Connection between harmonic potential and particle intepretation I just finished a quantum mechanics course, but I still have some problems. In the simple harmonics potential well, energy between two adjacent states is always $\hbar\omega$. I read that this can be interpreted as adding a quanta to get to the next energy sate, and these quanta behave like particles with energy $\hbar\omega$. If I consider an infinite square potential well now, the quantized energy is $\frac{n^{2}\pi^{2}\hbar^{2}}{2mL^{2}}$, and the energy difference beween two states are not equal. In the infinite potential well case, is the particle interpretation still correct?
 A: Particle interpretation indeed can be understood from the point of quantization of electromagnetic field. Basically, as the Maxwell equations are linear, each Fourier mode of the electromagnetic 4-potential $A^\mu$ is independent from others. This greatly simplifies the description, as the system is effectively diagonalized.
Each Fourier mode satisfies the wave equation which, essentially, makes each mode and independent harmonic oscillator. After canonical quantization of these modes, one obtains the quantum oscillators with even (mode-dependent) level spacing.
As in some processes it is fairly easy to measure the energy transferred from electromagnetic field (say, in photoelectric effect), one can imagine that the field consists of particles with different energy.

As to a question, why the harmonic potential, it is actually the simplest one both in terms of position and momentum space beside the trivial ones (the square box looks really bad in k-space). Luckily, quadratic Hamiltonians are both abundant in nature and are almost the only ones we can exactly solve.
I guess there might be a different particle-like interpretation for another evenly-spaced potential, but it definitely won't be as natural.
A: The interpretation that you cite dates from the early days of QED (prior to Feynman and renormaluzation).  Excitation of the field were treated by purturbation theory based upon small oscillations which leads to an HO spectrum. The excitation were then interpreted as single photons.  It was the small oscillations approximation that leads to evenly spaced levels.
