Double slit experiment and single particles. Is the wave function just a mathematical model? I really do want to apologize in advance, I know this question has been 'answered' before. I have this 'problem' of feeling negatively toward much of today's 'mystical' interpretations of physics, quantum physics in particular.
I am going through the Solid State Chemistry course provided by MIT's open-course-ware and I really like how the professor goes into some of the mathematical models. The current discussion is over Niels Bohr's quantization of the electron to support the planetary model of atoms. Very interesting.
So, this naturally brought me back to some of my pondering about quantum mechanics. I want to know: if one single photon, not a collection of them, is shot from a laser through the double slits, does it indeed physically go through both slits as a wave, only to "collapse" as a particle when it hits the screen? Will one single photon produce the interference pattern? Does it take many photons to build up the interference pattern?
See, I have this issue where I think that the wave function is just a mathematical model to represent the probability a particle will be in any given spot when measured. Basically, that it is not to say that a particle is actually in all places at once (as a wave), rather we can not know where it is because of the uncertainty of measurement at those scales, so we represent this as a mathematical model, on paper. Is this a correct interpretation, or do these mystical "quacks" actually have something right?
Update 1:
This answer here, https://physics.stackexchange.com/a/22934/46693 is what my question is about. There is the claim that "the photon simply doesn't have a position...because the photon's position is ill defined it occupies the whole experimental apparatus", and to me that seems like a logical fallacy, basically saying, "I don't know the location so it must not have one." Which interpretation is correct? I am saying that it does have a position, and that if we try to determine which slit the particle travels through, the measurement will have an impact on the outcome of the experiment, thus disrupting the interference pattern. While he is saying, that "it doesn't have a position, because I don't know what it is, so it must go through both slits."
I would be very grateful if this was clarified, thank you.
Update 2:
I understand that it is necessary for us to calculate things with the use of probability, and that assuming for the sake of equation that there are many worlds can be helpful. What I am asking is: Do professional physicists really use probabilistic models like these to "prove" that photons do not have location, or that there actually are "many worlds"? It seems to me that the fact that we use probability at all shows a lacking on our part as scientists, and not that nature is truly as strange as some hype it up to be. For instance, if movement caused time travel, people ought to be popping in and out of my experience, but that never happens.
 A: It's all theory. The measure is how well it predicts. If you're looking for a concrete epistemology of what a photon is, you will not find it.
The way I think about your photon is the many-worlds interpretation, where instead of each "world" having a probability, it has a probability amplitude, which is a complex number.
If a world had only a probability, and you didn't know which world you were in, but knew you were in some set of possible worlds, then you would get the probability of that set by just adding the individual probabilities.
However, since worlds don't have probabilities, but amplitudes (which are square roots of probabilities), then to find the amplitude of a set of possible worlds, you add their amplitudes, and since those are complex numbers, they can reinforce, or cancel.
So you don't know if the world you are in has the photon going through slit A, or slit B, and landing at location C.
But if you add the amplitudes of those two possibilities, you get a combined amplitude, when squared, that can be more, or less, than simply adding the probabilities.
So the question isn't, which slit did the photon come through, but what's the amplitude, and therefore the probability, of the set of possible worlds we are in.
It's just a mathematical model, but it's the one that nature seems to follow.
A: 
There is the claim that "the photon simply doesn't have a position...because the photon's position is ill defined it occupies the whole experimental apparatus" , and to me that seems like a logical fallacy, basically saying, "I don't know the location so it must not have one."

That's what many people have thought and many people still think, when they enter quantum mechanics. So, let's say that any particle really has a position and we only lack the knowledge for some reason. Maybe our instruments are not good enough? Or maybe, there is a fundamental reason why we can't build our instruments strong enough to measure the particle at a precise position, but nevertheless, this position exists?
People have asked these questions and the answer is: Probably not. If we assume that any particle has a well-defined position (and momentum) at any time, i.e. there is one (or maybe more) "hidden variables" that determine the exact position, but that we don't know, then we run into different problems. 
To make a long story short: Those models cannot exist, unless we allow for some different things, e.g. nonlocality. If in the double slit experiment we want a clear and precise trajectory of our particle, then we must allow that this trajectory is determined by the mere existence of a second slit, through which the particle does not pass, i.e. the trajectory through the open slit is different depending on whether there are other slits or not. That's essentially what Bohmian mechanics gives you (which is also alluded at in liquidspacetime's answer; however, even Bohmian mechanics cannot get rid of the fundamental probabilities in quantum mechanics, it just gives a different answer on how they are obtained, anyway, here's a link: http://www.bohmian-mechanics.net/whatisbm_pictures_doubleslit.html). If you don't like that (and most people don't), well then you are left to conclude that the particles cannot have a well-defined position. 
The problem you have here, is not physical, but rather philosophical. Your reluctancy to accept quantum mechanics is based on your intuition gained from living in a "classical world". Since this works extremely well in your daily life, you naturally extend this to anything you encounter, however, this just might be a false assumption. What seems to you like a "logical fallacy" might, to a person living in a macroscopic quantum world, seem as a completely illogical statement on your behalf. 
In this vein, also the answer to your question "Is the wave function just a mathematical object or does it present reality?" will be more philosophy than physics. Some people would say that yes, of course, it's just a mathematical object, since all of physics is just a mathematical way of trying to compute the universe and you aren't to take it literally, while others might say that yes, maybe, it is reality, insofar the theory is already the ultimate theory. 
However, it is certainly not used to prove that the wave function doesn't have a position. For that, you have to start at a much more fundamental level, which tries to encompass all theories that might, in some way, account for the observed experimental facts. The Wikipedia article http://en.wikipedia.org/wiki/Hidden_variable_theory provides some references. 
A: This double slit experiment gives a good intuition of what happens when the interference pattern disappears when closing one slit: The boundary values of the problem change and a different solution appears.
They progressively made one slit less and less transparent. This reduced the intensity of the interference pattern, which appeared while the electrons were shot one by one. At the limit of course, when th slit is closed there will be no interference pattern appearing.
There need be no mystical or many worlds  explanation of why.

“When the electron suffers inelastic scattering, it is localized; this means that its wavefunction collapses and after the measurement act, it propagates roughly as a spherical wave from the region of interaction, with no phase relation at all with other elastically or inelastically scattered electrons,”

Although I do not like the "collapse" terminology, it does reflect the fact that a different solution of the "electron two slit" problem is picked up if the electron goes through after being scattered in the filter material.
In a sense the two slit experiment is "scattering of electron by two slits". It has a probability to scatter from the edges of one slit or the other and depending where the quantum uncertainty in space   takes the individual electron an interference pattern reflecting the probability (given by the square of the wavefunction describing the setup) appears.
A: 
If one single photon, not a collection of them, is shot from a laser through the double slits, does it indeed physically go through both slits as a wave, only to "collapse" as a particle when it hits the screen? Will one single photon produce the interference pattern?

Yes, one single photon interferes with itself while passing through the double slit. And no, one single photon cant produce the interference pattern simply because of limited statistics. This is easily observed experimentally. Take a light source and dim it to such extent that one photon is emitted per second on an average. Place a double slit in front of it and a photographic plate after that. Over time one will see the interference pattern emerge. Since all the photons are temporally and spatially separated, they cannot interfere with each other. The only logical explanation remains that the photons interfered with itself. There is a whole class of experiments on this matter called: low-intensity double slit experiments. 
As a side note: it is best not to think of the photons as either wave or particle. Instead, assume that it has a dual nature. Also, the photons do not need to collapse as a particle on the screen. The particle picture of the photon is not necessary to explain its detection. There are a number of semiclassical models of photodetection (See: Optical Coherence and Quantum Optics, Leonard Mandel).

I have this issue where I think that the wave function is just a mathematical model to represent the probability a particle will be in any given spot when measured. Basically, that it is not to say that a particle is actually in all places at once (as a wave), rather we can not know where it is because of the uncertainty of measurement at those scales, so we represent this as a mathematical model, on paper. Is this a correct interpretation, or do these mystical "quacks" actually have something right?

The above interpretation is quite correct according to me. 

This answer here, https://physics.stackexchange.com/a/22934/46693 is what my question is about. There is the claim that "the photon simply doesn't have a position...because the photon's position is ill defined it occupies the whole experimental apparatus", and to me that seems like a logical fallacy, basically saying, "I don't know the location so it must not have one." Which interpretation is correct? I am saying that it does have a position, and that if we try to determine which slit the particle travels through, the measurement will have an impact on the outcome of the experiment, thus disrupting the interference pattern. While he is saying, that "it doesn't have a position, because I don't know what it is, so it must go through both slits."

Chapter 1 of Introduction to Quantum Mechanics, David Griffiths is an excellent place to resolve this issue. There were 3 different viewpoints regarding the values of different parameters before measurement:


*

*Realists: They, like the OP claimed that the particle had a definite parameter (let's say position) before it was measured.

*Orthodox: The particles definitely did not have a definite position before it was measured. The act of measurement forced the particle to take a stand at a particular position.

*Agnostic: The question is moot since we don't really have any information prior to measurement.


The orthodox view is the currently most widely accepted view. The realists assume that QM is incomplete because $\Psi$  does not contain all information (read: information prior to measurement). There must be some hidden variable that determines the position (or the value of a particular parameter) before measurement. John Bell, in his 1964 paper, On EPR Paradox derived an upper limit for the amount of correlations in the measurement performed on two entangled particles subjected to hidden variables. Real world experiments always violate this upper limit thus proving that realism does not occur in nature. 
Thus since the particle is nowhere and everywhere, it is prudent to assume that the particle passed through both the slits.

Do professional physicists really use probabilistic models like these to "prove" that photons do not have a location, or that there actually are "many worlds"?

If one comes to term with Heisenberg's Uncertainty Principle, it is quite easy to prove that Photons are not localized. Since its velocity is known ($c$) it's position is completely uncertain. There are other ways too to show that Photons are not localized.
A brief study on the subject will convince one that local realism is indeed violated.
The many-worlds interpretation is rooted in the path integral formalism of quantum mechanics. It is very difficult to check the consistency of the many world theory given the fact that we are limited to "one" universe for the moment.
TL;DR: Yes photon indeed does not have a position. Systems do not have a particular position (or value for any parameter) before measurement.
As a side note, one sometimes limit the photon to within a particular volume. This gives rise to discrete photon modes. Once the volume is increased to infinity, the photon mode spectra become continuous. This is also experimentally proven as only a finite number of modes survive in a cavity.
A: In a double slit experiment the particle always travels through a single slit. It is the associated wave in the dark matter which passes through both.
'1st place: Shifting the morals of quantum measurement'
http://physicsworld.com/cws/article/news/2011/dec/16/physics-world-reveals-its-top-10-breakthroughs-for-2011
"Using an emerging technique called "weak measurement", the team is the first to track the average paths of single photons passing through a Young's double-slit experiment – something that Steinberg says physicists had been "brainwashed" into thinking is impossible."
'Quantum mechanics rule 'bent' in classic experiment'
http://www.bbc.co.uk/news/science-environment-13626587
'For his part, Professor Steinberg believes that the result reduces a limitation not on quantum physics but on physicists themselves. "I feel like we're starting to pull back a veil on what nature really is," he said. "The trouble with quantum mechanics is that while we've learned to calculate the outcomes of all sorts of experiments, we've lost much of our ability to describe what is really happening in any natural language. I think that this has really hampered our ability to make progress, to come up with new ideas and see intuitively how new systems ought to behave."'
'New 'Double Slit' Experiment Skirts Uncertainty Principle'
scientificamerican.com/article.cfm?id=new-double-slit-experiment-skirts-uncertainty-principle
"Intriguingly, the trajectories closely match those predicted by an unconventional interpretation of quantum mechanics known as pilot-wave theory, in which each particle has a well-defined trajectory that takes it through one slit while the associated wave passes through both slits."
A particle physically displaces the dark matter. A moving particle has an associated displacement wave in the dark matter. In a double slit experiment the particle travels through a single slit. It is the associated  wave which passes through both. As the wave exits the slits it creates wave interference. As the particle exits a single slit the direction it travels is altered by the wave interference. This is the wave guiding the particle. Strongly detecting the particle causes a loss of cohesion between the particle and its associated wave.
What waves in a double slit experiment is the dark matter.
