We can't. Nobody knows how to measure a wavefunction. Instead, what you measure is one of the eigenvalues of a linear operator that acts on a wavefunction (which particular eigenvalue? You can't always predict this; hence the randomness in QM). For each "observable" you want to measure (position, momentum, spin), there is a specific linear operator associated with it. For instance, in position-space, the position operator is just the identity operator.
If you haven't taken linear algebra, I can clarify this further.
When a measurement is made, the wave function, which gives the probability of finding the system in any particular state, "collapses" so all the probability mass is on the state you just measured.
If you keep measuring the system quickly (relative to the time it would take to change significantly) you actually stop if from evolving the way it would have if you'd left it alone, as if you dropped a ball and kept staring at it and it dangled in the air instead of falling.
The reason it's difficult to know the precise situation of a photon, in-flight, is because it's been emitted from a moving target, whose precise state is difficult to control perfectly: a live atom.
Without having perfect control of the atom that sends off the spinning, oscillating particle, we have a hard time discerning the exact circumtances of the spin/oscillation imparted upon it. We can hazard guesses, but when we're right, it's only because we get lucky. Not because we controlled it.
EDIT: The reason is there are two factors driving the emission:
1. The external behavior of the electon shells
2. The internal nuclear behavior of the protons and neutrons (which do not behave simply as a clump of spherical marbles, the nucleus is it's own volatile, nebulous construct)
And, oh yeah... 3. Any other random cosmological events that might sail in from outer space (literally), as fortuitous signals
Really? Measuring a particle's state at a street corner in Columbus, Ohio instantly causes its partner particle in the Andromeda Galaxy to assume the opposite identity, isn't spooky?
http://en.wikipedia.org/wiki/Quantum_entanglement
Opening a box containing a cat, a vial of poison, and a radioactive sample instantly causes the cat to become either alive or dead, where before the cat was in a superposition of states, i.e. neither alive nor dead. Not spooky?
http://en.wikipedia.org/wiki/Schr%C3%B6dinger's_cat
> Maybe without a math degree I never will.
No need for a math degree. The weirdness of quantum theories is obvious without a knowledge of advanced mathematics.
"If you think you can talk about quantum theory without feeling dizzy, you haven't understood the first thing about it." -- Niels Bohr.
Things like: https://en.wikipedia.org/wiki/Quantum_nonlocality (aka: spooky interactions at a distance)
Quantum Non-locality seems (seems) to produce scenarios where two particles are stimulated using ostensibly unrelated mechanisms, but then also seem to exhibit cause-and-effect behavior where influencing one particle inexplicably influences the other particle, even though they may be (presumptively) incapable of influencing each other at the point in time and space when the correlating behavior is observed.
How did particle A affect particle B? Did you touch it?
What happened? No one touched it?!
This kind of interference pattern holds up, even at atomic scales, when diffracting x-rays through barriers of crystalline samples. Recently, there has been an experiment developed which may finally explain this behavior:
http://web.mit.edu/newsoffice/2010/quantum-mechanics-1020.ht...
http://www.youtube.com/watch?v=fnUBaBdl0Aw
https://en.wikipedia.org/wiki/Pilot_wave
...but this would suggest that there are unseen, unmeasuable forces (from parallel dimensions?) influencing our particles in this universe, at all times, and in unexpected ways. If this is true, what does it all mean? Is the sky falling?! Who knows?!
These sorts of unexplained behaviors, made physicists very nervous in the early half of the 20th century, since maybe attempts to build city destroying weapons might have horrible, unpredictable triggers or uncontrollable, world-destroying, runaway chain-reaction outcomes, never to be understood (not unlike the black hole fear mongering surrounding the LHC's publicity). Keep in mind that these ideas were kicking around before an atom bomb was anything other than science fiction. The fears and impatience often strike a chord like Dr. Emmett L. Brown's fear of a universe-destroying temporal paradox in Back to the Future.
Here's a bit from my own research. I start with a neutron whose spin is pointed up. I then use magnets to bend the neutron's trajectory depending on whether it's spinning to the left or the right. Since the spinning in the up state is a linear combination of spinning in the left or right states, it just so happens that I don't know whether the neutron bends to the left or the right.
For most of the neutrons, that's all I do. Another set of magnets pulls the beams back together. They get to the end of the instrument and I measure whether they're spinning up. Since I haven't done anything besides split the beam, they're all still spinning up.
However, for a small fraction of the neutrons, I measure which way they bent. I know know whether it is spinning left or spinning right. Let's say it's spinning left. Well, just as spinning up is a linear combination of spinning left and right, spinning left is a linear combination of spinning up and down. Since I know that the beam is spinning left, I no longer know whether it's spinning up or down. When I measure this neutron at the end of the instrument, it only has a fifty percent chance of pointing up.
This isn't a couple of photons hitting a particle and changing the velocity slightly. I've gone from all of my neutrons pointing up to only half of them pointing up.
You might immediately assume that the problem is that I did something to the neutron when I measured whether it was pointing left or right. The "spooky" thing about this is that I didn't do a single thing to that neutron to measure which way it was spinning. The way I perform the measurement is that, after I split the neutron beam, I put a wall in one of the beams. The other path, however, is just the same as it was before. I haven't touched any of these neutrons. Each of these neutrons goes through the exact same fields and paths as they did in the experiment where all the neutrons were pointed up, except no half of them are pointed down.
Quantum mechanics doesn't just say that you have to touch something to measure it and that will effect the outcome. It says that performing a measurement will effect the outcome, even if you don't touch it. That's where some people get spooked.
---- To preemptively answer a couple of questions: 1) No, you can't use magnets to bend neutron beams based on the Lorentz force. However, the neutron does have a magnetic moment which produces a potential with a magnetic field. This leads to an index of refraction change that causes a slight angular splitting of the neutron beam. Nothing ever gets separated by more than a micron.
2) No, I don't really put in a wall to measure just the left spinning neutrons. There's a solid material whose density fluctuations over the micro length scale serve as my effective wall. Thus, I don't actually get half the neutrons flipping at the end of the experiment, but rather a fraction proportional to how much information I get about the left right spin from my single wall.