Double Resonance and Quantized Oscillatory Particles?

Some quantum mechanical scenarios may be mathematically simplified by making use of a simpler quantum problem. For example, the problem of a particle in a box with an infinite potential at the wall can be solved and the wavefunctions used as a basis for a new problem created by adding a finite potential to the interior. For the harmonic oscillator, one may write the general wavefunction as a product of the ground state wavefunction and another function (which is found to be a Hermite polynomial). If a Hamiltonian may be written in terms of a simpler Hamiltonian with known solutions plus a small perturbation term, the new wavefunction may be described as a mixed state (in terms of the simpler Hamiltonian). Finally, even traditional quantum statistical mechanics makes use of single particle resonance states and the photons which allow one to jump from one to another, although one does not create a mixed state in such a case. Given that one solves a simpler quantum problem first, quantized wavefunction solutions appear with specific energy gaps. In the new, more complicated problem, the simpler wavefunctions serve as a basis and “on the surface” it appears there is already quantization in the problem based on the energy level gaps of the simpler states. These may perhaps be thought of in terms of oscillatory quantized particles, say phonons or other particles. Thus, the new problem contains a potential or perturbative potential which may operate together with these oscillatory particles, or the oscillatory particles operate directly with the wavefunction as in the case of the oscillator. In such cases, it is as if one has a double resonance. The “simpler” problem creates quantized gaps and possibly oscillatory particles which are then used in the more complicated problem. A question arises as to whether this double resonance is a mathematical convenience or represents physical phenomena in the sense of the first resonance physically existing and restricting quantization of the second. In this note, we try to investigate these ideas