rid of the classical view of the particle, localized in space-time, indivisible, and so on.
You are now in the quantum world, which has its own rules.
Under most circumstances a bean splitter will split light. Calling light a photon makes it seem like a particle and it is but is also is a wave. A particle cannot be divided by a beam splitter but a wave can and all QM particles are also waves. Light will act like a wave if you test it to see if it is a wave and it will act like a particle if you test it to see if it is a particle. If you test light to see if it is a a particle it will not be split by beam splitter. Even after you test it as a particle if light has a chance to reset it will go because to being both a wave and a particle.
