system
Given the following post, respond as if you're commenting on Reddit:
human
Why are harmonic oscillators so important?
gpt
Because any physical system sufficiently close to a stable equilibrium can be approximated as a simple harmonic oscillator.

If you have some smooth potential energy function V(x) with a stable equilibrium point x*_0_*, it can be expanded in a Taylor series about x*_0_*.

V(x) = V(x*_0_*) + V’(x*_0_*)(x - x*_0_*) + V’’(x*_0_*)(x - x*_0_*)^(2)/2 + ...

The constant term doesn’t matter, because only differences in potential energy are physical. The linear term is zero, because x*_0_* is assumes to be an equilibrium point, so V’ is zero there. The quadratic term is positive, because the equilibrium is assumed to be stable. So if you can drop higher-order terms, this boils down to

V(x) = k(x - x*_0_*)^(2)/2, where is a constant. This is a harmonic oscillator potential.

Many quantum systems can be approximated as harmonic oscillators too. For example, harmonic oscillator wavefunctions are convenient to use as a nuclear mean-field basis. There are more reslistic mean field potentials based on the shape of the nuclear density, but they don’t have analytic solutions like the QHO does. Additionally in QFT, when you quantize some field, for example the electromagnetic field, you can write the Hamiltonian in terms of creation and destruction operators, which looks exactly like a harmonic oscillator Hamiltonian in terms of oscillator raising and lowering operators.