Few classic quantum mechanic models will be particle in a box (infinite depth), particle in a box (finite depth), harmonic oscillator (HO) and mechanical rotation (MR). The wave functions of a QM-HO are similar to those of particle in a box (finite depth). I understand that this similarity is due to a) same kinetic term and b) finite potential energy (which leads to the decay of the function at large value of x). However, I cant really compare the idea of a box to QM-HO because I'm not sure what would physically represent the boundaries.
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Consider the HO in 1D classically: depending on the energy of the (classical) particle, there will be classical turning points, i.e., points, where the harmonic potential is exactly equal to the energy of the particle. At these points, the kinetic energy is zero.
You may see these classical turning points as the 'walls' of the box, if you treat the problem quantum mechanically. Between the turning points, in the classically allowed region, the particle has a finite kinetic energy, and the wave function will tend to oscillate. Outside the classical turning points (classically forbidden region), the wave function will have to decay exponentially.
Unlike in the hard-wall particle-in-a-box problem, in the HO-case, the position of the 'wall' depends on the total energy of the particle.