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Let's say we solve the Schrodinger equation with infinite well. We can quantize the field by the resonance state and can get the annihilation and creation operator. So we will get the sort of particle corresponding to the resonance state. The resonance state would have mass corresponding to the energy and the mass will exist as the multiple of the minimum energy level.

Is the resonance state a new type of particle? If it's not, what does the creation operator create?

Let's consider we put a photon inside the infinite well. If the photon is not in the resonance state, the photon will be represented by the superposition of the resonance states. So when it goes inside the well, photon seems not a fundamental particle but a complex particle constructed from more basic states.

And if the resonance is the particle, what happens to the photon then? If we put a photon detection device inside the infinite well, can we detect the photon? Or is the photon's identity vanished completely and it's meanless to talk of the photon anymore?

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  • $\begingroup$ The problem is that the Schrödinger equation in QFT is only applicable to systems of states of fields. Photons are quantization of the electromagnetic field, there is nothing such as potential for the photons, because the equation describing the photons is not the Schrödinger equation, but instead, $\square A_\mu=J_\mu$ (where $A_\mu$ is the electromagnetic quadri-potential field and $J_\mu$ a source of this field). $\endgroup$
    – Jeanbaptiste Roux
    Commented Jul 15, 2020 at 8:56
  • $\begingroup$ @JeanbaptisteRoux Thanks! It's no wonder I was talking something nonsense. Could I replace the photons with the electrons to make it a bit more sense? $\endgroup$
    – hbadger19042
    Commented Jul 15, 2020 at 9:33
  • $\begingroup$ I am not the expert of the experts, but I don't think it will have a sense even with a Dirac bispinor field: there is no meaning to quantization when we are in presence of interactions (because of the potential). We can do a series expansion of the S matrix and then we can talk about virtual particles, but there is no "quantization of an interacting field" only interactions via quantization of fields, due to a small structure constant (compared to 1). $\endgroup$
    – Jeanbaptiste Roux
    Commented Jul 15, 2020 at 9:47

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There is not one quantum field theory. One can use an infinite potential with a harmonic oscillator and create a theory of creation and annihilation operators, but to identify the mathematical states with physical states, and in your question, particles, needs clear definitions.

With the quantum field theory used in elementary particles, there are no infinite well solutions. The quantum fields are the free particle solutions of the corresponding equations, no potentials. For electrons it is the Dirac equation, for the photons it is the quantized Maxwell. Creation operators work in these plane wave functions, as well as annihilation operators, and that is how the Feynman diagrams for the interactions are decided upon and then calculated.

Potential wells are another use of quantum field theory. but not for electrons and photons which are elementary particles. (BTW the physical solution of the Schrodinger equation energy levels for the hydrogen atom has the 1/r potential of the coulomb interaction.)

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