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So I'm reading about bound systems right now in my quantum text. It is beginning to explain why energy must be quantized, and is doing so by introducing the reader to the one dimensional "quanton in a box". Essentially it shows this:

For a region of 0 potential energy, with infinite potential energy everywhere else, a particle is "free" within the region of 0 potential energy. This means that the particles wave function must go to zero at these "boudaries", and therefore resemble a standing wave between the potential energy bounds.

Because the wavefunction must be zero at the ends, the Dr Broglie wavelength must be so that an integer number of half-wavelengths fits inbetween the regions of infinite potential. Therefore, the energy of a particle (quanton) must be so to satisfy this, and cannot be any arbitrary value.

First of all, the more board question: why, then would an electron that is outside of a bound system be quantized? Or any other particle that is outside of a bound system (atomic nucleus, etc.)?

Secondly, the question that I am more interested in: the only reason that a classical standing wave in a string results in the way it does is be cause f resonance; the string resists frequencies that do not match a natural harmonic of the material, and accepts very well frequencies which do match. So then isn't the fact of energy quantization a direct consequence of the phenomenon of resonance? It must be. But then, what is the material that is experiencing this resonance? For an electron, which could be described as a "matter-wave", would the Higgs field, itself, be resonating? Or something else?

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  • $\begingroup$ Related: physics.stackexchange.com/q/39208/2451 , physics.stackexchange.com/q/65636/2451 and links therein. $\endgroup$ – Qmechanic Nov 4 '15 at 19:04
  • $\begingroup$ Analogy between quantum behavior and classical standing waves is a weak one. Although Schrodinger equation is similar to wave equation it is not its solution that features in measurements, but its square modulus, and the nature of measurement is completely different. Bottom line: there is no material, and there is no resonance except as a mathematical metaphor. But why there is a universal quantum of energy for all systems is a good question. $\endgroup$ – Conifold Nov 4 '15 at 20:48
  • $\begingroup$ @Conifold To say that there is no material is simply wrong. You must know this, yet you are afraid of the question of what that material might be. Something is waving when we speak of matter waves, light waves, anything. $\endgroup$ – user97626 Nov 4 '15 at 23:15
  • $\begingroup$ Not afraid. For light "something" used to be called ether, but the theory implied more and more properties of it (transparent but rigid, absolutely undetectable) that made whatever advantages intuition of it had get outweighed by the absurdity of the concept. Correct intuitions are dictated by mathematics of the theory, many of them have no familiar ready made prototypes, like little balls or waves, and have to be developed deliberately piece by piece. $\endgroup$ – Conifold Nov 5 '15 at 0:53