The quantum harmonic oscillator

The oscillator Hamiltonian is $H=p^2/(2m)+m\omega^2x^2/2$, with mass $m>0$ and angular frequency $\omega>0$. Define dimensionless lowering and raising operators from position and momentum. Their order matters because $[x,p]=i\hbar$. This gives $[a,a^\dagger]=1$ and an extra one-half in the Hamiltonian. Treating $x$ and $p$ as ordinary commuting numbers would miss that term.

The number operator $N=a^\dagger a$ is nonnegative because its expectation is the squared norm of $a|\psi\rangle$. A lowest state satisfies $a|0\rangle=0$. Raising repeatedly constructs normalized number states $|n\rangle$, and lowering cannot continue below zero. The argument uses normalizability and positivity, not a postulate that an oscillator follows a classical orbit whose amplitude happens to be discrete.

The energies are $E_n=\hbar\omega(n+1/2)$ for nonnegative integers $n$. Adjacent levels have the same gap $\hbar\omega$, unlike the square well's increasing gaps. The lowering amplitude is $\sqrt{n}$ and the raising amplitude is $\sqrt{n+1}$. These factors follow from the ladder algebra in a normalized basis; an operator's output generally needs renormalizing. They are amplitudes, not probabilities for spontaneous transitions without a specified interaction.

The ground energy is $\hbar\omega/2$. A normalized state cannot have both perfectly definite zero position and zero momentum, so the lowest energy balances kinetic and potential contributions. This zero-point energy belongs to the chosen Hamiltonian. Energy differences are what a simple isolated oscillator spectroscopy experiment measures; interpreting an infinite sum of field-mode zero-point energies involves additional physics and regularization, which this single-oscillator result does not settle.

Every number state has $\langle x\rangle=\langle p\rangle=0$ and stationary probability density. Nevertheless, its position spread is nonzero: $(\Delta x)^2=\hbar(2n+1)/(2m\omega)$. Calling that spread an orbit would assign a trajectory the state does not specify. An energy eigenstate does not trace the classical sinusoid in time simply because the potential is harmonic.

A coherent superposition of number states can instead have an oscillating position expectation. Coherent states give a particularly useful link to classical motion: their center follows the classical undamped oscillator while the minimal Gaussian uncertainty remains. A damped classical animation is still only a comparison of restoring-force time scales. Modeling quantum damping requires coupling to an environment and a suitable open-system description; a shrinking drawn trajectory is not a wavefunction collapse law.