quantum mechanics - 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?