What does it mean to measure a photon? In a lab at the university where I'm studying, there's a detector of gamma rays. We bring a radioactive caesium source near the detector, and the detector clicks now and then. The closer the source is to the detector, the more clicks we hear from the detector every second. My lab instructor tells me that the caesium source emits invisible particles called photons. These particles have a relatively well-defined momentum and position, he says; sometimes, one of them moves, for a time roughly equal to the distance between the source and the detector divided by the speed of light; during this motion, this photon is positioned roughly along the line connecting the detector to the source. 
When the particle reaches the detector, it causes an electron in the detector to get excited, knocking it from its previous position; the electron then reaches an electrode, causing a small fluctuation in the current, that is enough to make the detector tick. This tick, I can say with confidence, definitely has a well-defined time and place - which makes me think that the original collision between the photon and the electron also had a definite time and place.
My Quantum Mechanics instructor, however, tells me a completely different story. In my (undergraduate) advanced quantum mechanics class, we wrote equations for the quantized electromagnetic field. We saw that the electromagnetic field has many different modes, parametrized by a parameter $\omega$. Each of these modes can vibrate, thus allowing an amount of energy to be stored in the field. The amount of energy stored in a specific mode, however, can only come in discrete quantities - $n\hbar\omega$, to be exact. And when a mode parametrized by $\omega$ has energy equal to exactly $n\hbar\omega$, we say in my quantum mechanics class that the field has $n$ photons of frequency $\omega$ in it.
The picture just described does not at all seem to me to resemble anything with a definite position or momentum. My question, of course, is how are these two pictures connected? How is it possible that I'm getting localized ticks in my detector? I did not have this same problem with the quantum mechanics of electrons, because there I had a wavefunction, and I could believe that when the wavefunction looks somewhat localized, this means the electron has a somewhat definite position - and the same goes for momentum for the wavefunction in momentum space. I can't seem to be able to apply this idea to photons, however - so what's going on?
 A: You have provided us with a near-perfect example of wave/particle duality, about which much has been written over the years. A quick on-line search on that term will turn up lots for you to study; a very brief introductory description follows and I invite the experts here to provide more detail.
Roughly speaking, wave/particle duality means that in the quantum world, a thing like a photon or an electron is like a two-sided coin, that admits of two different descriptions: in one set of circumstances, it looks like a particle and in another it looks like a wave. Here are the the classic examples:
First we have a beam of electrons striking a single crystal of nickel. At the right angle, that beam will get diffracted just as if consisting of waves, but if the electrons are emitted singly into a detector, the detector registers their presence with clicks one by one just like they were little bullets. 
And if a beam of light that is so weak that it consists of individually-countable photons is sent towards a pair of optical slits, the photons will form interference fringe patterns on a screen behind the slits just as if they were waves- even though as single photons we would expect each to go through either one or the other of the slits and hence have no other photons to interfere with while doing so. 
From the standpoint of convenient description, the higher the energy (i.e., the shorter the wavelength) of the photon, the more it begins behaving like a speeding particle and the less like a wave uniformly spreading out through space. And the lower its energy (the longer its wavelength) the more it behaves like a wave spreading out in space and the less like a single speeding particle. 
This means that when a nucleus undergoes a decay process that releases energy, that energy emerges as a gamma-ray photon in a certain direction and produces a single click when it hits a detector, and when an AM radio station goes on the air, the signal leaving its antenna is radiated into space as a three-dimensional wave with a measurable polarization and a strength that follows the inverse-square law. 
