This is a system for which one can obtain mathematically lovely solutions. These closed-form solutions are appealing; they are also nice to work with in calculations of expectation values, time-dependent behavior and other things. The harmonic oscillator is a very important system in the lore of quantum mechanics.
A really amazing and important thing we see in the 1D HO is the emergence of a quantum length scale. The \(k x^2/2\) potential has no length scale; it looks the same whether you zoom in or out. (In the language of fractals and Mandelbrot, it is self-similar.) However, when you put a particle of a particular mass in that potential, a quantum length scale which depends on \(\hbar\), m and k emerges. This manifests itself as the size of an electron in the ground state, and is present in all excited states as well.
Homework 2 will give you a nice opportunity to consider and explore the nature of the ground state (and some excited state behavior as well). Please take time to savor it; I hope you don't have to rush. I would recommend doing the extra-credit (problem 6). I may have underestimated how difficult to make this homework. In fact, I think I will add a 2nd extra credit problem right now which will aslo be fun and edifying :)
Here are some states in the notation that I would like you to use. The top part shows the lowest 3 energy eigenstates of an infinite square well, the lower part shows the lowest 3 energy eigenstates of a 1D HO along with the equations for the spatial part of these wavefunctions.
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