Simple quantum mechanical oscillatory behavior

Well, yes, it is time to get something posted.  Here is a thought to ponder before any equations are written down:  How many different faces does the harmonic oscillator have in elementary quantum mechanics?  That is, in how many different cases does simple harmonic oscillator (SHO) behavior show up? 

In answering this question, I'm going to leave out the usual SHO Hamiltonian, which is the normal starting point for creating the quantum harmonic oscillator raising and lowering operators--that is, for quantizing the energy of the oscillator. Instead, I'm just looking for oscillatory behavior, meaning either sine waves themselves or equations whose solutions are sine waves.

The first one that pops up is in the post below, from my Weinberg class notes of 15 Sept. 1998. In parentheses next to the word "integrable!" Nope, those are exponential functions, non-oscillating, no i in the exponent.  But it is true that the wave function for the particle in a box (infinite square well potential) is a sine wave. Wave functions are not "real," however, so this is not called  a harmonic oscillator. Squaring the wave function gives the probability of finding the particle at different places in the box, but the particle hitting the walls of the box is in a way the opposite behavior of a particle in simple harmonic motion.  Or instead of opposite, let's call it complementary to the behavior of a harmonic oscillator: A constant velocity that instantly reverses itself at the walls, whereas SHO motion has the particle slowing down and then reaching zero velocity at the (unnecessary) walls, where the restoring force on the particle reaches a maximum and the particle reverses its direction.

But, in fact, since we have elastic collisions of the particle with the walls, we have the equivalent of little springs causing the interaction between the particle and the walls, and the restoring force of these reaches a maximum as the particle reverses direction.

Next, as far as simple oscillations are concerned, the time-dependent superposition of two stationary states shows sine wave behavior, even when the stationary states are not those of the harmonic oscillator.

AND, in solving the Schrodinger equation by the separation of variables method, the resulting time-dependent equation is a simple harmonic motion differential equation. Yes, it's a first order equation, but the factor of -i makes it the first-order version of the usual second-order SHO equation of motion.  This is something that is rarely pointed out--I don't recall seeing it ever pointed out--so it may be considered unimportant, or may be considered important only from a more esoteric point of view, such as  the quantum propagator point of view.  But I consider it important and will be discussing it again, soon!

So the answer to the question posed at the beginning of this post is: three. But it seems like I left one out...  Now it's December 1, and I do have to mention the coherent state, the Gaussian wave packet solution to the Schro equation that oscillates back and forth in the SHO potential energy well.  And now December 11, how could I forget to mention normal modes, since a "mode" is an abstraction of a harmonic oscillator or standing wave? I guess because normal mode analysis is not unique to quantum mechanics. But it is what Planck used when he analyzed the electromagnetic waves in a heated cavity in thermal equilibrium with the atoms in the cavity walls, thereby discovering energy quantization.