Mechanistic approach to quantum mechanics It seems we attribute effects to particles, but why does it seem that we don't try to answer the question: how does it have this effect?
For example, in modelling the hydrogen atom with quantum mechanics, one simply posits that the Hamiltonian contains a Coulomb potential describing the force between the proton and the electron. Can we understand the origin of that force with QM? 
 A: I think you are confused as to how the quantum world really works:
The basic insight of quantum mechanics is that the world is non-deterministic, there are no local hidden variables (and if you believe that non-local hidden variables are a good idea, you better had a damn good reason). The Bell inequalities show us that no mechanistic theory can ever reproduce the predictions of quantum mechanics (which are experimentally found to be indeed correct).
Now, as pointed out in the comments, it is not the time evolution of the world that is non-deterministic. The Hamiltonian is still the generator of time translation, and we are still (in principle) able to fully predict the quantum state of a system at any given time if we are provided with its initial state. But even the full quantum state does not allow a deterministic prediction what a specific measurement will result in. Unless it is an eigenstate of the observable measured, the quantum state comes equipped with precise probabilities for each possible result of the measurement, which are emphatically not due to imperfections in the ways we measure. So, the world in the abstract sense as described by physical law is still deterministic, but what we observe is not. We can predict the probability with which a result occurs, but we cannot predict the result itself. The essence of Bell's theorem is that it is impossible to produce a local theory that predicts the results of measurements deterministically. 
While quantum mechanics and quantum field theories are only effective theories, i.e. there may well be an underlying theory that is different, we know that it will also only be probabilistic. The basic building blocks of nature are neither particles nor waves, as quantum mechanics tells us that the objects that constitute reality behave as if they are states in a Hilbert space, which possess some qualities of particles (they can have momenta and positions) and some of waves (they can "interfere"). But they are neither.
Your "mechanisms" can't exist because the way the world works on the most fundamental level that is as of yet known to us does not work intuitively. You cannot really understand QM by thinking about point-like particles or interfering waves, you have to accept that reality is stranger than we can imagine. Our minds are not able to really grasp what states in a Hilbert space are since nothing on a macroscopic scale is like that, and all of our intuition comes from the macroscopic world.
I unfortunately don't understand what you mean with a "geometric" approach you desire in your comments, but I suspect it ultimately also relies on the wishful thinking that observables be deterministic. They are not.
The origin of the electromagnetic force can not be understood with QM. It is only apparent if we take the full classical gauge theory of electromagnetism and quantize it (either by Gupta-Bleuler quantization or through the BRST formalism in the path integral approach), thus creating a quantum field theory, that the Coulamb potential indeed arises in the non-relativistic limit by considering the relevant Feynman diagrams. I am not certain that you wish to see the explicit calculation, but tell me if you want to.
A: Ok the answer already posted by CuriousMind, explains in a nutshel the (official) interpretation of QM plus some application aspects.
However since i think i understand the motivation of the question, i will try to give another answer.
First although Bell's theorem of no-local realism is indeed important, there are in fact other formulations (or more correctly interpretations) of QM, which arrive at the same central equation (i.e Schrodinger equation) albeit though different premises and interpretation, but nevertheless explain and predict exactly the same things conventional QM does (e.g Bohmian Mechanics and others). In order not to feel ugly if one does not like the official QM too much, remember that Einstein himself famously did not subscribe to that interpretation (of course even Einstein can be wrong).
Now concerning the nature of "forces", this is an issue that has multiple facets. For example gravity, in General Relativity, is just curvature of space, whereas by Newton's original formulation can be assigned to the (gravitational) mass of objects.
Electromagnetism similarly is assigned to (the motion of) charged particles (among others).
There are even forces (or apparent forces) which although real (and experienced) are not assigned to a particle or object but rather to a relative motion between systems of objects (will leave that for now).
Similarly in QM, many forces just carry from the classical description into a quantum one by the process known as quantization. This is based on a principle of QM that it should re-produce classical results if taken to this limit (known as principle of correspondence).
When no classical description is available to apply to a quantum system, a suitable generalization is sought (usually) that generalises a known classical description and when quantized gives correct results (an example is Yang-Mills theory which generalises electromagnetism in multiple dimensions). As a result then any (conserved generalised) charges are taken to be the source  of the associated generalised forces (which make the objects interact) that also are described (exactly like electromagnetism) by the theory.
UPDATE:
As one can see in the answer (and in how physics descriptions are made and used), there is an equivalence between these 3 concepts:
"CHARGE"  - "MOTION / INTERACTION" - "FORCE"
Now one can use one or two of these concepts and have the third as a derivative concept, for example one can use "charge" and "motion" of objects and derive a concept of "force" which unites these concepts in various compatibility conditions for a given system.
Alternatively one can use "force" and "motion" and derive a "charge" concept which similarly is used to define compatibility conditions for a given system that relate "force" and "interaction".
The fact remains that these concepts (or more correctly descriptions) are related in various ways and describe conditions on the evolution of given systems. One can use words like "force" or "charge" (which are generalizations or extarpolations of other cases), one can as well use words with no previous meaning (like "quarks" or "strangeness" etc..). It is not of the essense although in many cases it can imply meaningful generalisations
