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?