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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.

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Personally I don't think that "a quantum field can be thought of as a tiny bowl at every point in space with a ball doing SHM" is a particularly good or useful analogy. At a minimum it will get the wavelength variation of the density of states wrong. Certainly you should not try to extend that model to a more general situation. –  dmckee Feb 4 '12 at 17:00
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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.

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Apart from the fact that I don't think OPs questions has too much to do with quantum fields, but rather with what is actually quantized and what is oscillating in QM and what this says about the other observables, I would hesitate with absolutist statements like "quantum field theory is a theory about fields, not particles". Both aspects are certainly important. –  NikolajK Feb 4 '12 at 21:11
    
Yup, I was more of confused as to what was quantized and what was oscillating. But yes, I would need to understand quantum fields more. I'm quite sure that the field and particle aspects are interchangeable. –  Manishearth Feb 5 '12 at 3:11
    
Sorry, I must have not understood your question. I still hope you got something out of the answer. I disagree with you (plural) as to the relative importance of particles in QFT, I would not say they are unimportant (I'd be shooting myself in the foot), but I think they are relatively less important then the field quantity, which I consider fundamental and paramount. –  kηives Feb 5 '12 at 3:22
    
@kηives Alright, I'll rephrase it: What is a fermion field, and what is its relations with wavefunctions (the same psi is used)? What exactly is oscillating? What type of field is a quantum field (as in, given the coordinates, what does it spit out? Scalar? Vector? Tensor? And what is the significance of the value that it has spit out?). The point is, I've seen explanations involving QHO (for example, symmetry breaking in Higgs), which I sort of understand but not quite, as i'm confused about what the field is. –  Manishearth Feb 5 '12 at 11:36
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@Manishearth The fermion field was a response to the wave function. The wave function could be used in non-relativistic scattering theory, but could not be used in particle creation and annihilation scenarios. Dirac's fermion field, when decomposed into its Fourier components and promoted to operators on a Fock space, can change particle number (are is Lorentz invariant to-boot). When this is done, just like it is with the electromagnetic field, then the field is a "quantum field." Quantum fields can come in all kinds, Dirac's field is a bilinear spinor field. Hope this helps. –  kηives Feb 5 '12 at 17:00
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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.

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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.

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By this, do you mean that each bowl in the field has a wavefunction? I thought that the field is the wavefunction. (Atleast it is represented by the same variable as wavefunctions) –  Manishearth Feb 5 '12 at 3:03
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In quantum field theory, the field is not a wavefunction. Rather, the field has a wavefunction, i.e., for every possible state of the field there is a probability amplitude. (Note that this means that the wavefunction is infinite-dimensional, with one dimension for each point in the field.) –  Harry Johnston Feb 6 '12 at 18:56
    
Thanks, that helped too.. –  Manishearth Feb 7 '12 at 6:20
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