Hermite polynomials
The Hermite polynomials are a family of mathematical functions, one for each whole number, that shape the wavefunctions of the harmonic oscillator. When you solve the Schrödinger equation for the parabolic potential by calculus, each allowed state turns out to be a Hermite polynomial multiplied by a bell-shaped Gaussian envelope. The polynomial supplies the wiggles; the Gaussian makes the whole thing fade smoothly to zero far from the centre.
Each successive Hermite polynomial has one more wiggle than the last, which is why the state with quantum number n has exactly n nodes. The lowest, for the ground state, is just a constant, giving the simple Gaussian bump. The next adds a single zero-crossing, the one after that adds two, and so the wavefunctions grow steadily more structured as the energy climbs the ladder.
These polynomials were studied by mathematicians, notably Charles Hermite, long before quantum mechanics needed them. They share an elegant property called orthogonality, which guarantees that the oscillator's states are mutually independent and form a complete basis. The same functions reappear across mathematics and physics, from probability theory to the modes of laser beams.
Each oscillator wavefunction is a Hermite polynomial dressed in a Gaussian envelope.
The wavefunction ψ_n is a probability amplitude, not a physical wave in space; the polynomial does not 'wiggle' anything real. What you can measure is |ψ_n|² — the probability density of finding the particle, via the Born rule.