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The Quantum Harmonic Oscillator

The harmonic oscillator is the model every physicist returns to, because almost anything near a stable equilibrium looks like a parabola, and its quantum version is exactly solvable. In the well $V(x) = \tfrac{1}{2}m\omega^2 x^2$ the allowed energies come out perfectly evenly spaced, $E_n = (n + \tfrac{1}{2})\hbar\omega$, a ladder of identical rungs unlike the crowding levels of the square well or the hydrogen atom. The lowest rung is not at the bottom: even the ground state carries $\tfrac{1}{2}\hbar\omega$ of zero-point energy, because confinement and the uncertainty principle forbid a particle from sitting still at the minimum. The scene draws the parabolic well with those evenly spaced levels, and the chosen eigenstate riding on its own level, oscillating in time, a Hermite-Gauss function $\psi_n(x) = H_n(x)e^{-x^2/2}$ with exactly $n$ nodes and a gentle leak past the classical turning points into the forbidden region. The bottom panel sets the quantum probability $|\psi_n|^2$ against the classical one, the chance of catching an ordinary oscillating mass at each point, which is largest at the turning points where it moves slowest. For small $n$ they look nothing alike, the quantum density bunched in the middle for the ground state; but slide $n$ up and the rapid quantum wiggles begin to average out into the classical curve, the correspondence principle emerging level by level.

Figure 1. The quantum harmonic oscillator. Top: the parabolic well (grey) with equally spaced levels E_n = (n + 1/2) hbar omega and the selected eigenstate (blue) on its level, with n nodes and the classical turning points (orange). Bottom: the quantum probability |psi_n|^2 (blue) against the classical density (orange), which diverges at the turning points; the two converge at high n. Method: Hermite-Gauss eigenstates by stable recurrence. Source: Griffiths, Introduction to Quantum Mechanics, 2nd ed., Sec. 2.3.
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WHAT TO TRY

  • Step the level n: each rung adds one node to the wavefunction, and the levels stay evenly spaced by $\hbar\omega$, unlike the well or the atom.
  • Look at the ground state (n = 0): it sits at $\tfrac{1}{2}\hbar\omega$, the zero-point energy, a Gaussian with no nodes.
  • Push n high and watch $|\psi_n|^2$ start to follow the classical curve, piling up toward the turning points (correspondence).
  • Notice the wavefunction leaking past the orange turning points into the classically forbidden region.