The quantum harmonic oscillator

Why the harmonic oscillator matters

The harmonic oscillator is one of the most important systems in physics. Any smooth potential near a stable minimum can be approximated as a parabola. If $x=0$ is an equilibrium point, then

V(x)approx rac{1}{2}momega^2x^2

This model describes molecular vibrations, lattice vibrations, electromagnetic field modes, and many small oscillations.

Schrödinger equation

The Hamiltonian for the one-dimensional quantum harmonic oscillator is

hat{H}=rac{hat{p}^2}{2m}+rac{1}{2}momega^2hat{x}^2

In position space,

hat{p}=-ihbarrac{d}{dx}

so the time-independent Schrödinger equation becomes

-rac{hbar^2}{2m}rac{d^2psi}{dx^2}+rac{1}{2}momega^2x^2psi=Epsi

Energy levels

The allowed energies are

ight)$$ where $$n=0,1,2,ldots$$ The levels are equally spaced: $$Delta E=hbaromega$$ Unlike the infinite square well, where spacing grows with $n$, harmonic oscillator levels have constant spacing. ## Zero-point energy The ground state has energy $$E_0=rac{1}{2}hbaromega$$ This is zero-point energy. The oscillator cannot have zero energy because that would require both definite position at the potential minimum and definite zero momentum, violating the uncertainty principle. ## Ladder operators The harmonic oscillator can be solved elegantly using ladder operators. Define annihilation and creation operators $hat{a}$ and $hat{a}^dagger$ such that $$hat{H}=hbaromegaleft(hat{a}^daggerhat{a}+rac{1}{2} ight)$$ The number operator is $$hat{N}=hat{a}^daggerhat{a}$$ with eigenvalues $n$. The ladder operators act as $$hat{a}|n angle=sqrt{n}|n-1 angle$$ and $$hat{a}^dagger|n angle=sqrt{n+1}|n+1 angle$$ ## Ground-state wavefunction The ground-state wavefunction is a Gaussian: $$psi_0(x)=left(rac{momega}{pihbar} ight)^{1/4}e^{-momega x^2/(2hbar)}$$ It has no nodes and is centered at the equilibrium position. Excited-state wavefunctions involve Hermite polynomials multiplied by the same Gaussian envelope. The $n$th state has $n$ nodes. ## Classical comparison A classical oscillator has continuous energy and spends more time near turning points where its speed is low. Quantum probability distributions differ strongly at low $n$, but for large $n$, they begin to resemble classical behavior. This is an example of the correspondence principle. ## The big idea The quantum harmonic oscillator has equally spaced energy levels and unavoidable zero-point energy. Its ladder-operator structure appears throughout quantum physics, from molecular vibrations to quantum field theory. It is both exactly solvable and deeply universal.