Quantum harmonic oscillator I read somewhere that a quantum field can be thought of as a tiny bowl at every point in space with a ball doing SHM (quantum harmonic oscillator). It was given that the amplitude of this SHM is quantized, and each quantum signifies a particle. (i.e. if the ball rolls with minimum amplitude, there are no particles in that point of space. If it has the next amplitude, then there is one particle and so on).
What I don't get is how this analogy relates to quantum fields which are not exactly quantized at every point of space. For example, a single electron has a wavefunction spread out over some space. At every point in this space, we can say that "there is a fraction of the electron over here". But, If I model this as a bunch of oscillators, I can't have a fraction of an electron as the amplitude of SHM, as its supposed to be quantized.
I'm quite sure there's a flaw in my interpretation, but I can't figure it out. Could someone give a more detailed explanation of quantum harmonic oscillators?
Note that I do not understand the mathematics behind quantum mechanics, so though I don't need layman's terms, I would rather stay away from the equations.
 A: The hypothetical balls are part of a single quantum system, i.e., there can be (and indeed are) quantum mechanical correlations between them.  
If the system is in a state representing a single particle, then it is known that only one ball is excited, but it is uncertain which ball it is.
For each ball, there is a probability amplitude that it is the one that is excited.  If you write a function for the probability amplitude that the ball at a particular position is excited, that gives you the quantum wavefunction of the particle.
A: I think the analogy with the bowls is not really appropriate. If one thinks of things oscillating at each point in space, these oscillations are heavily correlated, due to the field equations. 
Independent harmonic oscillators are not associated with points in space but with directions in space, and what oscillates are the Fourier modes of the quantum field in each such direction (momentum vector p). A free particle with momentum p is associated with such a wave vector. Multiple excitations correspond to multiple particles. 
If one disregards the small-scale structure, only the mean behavior of the quantum fields is visible, and this just gives classical fields. In QED we get the electromagnetic field and a matter/charge field for the electrons.
The other microscopic fields from the standard model leave as macroscopic traces the various chemicla compounds and their concentration fields.
Continuously generated bundles of localized particles are seen in this coarse picture as beams of light or electrons. As one increases the resolution, quantum effects become noticeable, and with it the statistical nature of quantum fields and quantum particles.
A: It's important to remember that quantum field theory is a theory about fields, not particles.  I know you said shy away from equations, so I'm just going to reference one part of one, and you can see this equation on any o'l web site, like wikipedia.  Take the Dirac equation, here there is a quantity $\psi$ that shows up.  And part of the history of this $\psi$ was what it meant.  Ultimately, it was determined to be a field: the Fermion field.  This is our fundamental understanding as of now about the world, that there are fields, and that interactions take place between fields, mediated by quantum excitations of these fields.
In light of this, The wave function you talked about corresponding to the electron is not the fermionic field I mentioned above.  The fermion field can be excited either to produce or destroy certain fermions like electrons and positrons.
As far as how deep the oscillator analogy runs, I'll just say this: How deep or how far it runs is debatable, but I don't think anyone will argue its fundamental role in developing QFT.  Quantizing fields and placing field variables in terms of canonical field variables is pivotal for an understanding of QFT, and before even getting to QFT, a good understanding of the SHO in quantum mechanics is indispensable.  This is because the creation and annihilation of excitations in QFT is analogous to the creation and annihilation of energy states in the non-relativistic quantum SHO. 
I hope this helps.
